TPTP Problem File: SCT169_2.p
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%------------------------------------------------------------------------------
% File : SCT169_2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 29, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : arrow_500_tff_l29 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.5.0, 0.75 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 904 ( 287 unt; 188 typ; 0 def)
% Number of atoms : 1432 ( 813 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 951 ( 235 ~; 38 |; 87 &)
% ( 161 <=>; 430 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 46 ( 45 usr)
% Number of type conns : 167 ( 101 >; 66 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 142 ( 142 usr; 42 con; 0-2 aty)
% Number of variables : 1570 (1515 !; 55 ?;1570 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:15:00
%------------------------------------------------------------------------------
%----Should-be-implicit typings (45)
tff(ty_ty_tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
arrow_931160158le_alt: $tType ).
tff(ty_ty_tc__HOL__Obool,type,
bool: $tType ).
tff(ty_ty_tc__Int__Oint,type,
int: $tType ).
tff(ty_ty_tc__List__Olist_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_J,type,
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tff(ty_ty_tc__List__Olist_Itc__Int__Oint_J,type,
list_int: $tType ).
tff(ty_ty_tc__List__Olist_Itc__Nat__Onat_J,type,
list_nat: $tType ).
tff(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
tff(ty_ty_tc__fun_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_Mtc__HOL__Obo,type,
fun_Ar1091019988t_bool: $tType ).
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fun_Ar1553825290le_alt: $tType ).
tff(ty_ty_tc__fun_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_Mtc__List__Ol_001,type,
fun_Ar848716726st_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_Mtc__Nat__Ona,type,
fun_Ar518461033lt_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_Mtc__Option__,type,
fun_Ar237091702on_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_Mtc__fun_Itc_,type,
fun_Ar1414184139t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
fun_bool_bool: $tType ).
tff(ty_ty_tc__fun_Itc__HOL__Obool_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
fun_bo1549164019l_bool: $tType ).
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tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
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tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_J,type,
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tff(ty_ty_tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
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fun_li954148211t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__List__Olist_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__O_003,type,
fun_li1393995935t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__List__Olist_Itc__Int__Oint_J_Mtc__HOL__Obool_J,type,
fun_list_int_bool: $tType ).
tff(ty_ty_tc__fun_Itc__List__Olist_Itc__Nat__Onat_J_Mtc__HOL__Obool_J,type,
fun_list_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__List__Olist_Itc__Nat__Onat_J_Mtc__fun_Itc__List__Olist_Itc__A,type,
fun_li2096081351t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__List__Olist_Itc__Nat__Onat_J_Mtc__fun_Itc__List__Olist_Itc__N,type,
fun_li981111539t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oal,type,
fun_na151831313le_alt: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Int__Oint_J,type,
fun_nat_int: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__List__Olist_Itc__Arrow____Order____Mirabelle__,type,
fun_na1941391286le_alt: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__List__Olist_Itc__Nat__Onat_J_J,type,
fun_nat_list_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
fun_nat_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Option__Ooption_Itc__Arrow____Order____Mirabel,type,
fun_na1357632374le_alt: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Arrow____Order____Mirabelle____ctuiws,type,
fun_na81669751t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_J,type,
fun_na1469252690l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_J,type,
fun_nat_fun_nat_bool: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_J,type,
fun_nat_fun_nat_nat: $tType ).
tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
fun_fu819804446l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_Itc__I,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
fun_fu425979586l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__Nat__Onat_,type,
fun_fu821463397t_bool: $tType ).
tff(ty_ty_tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_Itc__N,type,
fun_fu1217155507l_bool: $tType ).
tff(ty_ty_tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__fun_Itc__fun_Itc__N_004,type,
fun_fu915923926t_bool: $tType ).
%----Explicit typings (143)
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat,type,
cOMBB_bool_bool_nat: ( fun_bool_bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc_,type,
cOMBB_1015721476ol_nat: ( fun_bo1549164019l_bool * fun_nat_bool ) > fun_na1469252690l_bool ).
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_nat_bool ) > fun_nat_bool ).
tff(sy_c_COMBK_000tc__HOL__Obool_000tc__Nat__Onat,type,
cOMBK_bool_nat: bool > fun_nat_bool ).
tff(sy_c_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
cOMBS_nat_bool_bool: ( fun_na1469252690l_bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: fun_int_fun_int_int ).
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__Nat__Onat_Mtc__HOL__Obool_J,type,
minus_2067140911t_bool: fun_fu915923926t_bool ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
tff(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: fun_int_fun_int_int ).
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__Int__Oint,type,
zero_zero_int: int ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_HOL_Oequal__class_Oequal_000tc__List__Olist_Itc__Arrow____Order____Mirabell,type,
equal_117954737le_alt: fun_li954148211t_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_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
tff(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
tff(sy_c_Int_Oring__1__class_OInts_000tc__Int__Oint,type,
ring_1_Ints_int: fun_int_bool ).
tff(sy_c_List_OListMem_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
listMe533848314le_alt: arrow_931160158le_alt > fun_li1323594581t_bool ).
tff(sy_c_List_Oappend_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
append634883619le_alt: ( list_A1748581779le_alt * list_A1748581779le_alt ) > list_A1748581779le_alt ).
tff(sy_c_List_Oappend_000tc__Nat__Onat,type,
append_nat: ( list_nat * list_nat ) > list_nat ).
tff(sy_c_List_Obutlast_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
butlas730749018le_alt: list_A1748581779le_alt > list_A1748581779le_alt ).
tff(sy_c_List_Obutlast_000tc__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
tff(sy_c_List_Odistinct_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
distin692125441le_alt: fun_li1323594581t_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____ctuiwsvqnm__Oalt,type,
dropWh1772583087le_alt: ( fun_Ar1091019988t_bool * list_A1748581779le_alt ) > list_A1748581779le_alt ).
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____ctuiwsvqnm__Oalt,type,
drop_A1802510926le_alt: ( nat * list_A1748581779le_alt ) > list_A1748581779le_alt ).
tff(sy_c_List_Odrop_000tc__Nat__Onat,type,
drop_nat: ( nat * list_nat ) > list_nat ).
tff(sy_c_List_Ofoldr_000tc__Int__Oint_000tc__Int__Oint,type,
foldr_int_int: ( fun_int_fun_int_int * list_int ) > fun_int_int ).
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____ctuiwsvqnm__Oalt,type,
hd_Arr274000865le_alt: list_A1748581779le_alt > arrow_931160158le_alt ).
tff(sy_c_List_Ohd_000tc__Nat__Onat,type,
hd_nat: list_nat > nat ).
tff(sy_c_List_Oinsert_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
insert428884260le_alt: ( arrow_931160158le_alt * list_A1748581779le_alt ) > list_A1748581779le_alt ).
tff(sy_c_List_Oinsert_000tc__Nat__Onat,type,
insert_nat: ( nat * list_nat ) > list_nat ).
tff(sy_c_List_Olast_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
last_A1673116455le_alt: list_A1748581779le_alt > arrow_931160158le_alt ).
tff(sy_c_List_Olast_000tc__Nat__Onat,type,
last_nat: list_nat > nat ).
tff(sy_c_List_Olinorder__class_Osorted_000tc__Int__Oint,type,
linorder_sorted_int: fun_list_int_bool ).
tff(sy_c_List_Olinorder__class_Osorted_000tc__Nat__Onat,type,
linorder_sorted_nat: fun_list_nat_bool ).
tff(sy_c_List_Olinorder__class_Osorted__list__of__set_000tc__Nat__Onat,type,
linord810813188et_nat: fun_nat_bool > list_nat ).
tff(sy_c_List_Olist_OCons_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
cons_A684544190le_alt: ( arrow_931160158le_alt * list_A1748581779le_alt ) > list_A1748581779le_alt ).
tff(sy_c_List_Olist_OCons_000tc__Nat__Onat,type,
cons_nat: ( nat * list_nat ) > list_nat ).
tff(sy_c_List_Olist_ONil_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
nil_Ar1741995278le_alt: list_A1748581779le_alt ).
tff(sy_c_List_Olist_ONil_000tc__Int__Oint,type,
nil_int: list_int ).
tff(sy_c_List_Olist_ONil_000tc__Nat__Onat,type,
nil_nat: list_nat ).
tff(sy_c_List_Olist_Olist__size_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oa,type,
list_s706097040le_alt: ( fun_Ar518461033lt_nat * list_A1748581779le_alt ) > 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__ex1_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
list_e695046097le_alt: fun_Ar1091019988t_bool > fun_li1323594581t_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_Olist__ex_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
list_e1708603724le_alt: fun_Ar1091019988t_bool > fun_li1323594581t_bool ).
tff(sy_c_List_Olist__ex_000tc__Nat__Onat,type,
list_ex_nat: fun_nat_bool > fun_list_nat_bool ).
tff(sy_c_List_Olist__update_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
list_u518279030le_alt: ( list_A1748581779le_alt * nat ) > fun_Ar1553825290le_alt ).
tff(sy_c_List_Olist__update_000tc__Nat__Onat,type,
list_update_nat: ( list_nat * nat ) > fun_nat_list_nat ).
tff(sy_c_List_Olistrelp_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
listre1668963176le_alt: fun_Ar1414184139t_bool > fun_li954148211t_bool ).
tff(sy_c_List_Olistrelp_000tc__Nat__Onat,type,
listrelp_nat: fun_nat_fun_nat_bool > fun_li981111539t_bool ).
tff(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_00,type,
map_fi747840374lt_nat: ( fun_Ar237091702on_nat * list_A1748581779le_alt ) > list_nat ).
tff(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____ct,type,
map_fi1918498462le_alt: ( fun_na1357632374le_alt * list_nat ) > list_A1748581779le_alt ).
tff(sy_c_List_Omaps_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_000tc__Ar,type,
maps_A1570769089le_alt: ( fun_Ar1553825290le_alt * list_A1748581779le_alt ) > list_A1748581779le_alt ).
tff(sy_c_List_Omaps_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_000tc__Na,type,
maps_A413099501lt_nat: ( fun_Ar848716726st_nat * list_A1748581779le_alt ) > list_nat ).
tff(sy_c_List_Omaps_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____ctuiwsvqn,type,
maps_n1583757589le_alt: ( fun_na1941391286le_alt * list_nat ) > list_A1748581779le_alt ).
tff(sy_c_List_Omember_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
member1960769315le_alt: list_A1748581779le_alt > fun_Ar1091019988t_bool ).
tff(sy_c_List_Omember_000tc__Nat__Onat,type,
member_nat_1: list_nat > fun_nat_bool ).
tff(sy_c_List_Onth_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
nth_Ar581680143le_alt: list_A1748581779le_alt > fun_na151831313le_alt ).
tff(sy_c_List_Onth_000tc__Int__Oint,type,
nth_int: list_int > fun_nat_int ).
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____ctuiwsvqnm__Oalt,type,
null_A1976766230le_alt: fun_li1323594581t_bool ).
tff(sy_c_List_Onull_000tc__Nat__Onat,type,
null_nat: fun_list_nat_bool ).
tff(sy_c_List_Oreplicate_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
replic1967339976le_alt: nat > fun_Ar1553825290le_alt ).
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____ctuiwsvqnm__Oalt,type,
rev_Ar1562208110le_alt: list_A1748581779le_alt > list_A1748581779le_alt ).
tff(sy_c_List_Orev_000tc__Int__Oint,type,
rev_int: list_int > list_int ).
tff(sy_c_List_Orev_000tc__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
tff(sy_c_List_Orotate1_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
rotate791150427le_alt: list_A1748581779le_alt > list_A1748581779le_alt ).
tff(sy_c_List_Orotate1_000tc__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
tff(sy_c_List_Osplice_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
splice1976699617le_alt: ( list_A1748581779le_alt * list_A1748581779le_alt ) > list_A1748581779le_alt ).
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____ctuiwsvqnm__Oalt,type,
sublis356271123le_alt: ( list_A1748581779le_alt * fun_nat_bool ) > list_A1748581779le_alt ).
tff(sy_c_List_Osublist_000tc__Nat__Onat,type,
sublist_nat: ( list_nat * fun_nat_bool ) > list_nat ).
tff(sy_c_List_OtakeWhile_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
takeWh4609031le_alt: ( fun_Ar1091019988t_bool * list_A1748581779le_alt ) > list_A1748581779le_alt ).
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____ctuiwsvqnm__Oalt,type,
take_A1069848822le_alt: ( nat * list_A1748581779le_alt ) > list_A1748581779le_alt ).
tff(sy_c_List_Otl_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt,type,
tl_Arr921252325le_alt: list_A1748581779le_alt > list_A1748581779le_alt ).
tff(sy_c_List_Otl_000tc__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
tff(sy_c_Nat_OSuc,type,
suc: fun_nat_nat ).
tff(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Arrow____Order____Mirabelle_,type,
size_s1492124157le_alt: list_A1748581779le_alt > nat ).
tff(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,type,
bot_bot_bool: bool ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
bot_bot_nat: nat ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
bot_bot_fun_int_bool: fun_int_bool ).
tff(sy_c_Orderings_Obot__class_Obot_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
bot_bot_fun_nat_bool: fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__HOL__Obool,type,
ord_less_bool: fun_bo1549164019l_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: fun_int_fun_int_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: fun_nat_fun_nat_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
ord_le1912455174t_bool: fun_fu1151772339l_bool ).
tff(sy_c_Orderings_Oord__class_Oless_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
ord_le382113706t_bool: fun_fu1217155507l_bool ).
tff(sy_c_Orderings_Oord__class_Omax_000tc__Int__Oint,type,
ord_max_int: fun_int_fun_int_int ).
tff(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
ord_max_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: fun_nat_bool > fun_nat_bool ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun_bool_bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun_bo1549164019l_bool ).
tff(sy_c_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__O,type,
fequal414630996le_alt: fun_li954148211t_bool ).
tff(sy_c_hAPP_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_000tc__HOL__Obo,type,
hAPP_A674791462t_bool: ( fun_Ar1091019988t_bool * arrow_931160158le_alt ) > bool ).
tff(sy_c_hAPP_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_000tc__List__Ol,type,
hAPP_A1866301020le_alt: ( fun_Ar1553825290le_alt * arrow_931160158le_alt ) > list_A1748581779le_alt ).
tff(sy_c_hAPP_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_000tc__Nat__Ona,type,
hAPP_A846389249lt_nat: ( fun_Ar518461033lt_nat * arrow_931160158le_alt ) > nat ).
tff(sy_c_hAPP_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oalt_000tc__fun_Itc_,type,
hAPP_A1858212765t_bool: ( fun_Ar1414184139t_bool * arrow_931160158le_alt ) > fun_Ar1091019988t_bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,type,
hAPP_bool_bool: ( fun_bool_bool * bool ) > bool ).
tff(sy_c_hAPP_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_b589554111l_bool: ( fun_bo1549164019l_bool * bool ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool,type,
hAPP_int_bool: ( fun_int_bool * int ) > bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint,type,
hAPP_int_int: ( fun_int_int * int ) > int ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J,type,
hAPP_i1948725293t_bool: ( fun_int_fun_int_bool * int ) > fun_int_bool ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J,type,
hAPP_int_fun_int_int: ( fun_int_fun_int_int * int ) > fun_int_int ).
tff(sy_c_hAPP_000tc__Int__Oint_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_,type,
hAPP_i2112223885l_bool: ( fun_in938817141l_bool * int ) > fun_fu819804446l_bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oal,type,
hAPP_l713721587t_bool: ( fun_li1323594581t_bool * list_A1748581779le_alt ) > bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oal_005,type,
hAPP_l1173894719t_bool: ( fun_li954148211t_bool * list_A1748581779le_alt ) > fun_li1323594581t_bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oal_006,type,
hAPP_l931213803t_bool: ( fun_li1393995935t_bool * list_A1748581779le_alt ) > fun_list_nat_bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Int__Oint_J_000tc__HOL__Obool,type,
hAPP_list_int_bool: ( fun_list_int_bool * list_int ) > bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Nat__Onat_J_000tc__HOL__Obool,type,
hAPP_list_nat_bool: ( fun_list_nat_bool * list_nat ) > bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Nat__Onat_J_000tc__fun_Itc__List__Olist_Itc__A,type,
hAPP_l824545683t_bool: ( fun_li2096081351t_bool * list_nat ) > fun_li1323594581t_bool ).
tff(sy_c_hAPP_000tc__List__Olist_Itc__Nat__Onat_J_000tc__fun_Itc__List__Olist_Itc__N,type,
hAPP_l120351551t_bool: ( fun_li981111539t_bool * list_nat ) > fun_list_nat_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____ctuiwsvqnm__Oal,type,
hAPP_n2017047337le_alt: ( fun_na151831313le_alt * nat ) > arrow_931160158le_alt ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool,type,
hAPP_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint,type,
hAPP_nat_int: ( fun_nat_int * nat ) > int ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__List__Olist_Itc__Nat__Onat_J,type,
hAPP_nat_list_nat: ( fun_nat_list_nat * nat ) > list_nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__Nat__Onat,type,
hAPP_nat_nat: ( fun_nat_nat * nat ) > nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Arrow____Order____Mirabelle____ctuiws,type,
hAPP_n127720521t_bool: ( fun_na81669751t_bool * nat ) > fun_Ar1091019988t_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J,type,
hAPP_n1006566506l_bool: ( fun_na1469252690l_bool * nat ) > fun_bool_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
hAPP_n1699378549t_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
hAPP_nat_fun_nat_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_nat ).
tff(sy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
hAPP_n215258509l_bool: ( fun_na1436237685l_bool * nat ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f448129468l_bool: ( fun_fu819804446l_bool * fun_int_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__I,type,
hAPP_f284875647l_bool: ( fun_fu1151772339l_bool * fun_int_bool ) > fun_fu819804446l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool,type,
hAPP_f54304608l_bool: ( fun_fu425979586l_bool * fun_nat_bool ) > bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__Nat__Onat_,type,
hAPP_f800510211t_bool: ( fun_fu821463397t_bool * fun_nat_bool ) > fun_nat_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N,type,
hAPP_f103356543l_bool: ( fun_fu1217155507l_bool * fun_nat_bool ) > fun_fu425979586l_bool ).
tff(sy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc__N_007,type,
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__Int__Oint,type,
member_int: fun_in938817141l_bool ).
tff(sy_c_member_000tc__Nat__Onat,type,
member_nat: fun_na1436237685l_bool ).
tff(sy_v_a,type,
a: arrow_931160158le_alt ).
tff(sy_v_b,type,
b: arrow_931160158le_alt ).
%----Relevant facts (700)
tff(fact_0_alt3,axiom,
? [A_47: arrow_931160158le_alt,B_30: arrow_931160158le_alt,C: arrow_931160158le_alt] : hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,cons_A684544190le_alt(A_47,cons_A684544190le_alt(B_30,cons_A684544190le_alt(C,nil_Ar1741995278le_alt))))) ).
tff(fact_1_distinct_Osimps_I1_J,axiom,
hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,nil_Ar1741995278le_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_48: arrow_931160158le_alt,List_4: list_A1748581779le_alt] : nil_Ar1741995278le_alt != cons_A684544190le_alt(A_48,List_4) ).
tff(fact_4_list_Osimps_I2_J,axiom,
! [A_48: nat,List_4: list_nat] : nil_nat != cons_nat(A_48,List_4) ).
tff(fact_5_list_Osimps_I3_J,axiom,
! [A_46: arrow_931160158le_alt,List_2: list_A1748581779le_alt] : cons_A684544190le_alt(A_46,List_2) != nil_Ar1741995278le_alt ).
tff(fact_6_list_Osimps_I3_J,axiom,
! [A_46: nat,List_2: list_nat] : cons_nat(A_46,List_2) != nil_nat ).
tff(fact_7_neq__Nil__conv,axiom,
! [Xs_22: list_A1748581779le_alt] :
( ( Xs_22 != nil_Ar1741995278le_alt )
<=> ? [Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] : Xs_22 = cons_A684544190le_alt(Y_3,Ys_5) ) ).
tff(fact_8_neq__Nil__conv,axiom,
! [Xs_22: list_nat] :
( ( Xs_22 != nil_nat )
<=> ? [Y_3: nat,Ys_5: list_nat] : Xs_22 = cons_nat(Y_3,Ys_5) ) ).
tff(fact_9_list_Oexhaust,axiom,
! [Y_30: list_A1748581779le_alt] :
( ( Y_30 != nil_Ar1741995278le_alt )
=> ~ ! [A_47: arrow_931160158le_alt,List_3: list_A1748581779le_alt] : Y_30 != cons_A684544190le_alt(A_47,List_3) ) ).
tff(fact_10_list_Oexhaust,axiom,
! [Y_30: list_nat] :
( ( Y_30 != nil_nat )
=> ~ ! [A_47: nat,List_3: list_nat] : Y_30 != cons_nat(A_47,List_3) ) ).
tff(fact_11_not__Cons__self,axiom,
! [Xs_114: list_A1748581779le_alt,X_84: arrow_931160158le_alt] : Xs_114 != cons_A684544190le_alt(X_84,Xs_114) ).
tff(fact_12_not__Cons__self2,axiom,
! [X_83: arrow_931160158le_alt,Xs_113: list_A1748581779le_alt] : cons_A684544190le_alt(X_83,Xs_113) != Xs_113 ).
tff(fact_13_list_Oinject,axiom,
! [Aa: arrow_931160158le_alt,List: list_A1748581779le_alt,A_46: arrow_931160158le_alt,List_2: list_A1748581779le_alt] :
( ( cons_A684544190le_alt(Aa,List) = cons_A684544190le_alt(A_46,List_2) )
<=> ( ( Aa = A_46 )
& ( List = List_2 ) ) ) ).
tff(fact_14_splice_Osimps_I2_J,axiom,
! [V_4: arrow_931160158le_alt,Va: list_A1748581779le_alt] : splice1976699617le_alt(cons_A684544190le_alt(V_4,Va),nil_Ar1741995278le_alt) = cons_A684544190le_alt(V_4,Va) ).
tff(fact_15_splice_Osimps_I2_J,axiom,
! [V_4: nat,Va: list_nat] : splice_nat(cons_nat(V_4,Va),nil_nat) = cons_nat(V_4,Va) ).
tff(fact_16_insert__Nil,axiom,
! [X_82: arrow_931160158le_alt] : insert428884260le_alt(X_82,nil_Ar1741995278le_alt) = cons_A684544190le_alt(X_82,nil_Ar1741995278le_alt) ).
tff(fact_17_insert__Nil,axiom,
! [X_82: nat] : insert_nat(X_82,nil_nat) = cons_nat(X_82,nil_nat) ).
tff(fact_18_list__nonempty__induct,axiom,
! [P_1: fun_li1323594581t_bool,Xs_22: list_A1748581779le_alt] :
( ( Xs_22 != nil_Ar1741995278le_alt )
=> ( ! [X_52: arrow_931160158le_alt] : hBOOL(hAPP_l713721587t_bool(P_1,cons_A684544190le_alt(X_52,nil_Ar1741995278le_alt)))
=> ( ! [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt] :
( ( Xs_44 != nil_Ar1741995278le_alt )
=> ( hBOOL(hAPP_l713721587t_bool(P_1,Xs_44))
=> hBOOL(hAPP_l713721587t_bool(P_1,cons_A684544190le_alt(X_52,Xs_44))) ) )
=> hBOOL(hAPP_l713721587t_bool(P_1,Xs_22)) ) ) ) ).
tff(fact_19_list__nonempty__induct,axiom,
! [P_1: fun_list_nat_bool,Xs_22: list_nat] :
( ( Xs_22 != nil_nat )
=> ( ! [X_52: nat] : hBOOL(hAPP_list_nat_bool(P_1,cons_nat(X_52,nil_nat)))
=> ( ! [X_52: nat,Xs_44: list_nat] :
( ( Xs_44 != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(P_1,Xs_44))
=> hBOOL(hAPP_list_nat_bool(P_1,cons_nat(X_52,Xs_44))) ) )
=> hBOOL(hAPP_list_nat_bool(P_1,Xs_22)) ) ) ) ).
tff(fact_20_distinct__butlast,axiom,
! [Xs_112: list_A1748581779le_alt] :
( ( Xs_112 != nil_Ar1741995278le_alt )
=> ( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_112))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,butlas730749018le_alt(Xs_112))) ) ) ).
tff(fact_21_distinct__butlast,axiom,
! [Xs_112: list_nat] :
( ( Xs_112 != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_112))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,butlast_nat(Xs_112))) ) ) ).
tff(fact_22_butlast_Osimps_I1_J,axiom,
butlas730749018le_alt(nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_23_butlast_Osimps_I1_J,axiom,
butlast_nat(nil_nat) = nil_nat ).
tff(fact_24_distinct__insert,axiom,
! [X_81: arrow_931160158le_alt,Xs_111: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_111))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,insert428884260le_alt(X_81,Xs_111))) ) ).
tff(fact_25_splice_Osimps_I3_J,axiom,
! [X_80: arrow_931160158le_alt,Xs_110: list_A1748581779le_alt,Y_29: arrow_931160158le_alt,Ys_31: list_A1748581779le_alt] : splice1976699617le_alt(cons_A684544190le_alt(X_80,Xs_110),cons_A684544190le_alt(Y_29,Ys_31)) = cons_A684544190le_alt(X_80,cons_A684544190le_alt(Y_29,splice1976699617le_alt(Xs_110,Ys_31))) ).
tff(fact_26_splice__Nil2,axiom,
! [Xs_109: list_A1748581779le_alt] : splice1976699617le_alt(Xs_109,nil_Ar1741995278le_alt) = Xs_109 ).
tff(fact_27_splice__Nil2,axiom,
! [Xs_109: list_nat] : splice_nat(Xs_109,nil_nat) = Xs_109 ).
tff(fact_28_splice_Osimps_I1_J,axiom,
! [Ys_30: list_A1748581779le_alt] : splice1976699617le_alt(nil_Ar1741995278le_alt,Ys_30) = Ys_30 ).
tff(fact_29_splice_Osimps_I1_J,axiom,
! [Ys_30: list_nat] : splice_nat(nil_nat,Ys_30) = Ys_30 ).
tff(fact_30_butlast_Osimps_I2_J,axiom,
! [X_79: arrow_931160158le_alt,Xs_108: list_A1748581779le_alt] :
( ( ( Xs_108 = nil_Ar1741995278le_alt )
=> ( butlas730749018le_alt(cons_A684544190le_alt(X_79,Xs_108)) = nil_Ar1741995278le_alt ) )
& ( ( Xs_108 != nil_Ar1741995278le_alt )
=> ( butlas730749018le_alt(cons_A684544190le_alt(X_79,Xs_108)) = cons_A684544190le_alt(X_79,butlas730749018le_alt(Xs_108)) ) ) ) ).
tff(fact_31_butlast_Osimps_I2_J,axiom,
! [X_79: nat,Xs_108: list_nat] :
( ( ( Xs_108 = nil_nat )
=> ( butlast_nat(cons_nat(X_79,Xs_108)) = nil_nat ) )
& ( ( Xs_108 != nil_nat )
=> ( butlast_nat(cons_nat(X_79,Xs_108)) = cons_nat(X_79,butlast_nat(Xs_108)) ) ) ) ).
tff(fact_32_eq__Nil__null,axiom,
! [Xs_22: list_A1748581779le_alt] :
( ( Xs_22 = nil_Ar1741995278le_alt )
<=> hBOOL(hAPP_l713721587t_bool(null_A1976766230le_alt,Xs_22)) ) ).
tff(fact_33_eq__Nil__null,axiom,
! [Xs_22: list_nat] :
( ( Xs_22 = nil_nat )
<=> hBOOL(hAPP_list_nat_bool(null_nat,Xs_22)) ) ).
tff(fact_34_List_Onull__def,axiom,
! [Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(null_A1976766230le_alt,Xs_22))
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_35_List_Onull__def,axiom,
! [Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(null_nat,Xs_22))
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_36_null__rec_I2_J,axiom,
hBOOL(hAPP_l713721587t_bool(null_A1976766230le_alt,nil_Ar1741995278le_alt)) ).
tff(fact_37_null__rec_I2_J,axiom,
hBOOL(hAPP_list_nat_bool(null_nat,nil_nat)) ).
tff(fact_38_null__rec_I1_J,axiom,
! [X_78: arrow_931160158le_alt,Xs_107: list_A1748581779le_alt] : ~ hBOOL(hAPP_l713721587t_bool(null_A1976766230le_alt,cons_A684544190le_alt(X_78,Xs_107))) ).
tff(fact_39_butlast__snoc,axiom,
! [Xs_106: list_A1748581779le_alt,X_77: arrow_931160158le_alt] : butlas730749018le_alt(append634883619le_alt(Xs_106,cons_A684544190le_alt(X_77,nil_Ar1741995278le_alt))) = Xs_106 ).
tff(fact_40_butlast__snoc,axiom,
! [Xs_106: list_nat,X_77: nat] : butlast_nat(append_nat(Xs_106,cons_nat(X_77,nil_nat))) = Xs_106 ).
tff(fact_41_maps__simps_I2_J,axiom,
! [F: fun_na1941391286le_alt] : maps_n1583757589le_alt(F,nil_nat) = nil_Ar1741995278le_alt ).
tff(fact_42_maps__simps_I2_J,axiom,
! [F: fun_Ar848716726st_nat] : maps_A413099501lt_nat(F,nil_Ar1741995278le_alt) = nil_nat ).
tff(fact_43_maps__simps_I2_J,axiom,
! [F: fun_Ar1553825290le_alt] : maps_A1570769089le_alt(F,nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_44_sorted__single,axiom,
! [X_76: nat] : hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,cons_nat(X_76,nil_nat))) ).
tff(fact_45_sorted_ONil,axiom,
hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,nil_nat)) ).
tff(fact_46_append__assoc,axiom,
! [Xs_105: list_A1748581779le_alt,Ys_29: list_A1748581779le_alt,Zs_4: list_A1748581779le_alt] : append634883619le_alt(append634883619le_alt(Xs_105,Ys_29),Zs_4) = append634883619le_alt(Xs_105,append634883619le_alt(Ys_29,Zs_4)) ).
tff(fact_47_append__eq__append__conv2,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt,Zs: list_A1748581779le_alt,Ts: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,Ys_4) = append634883619le_alt(Zs,Ts) )
<=> ? [Us_1: list_A1748581779le_alt] :
( ( ( Xs_22 = append634883619le_alt(Zs,Us_1) )
& ( append634883619le_alt(Us_1,Ys_4) = Ts ) )
| ( ( append634883619le_alt(Xs_22,Us_1) = Zs )
& ( Ys_4 = append634883619le_alt(Us_1,Ts) ) ) ) ) ).
tff(fact_48_same__append__eq,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt,Zs: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,Ys_4) = append634883619le_alt(Xs_22,Zs) )
<=> ( Ys_4 = Zs ) ) ).
tff(fact_49_append__same__eq,axiom,
! [Ys_4: list_A1748581779le_alt,Xs_22: list_A1748581779le_alt,Zs: list_A1748581779le_alt] :
( ( append634883619le_alt(Ys_4,Xs_22) = append634883619le_alt(Zs,Xs_22) )
<=> ( Ys_4 = Zs ) ) ).
tff(fact_50_append__eq__appendI,axiom,
! [Ys_28: list_A1748581779le_alt,Us: list_A1748581779le_alt,Xs_104: list_A1748581779le_alt,Xs1_1: list_A1748581779le_alt,Zs_3: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_104,Xs1_1) = Zs_3 )
=> ( ( Ys_28 = append634883619le_alt(Xs1_1,Us) )
=> ( append634883619le_alt(Xs_104,Ys_28) = append634883619le_alt(Zs_3,Us) ) ) ) ).
tff(fact_51_maps__simps_I1_J,axiom,
! [F: fun_Ar1553825290le_alt,X_9: arrow_931160158le_alt,Xs_22: list_A1748581779le_alt] : maps_A1570769089le_alt(F,cons_A684544190le_alt(X_9,Xs_22)) = append634883619le_alt(hAPP_A1866301020le_alt(F,X_9),maps_A1570769089le_alt(F,Xs_22)) ).
tff(fact_52_append__Cons,axiom,
! [X_75: arrow_931160158le_alt,Xs_103: list_A1748581779le_alt,Ys_27: list_A1748581779le_alt] : append634883619le_alt(cons_A684544190le_alt(X_75,Xs_103),Ys_27) = cons_A684544190le_alt(X_75,append634883619le_alt(Xs_103,Ys_27)) ).
tff(fact_53_Cons__eq__appendI,axiom,
! [Xs_102: list_A1748581779le_alt,Zs_2: list_A1748581779le_alt,X_74: arrow_931160158le_alt,Xs1: list_A1748581779le_alt,Ys_26: list_A1748581779le_alt] :
( ( cons_A684544190le_alt(X_74,Xs1) = Ys_26 )
=> ( ( Xs_102 = append634883619le_alt(Xs1,Zs_2) )
=> ( cons_A684544190le_alt(X_74,Xs_102) = append634883619le_alt(Ys_26,Zs_2) ) ) ) ).
tff(fact_54_append__Nil,axiom,
! [Ys_25: list_A1748581779le_alt] : append634883619le_alt(nil_Ar1741995278le_alt,Ys_25) = Ys_25 ).
tff(fact_55_append__Nil,axiom,
! [Ys_25: list_nat] : append_nat(nil_nat,Ys_25) = Ys_25 ).
tff(fact_56_Nil__is__append__conv,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( nil_Ar1741995278le_alt = append634883619le_alt(Xs_22,Ys_4) )
<=> ( ( Xs_22 = nil_Ar1741995278le_alt )
& ( Ys_4 = nil_Ar1741995278le_alt ) ) ) ).
tff(fact_57_Nil__is__append__conv,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( nil_nat = append_nat(Xs_22,Ys_4) )
<=> ( ( Xs_22 = nil_nat )
& ( Ys_4 = nil_nat ) ) ) ).
tff(fact_58_append__Nil2,axiom,
! [Xs_101: list_A1748581779le_alt] : append634883619le_alt(Xs_101,nil_Ar1741995278le_alt) = Xs_101 ).
tff(fact_59_append__Nil2,axiom,
! [Xs_101: list_nat] : append_nat(Xs_101,nil_nat) = Xs_101 ).
tff(fact_60_self__append__conv,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( Xs_22 = append634883619le_alt(Xs_22,Ys_4) )
<=> ( Ys_4 = nil_Ar1741995278le_alt ) ) ).
tff(fact_61_self__append__conv,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( Xs_22 = append_nat(Xs_22,Ys_4) )
<=> ( Ys_4 = nil_nat ) ) ).
tff(fact_62_self__append__conv2,axiom,
! [Ys_4: list_A1748581779le_alt,Xs_22: list_A1748581779le_alt] :
( ( Ys_4 = append634883619le_alt(Xs_22,Ys_4) )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_63_self__append__conv2,axiom,
! [Ys_4: list_nat,Xs_22: list_nat] :
( ( Ys_4 = append_nat(Xs_22,Ys_4) )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_64_append__is__Nil__conv,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,Ys_4) = nil_Ar1741995278le_alt )
<=> ( ( Xs_22 = nil_Ar1741995278le_alt )
& ( Ys_4 = nil_Ar1741995278le_alt ) ) ) ).
tff(fact_65_append__is__Nil__conv,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( append_nat(Xs_22,Ys_4) = nil_nat )
<=> ( ( Xs_22 = nil_nat )
& ( Ys_4 = nil_nat ) ) ) ).
tff(fact_66_append__self__conv,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,Ys_4) = Xs_22 )
<=> ( Ys_4 = nil_Ar1741995278le_alt ) ) ).
tff(fact_67_append__self__conv,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( append_nat(Xs_22,Ys_4) = Xs_22 )
<=> ( Ys_4 = nil_nat ) ) ).
tff(fact_68_append__self__conv2,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,Ys_4) = Ys_4 )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_69_append__self__conv2,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( append_nat(Xs_22,Ys_4) = Ys_4 )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_70_eq__Nil__appendI,axiom,
! [Xs_100: list_A1748581779le_alt,Ys_24: list_A1748581779le_alt] :
( ( Xs_100 = Ys_24 )
=> ( Xs_100 = append634883619le_alt(nil_Ar1741995278le_alt,Ys_24) ) ) ).
tff(fact_71_eq__Nil__appendI,axiom,
! [Xs_100: list_nat,Ys_24: list_nat] :
( ( Xs_100 = Ys_24 )
=> ( Xs_100 = append_nat(nil_nat,Ys_24) ) ) ).
tff(fact_72_append__eq__Cons__conv,axiom,
! [Ys_4: list_A1748581779le_alt,Zs: list_A1748581779le_alt,X_9: arrow_931160158le_alt,Xs_22: list_A1748581779le_alt] :
( ( append634883619le_alt(Ys_4,Zs) = cons_A684544190le_alt(X_9,Xs_22) )
<=> ( ( ( Ys_4 = nil_Ar1741995278le_alt )
& ( Zs = cons_A684544190le_alt(X_9,Xs_22) ) )
| ? [Ys_23: list_A1748581779le_alt] :
( ( Ys_4 = cons_A684544190le_alt(X_9,Ys_23) )
& ( append634883619le_alt(Ys_23,Zs) = Xs_22 ) ) ) ) ).
tff(fact_73_append__eq__Cons__conv,axiom,
! [Ys_4: list_nat,Zs: list_nat,X_9: nat,Xs_22: list_nat] :
( ( append_nat(Ys_4,Zs) = cons_nat(X_9,Xs_22) )
<=> ( ( ( Ys_4 = nil_nat )
& ( Zs = cons_nat(X_9,Xs_22) ) )
| ? [Ys_23: list_nat] :
( ( Ys_4 = cons_nat(X_9,Ys_23) )
& ( append_nat(Ys_23,Zs) = Xs_22 ) ) ) ) ).
tff(fact_74_Cons__eq__append__conv,axiom,
! [X_9: arrow_931160158le_alt,Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt,Zs: list_A1748581779le_alt] :
( ( cons_A684544190le_alt(X_9,Xs_22) = append634883619le_alt(Ys_4,Zs) )
<=> ( ( ( Ys_4 = nil_Ar1741995278le_alt )
& ( cons_A684544190le_alt(X_9,Xs_22) = Zs ) )
| ? [Ys_23: list_A1748581779le_alt] :
( ( cons_A684544190le_alt(X_9,Ys_23) = Ys_4 )
& ( Xs_22 = append634883619le_alt(Ys_23,Zs) ) ) ) ) ).
tff(fact_75_Cons__eq__append__conv,axiom,
! [X_9: nat,Xs_22: list_nat,Ys_4: list_nat,Zs: list_nat] :
( ( cons_nat(X_9,Xs_22) = append_nat(Ys_4,Zs) )
<=> ( ( ( Ys_4 = nil_nat )
& ( cons_nat(X_9,Xs_22) = Zs ) )
| ? [Ys_23: list_nat] :
( ( cons_nat(X_9,Ys_23) = Ys_4 )
& ( Xs_22 = append_nat(Ys_23,Zs) ) ) ) ) ).
tff(fact_76_append1__eq__conv,axiom,
! [Xs_22: list_A1748581779le_alt,X_9: arrow_931160158le_alt,Ys_4: list_A1748581779le_alt,Y_9: arrow_931160158le_alt] :
( ( append634883619le_alt(Xs_22,cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt)) = append634883619le_alt(Ys_4,cons_A684544190le_alt(Y_9,nil_Ar1741995278le_alt)) )
<=> ( ( Xs_22 = Ys_4 )
& ( X_9 = Y_9 ) ) ) ).
tff(fact_77_append1__eq__conv,axiom,
! [Xs_22: list_nat,X_9: nat,Ys_4: list_nat,Y_9: nat] :
( ( append_nat(Xs_22,cons_nat(X_9,nil_nat)) = append_nat(Ys_4,cons_nat(Y_9,nil_nat)) )
<=> ( ( Xs_22 = Ys_4 )
& ( X_9 = Y_9 ) ) ) ).
tff(fact_78_butlast__append,axiom,
! [Xs_99: list_A1748581779le_alt,Ys_22: list_A1748581779le_alt] :
( ( ( Ys_22 = nil_Ar1741995278le_alt )
=> ( butlas730749018le_alt(append634883619le_alt(Xs_99,Ys_22)) = butlas730749018le_alt(Xs_99) ) )
& ( ( Ys_22 != nil_Ar1741995278le_alt )
=> ( butlas730749018le_alt(append634883619le_alt(Xs_99,Ys_22)) = append634883619le_alt(Xs_99,butlas730749018le_alt(Ys_22)) ) ) ) ).
tff(fact_79_butlast__append,axiom,
! [Xs_99: list_nat,Ys_22: list_nat] :
( ( ( Ys_22 = nil_nat )
=> ( butlast_nat(append_nat(Xs_99,Ys_22)) = butlast_nat(Xs_99) ) )
& ( ( Ys_22 != nil_nat )
=> ( butlast_nat(append_nat(Xs_99,Ys_22)) = append_nat(Xs_99,butlast_nat(Ys_22)) ) ) ) ).
tff(fact_80_sorted__butlast,axiom,
! [Xs_98: list_nat] :
( ( Xs_98 != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_98))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,butlast_nat(Xs_98))) ) ) ).
tff(fact_81_rev__induct,axiom,
! [Xs_22: list_A1748581779le_alt,P_1: fun_li1323594581t_bool] :
( hBOOL(hAPP_l713721587t_bool(P_1,nil_Ar1741995278le_alt))
=> ( ! [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(P_1,Xs_44))
=> hBOOL(hAPP_l713721587t_bool(P_1,append634883619le_alt(Xs_44,cons_A684544190le_alt(X_52,nil_Ar1741995278le_alt)))) )
=> hBOOL(hAPP_l713721587t_bool(P_1,Xs_22)) ) ) ).
tff(fact_82_rev__induct,axiom,
! [Xs_22: list_nat,P_1: fun_list_nat_bool] :
( hBOOL(hAPP_list_nat_bool(P_1,nil_nat))
=> ( ! [X_52: nat,Xs_44: list_nat] :
( hBOOL(hAPP_list_nat_bool(P_1,Xs_44))
=> hBOOL(hAPP_list_nat_bool(P_1,append_nat(Xs_44,cons_nat(X_52,nil_nat)))) )
=> hBOOL(hAPP_list_nat_bool(P_1,Xs_22)) ) ) ).
tff(fact_83_rev__cases,axiom,
! [Xs_97: list_A1748581779le_alt] :
( ( Xs_97 != nil_Ar1741995278le_alt )
=> ~ ! [Ys_5: list_A1748581779le_alt,Y_3: arrow_931160158le_alt] : Xs_97 != append634883619le_alt(Ys_5,cons_A684544190le_alt(Y_3,nil_Ar1741995278le_alt)) ) ).
tff(fact_84_rev__cases,axiom,
! [Xs_97: list_nat] :
( ( Xs_97 != nil_nat )
=> ~ ! [Ys_5: list_nat,Y_3: nat] : Xs_97 != append_nat(Ys_5,cons_nat(Y_3,nil_nat)) ) ).
tff(fact_85_equal__Nil__null,axiom,
! [Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(equal_117954737le_alt,Xs_22),nil_Ar1741995278le_alt))
<=> hBOOL(hAPP_l713721587t_bool(null_A1976766230le_alt,Xs_22)) ) ).
tff(fact_86_equal__Nil__null,axiom,
! [Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(equal_equal_list_nat,Xs_22),nil_nat))
<=> hBOOL(hAPP_list_nat_bool(null_nat,Xs_22)) ) ).
tff(fact_87_snoc__eq__iff__butlast,axiom,
! [Xs_22: list_A1748581779le_alt,X_9: arrow_931160158le_alt,Ys_4: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt)) = Ys_4 )
<=> ( ( Ys_4 != nil_Ar1741995278le_alt )
& ( butlas730749018le_alt(Ys_4) = Xs_22 )
& ( last_A1673116455le_alt(Ys_4) = X_9 ) ) ) ).
tff(fact_88_snoc__eq__iff__butlast,axiom,
! [Xs_22: list_nat,X_9: nat,Ys_4: list_nat] :
( ( append_nat(Xs_22,cons_nat(X_9,nil_nat)) = Ys_4 )
<=> ( ( Ys_4 != nil_nat )
& ( butlast_nat(Ys_4) = Xs_22 )
& ( last_nat(Ys_4) = X_9 ) ) ) ).
tff(fact_89_append__butlast__last__id,axiom,
! [Xs_96: list_A1748581779le_alt] :
( ( Xs_96 != nil_Ar1741995278le_alt )
=> ( append634883619le_alt(butlas730749018le_alt(Xs_96),cons_A684544190le_alt(last_A1673116455le_alt(Xs_96),nil_Ar1741995278le_alt)) = Xs_96 ) ) ).
tff(fact_90_append__butlast__last__id,axiom,
! [Xs_96: list_nat] :
( ( Xs_96 != nil_nat )
=> ( append_nat(butlast_nat(Xs_96),cons_nat(last_nat(Xs_96),nil_nat)) = Xs_96 ) ) ).
tff(fact_91_rotate__simps,axiom,
! [X_73: arrow_931160158le_alt,Xs_95: list_A1748581779le_alt] :
( ( rotate1_nat(nil_nat) = nil_nat )
& ( rotate791150427le_alt(cons_A684544190le_alt(X_73,Xs_95)) = append634883619le_alt(Xs_95,cons_A684544190le_alt(X_73,nil_Ar1741995278le_alt)) ) ) ).
tff(fact_92_rotate__simps,axiom,
! [X_73: arrow_931160158le_alt,Xs_95: list_A1748581779le_alt] :
( ( rotate791150427le_alt(nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt )
& ( rotate791150427le_alt(cons_A684544190le_alt(X_73,Xs_95)) = append634883619le_alt(Xs_95,cons_A684544190le_alt(X_73,nil_Ar1741995278le_alt)) ) ) ).
tff(fact_93_rotate__simps,axiom,
! [X_73: nat,Xs_95: list_nat] :
( ( rotate791150427le_alt(nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt )
& ( rotate1_nat(cons_nat(X_73,Xs_95)) = append_nat(Xs_95,cons_nat(X_73,nil_nat)) ) ) ).
tff(fact_94_last__snoc,axiom,
! [Xs_94: list_A1748581779le_alt,X_72: arrow_931160158le_alt] : last_A1673116455le_alt(append634883619le_alt(Xs_94,cons_A684544190le_alt(X_72,nil_Ar1741995278le_alt))) = X_72 ).
tff(fact_95_last__snoc,axiom,
! [Xs_94: list_nat,X_72: nat] : last_nat(append_nat(Xs_94,cons_nat(X_72,nil_nat))) = X_72 ).
tff(fact_96_not__distinct__decomp,axiom,
! [Ws: list_A1748581779le_alt] :
( ~ hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Ws))
=> ? [Xs_44: list_A1748581779le_alt,Ys_5: list_A1748581779le_alt,Zs_1: list_A1748581779le_alt,Y_3: arrow_931160158le_alt] : Ws = append634883619le_alt(Xs_44,append634883619le_alt(cons_A684544190le_alt(Y_3,nil_Ar1741995278le_alt),append634883619le_alt(Ys_5,append634883619le_alt(cons_A684544190le_alt(Y_3,nil_Ar1741995278le_alt),Zs_1)))) ) ).
tff(fact_97_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ hBOOL(hAPP_list_nat_bool(distinct_nat,Ws))
=> ? [Xs_44: list_nat,Ys_5: list_nat,Zs_1: list_nat,Y_3: nat] : Ws = append_nat(Xs_44,append_nat(cons_nat(Y_3,nil_nat),append_nat(Ys_5,append_nat(cons_nat(Y_3,nil_nat),Zs_1)))) ) ).
tff(fact_98_replicate__append__same,axiom,
! [I_27: nat,X_71: arrow_931160158le_alt] : append634883619le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(I_27),X_71),cons_A684544190le_alt(X_71,nil_Ar1741995278le_alt)) = cons_A684544190le_alt(X_71,hAPP_A1866301020le_alt(replic1967339976le_alt(I_27),X_71)) ).
tff(fact_99_replicate__append__same,axiom,
! [I_27: nat,X_71: nat] : append_nat(hAPP_nat_list_nat(replicate_nat(I_27),X_71),cons_nat(X_71,nil_nat)) = cons_nat(X_71,hAPP_nat_list_nat(replicate_nat(I_27),X_71)) ).
tff(fact_100_rev_Osimps_I2_J,axiom,
! [X_70: arrow_931160158le_alt,Xs_93: list_A1748581779le_alt] : rev_Ar1562208110le_alt(cons_A684544190le_alt(X_70,Xs_93)) = append634883619le_alt(rev_Ar1562208110le_alt(Xs_93),cons_A684544190le_alt(X_70,nil_Ar1741995278le_alt)) ).
tff(fact_101_rev_Osimps_I2_J,axiom,
! [X_70: nat,Xs_93: list_nat] : rev_nat(cons_nat(X_70,Xs_93)) = append_nat(rev_nat(Xs_93),cons_nat(X_70,nil_nat)) ).
tff(fact_102_rev__is__rev__conv,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( rev_nat(Xs_22) = rev_nat(Ys_4) )
<=> ( Xs_22 = Ys_4 ) ) ).
tff(fact_103_rev__is__rev__conv,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( rev_Ar1562208110le_alt(Xs_22) = rev_Ar1562208110le_alt(Ys_4) )
<=> ( Xs_22 = Ys_4 ) ) ).
tff(fact_104_rev__swap,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( rev_nat(Xs_22) = Ys_4 )
<=> ( Xs_22 = rev_nat(Ys_4) ) ) ).
tff(fact_105_rev__swap,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( rev_Ar1562208110le_alt(Xs_22) = Ys_4 )
<=> ( Xs_22 = rev_Ar1562208110le_alt(Ys_4) ) ) ).
tff(fact_106_rev__replicate,axiom,
! [N_45: nat,X_69: nat] : rev_nat(hAPP_nat_list_nat(replicate_nat(N_45),X_69)) = hAPP_nat_list_nat(replicate_nat(N_45),X_69) ).
tff(fact_107_rev__replicate,axiom,
! [N_45: nat,X_69: arrow_931160158le_alt] : rev_Ar1562208110le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_45),X_69)) = hAPP_A1866301020le_alt(replic1967339976le_alt(N_45),X_69) ).
tff(fact_108_rev__rev__ident,axiom,
! [Xs_92: list_nat] : rev_nat(rev_nat(Xs_92)) = Xs_92 ).
tff(fact_109_rev__rev__ident,axiom,
! [Xs_92: list_A1748581779le_alt] : rev_Ar1562208110le_alt(rev_Ar1562208110le_alt(Xs_92)) = Xs_92 ).
tff(fact_110_equal__list__def,axiom,
! [X_9: list_A1748581779le_alt,Y_9: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(equal_117954737le_alt,X_9),Y_9))
<=> ( X_9 = Y_9 ) ) ).
tff(fact_111_rev__is__Nil__conv,axiom,
! [Xs_22: list_A1748581779le_alt] :
( ( rev_Ar1562208110le_alt(Xs_22) = nil_Ar1741995278le_alt )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_112_rev__is__Nil__conv,axiom,
! [Xs_22: list_nat] :
( ( rev_nat(Xs_22) = nil_nat )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_113_Nil__is__rev__conv,axiom,
! [Xs_22: list_A1748581779le_alt] :
( ( nil_Ar1741995278le_alt = rev_Ar1562208110le_alt(Xs_22) )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_114_Nil__is__rev__conv,axiom,
! [Xs_22: list_nat] :
( ( nil_nat = rev_nat(Xs_22) )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_115_rev_Osimps_I1_J,axiom,
rev_Ar1562208110le_alt(nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_116_rev_Osimps_I1_J,axiom,
rev_nat(nil_nat) = nil_nat ).
tff(fact_117_rev__append,axiom,
! [Xs_91: list_nat,Ys_21: list_nat] : rev_nat(append_nat(Xs_91,Ys_21)) = append_nat(rev_nat(Ys_21),rev_nat(Xs_91)) ).
tff(fact_118_rev__append,axiom,
! [Xs_91: list_A1748581779le_alt,Ys_21: list_A1748581779le_alt] : rev_Ar1562208110le_alt(append634883619le_alt(Xs_91,Ys_21)) = append634883619le_alt(rev_Ar1562208110le_alt(Ys_21),rev_Ar1562208110le_alt(Xs_91)) ).
tff(fact_119_distinct__rev,axiom,
! [Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,rev_Ar1562208110le_alt(Xs_22)))
<=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22)) ) ).
tff(fact_120_distinct__rev,axiom,
! [Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,rev_nat(Xs_22)))
<=> hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_22)) ) ).
tff(fact_121_append__replicate__commute,axiom,
! [N_44: nat,X_68: arrow_931160158le_alt,K_7: nat] : append634883619le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_44),X_68),hAPP_A1866301020le_alt(replic1967339976le_alt(K_7),X_68)) = append634883619le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(K_7),X_68),hAPP_A1866301020le_alt(replic1967339976le_alt(N_44),X_68)) ).
tff(fact_122_rotate1__is__Nil__conv,axiom,
! [Xs_22: list_A1748581779le_alt] :
( ( rotate791150427le_alt(Xs_22) = nil_Ar1741995278le_alt )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_123_rotate1__is__Nil__conv,axiom,
! [Xs_22: list_nat] :
( ( rotate1_nat(Xs_22) = nil_nat )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_124_mem__def,axiom,
! [X_9: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_9),A))
<=> hBOOL(hAPP_nat_bool(A,X_9)) ) ).
tff(fact_125_mem__def,axiom,
! [X_9: int,A: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_9),A))
<=> hBOOL(hAPP_int_bool(A,X_9)) ) ).
tff(fact_126_Collect__def,axiom,
! [P_1: fun_nat_bool] : collect_nat(P_1) = P_1 ).
tff(fact_127_distinct1__rotate,axiom,
! [Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,rotate791150427le_alt(Xs_22)))
<=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22)) ) ).
tff(fact_128_singleton__rev__conv,axiom,
! [X_9: arrow_931160158le_alt,Xs_22: list_A1748581779le_alt] :
( ( cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt) = rev_Ar1562208110le_alt(Xs_22) )
<=> ( Xs_22 = cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt) ) ) ).
tff(fact_129_singleton__rev__conv,axiom,
! [X_9: nat,Xs_22: list_nat] :
( ( cons_nat(X_9,nil_nat) = rev_nat(Xs_22) )
<=> ( Xs_22 = cons_nat(X_9,nil_nat) ) ) ).
tff(fact_130_rev__singleton__conv,axiom,
! [Xs_22: list_A1748581779le_alt,X_9: arrow_931160158le_alt] :
( ( rev_Ar1562208110le_alt(Xs_22) = cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt) )
<=> ( Xs_22 = cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt) ) ) ).
tff(fact_131_rev__singleton__conv,axiom,
! [Xs_22: list_nat,X_9: nat] :
( ( rev_nat(Xs_22) = cons_nat(X_9,nil_nat) )
<=> ( Xs_22 = cons_nat(X_9,nil_nat) ) ) ).
tff(fact_132_replicate__app__Cons__same,axiom,
! [N_43: nat,X_67: arrow_931160158le_alt,Xs_90: list_A1748581779le_alt] : append634883619le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_43),X_67),cons_A684544190le_alt(X_67,Xs_90)) = cons_A684544190le_alt(X_67,append634883619le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_43),X_67),Xs_90)) ).
tff(fact_133_last__ConsL,axiom,
! [X_66: arrow_931160158le_alt,Xs_89: list_A1748581779le_alt] :
( ( Xs_89 = nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(cons_A684544190le_alt(X_66,Xs_89)) = X_66 ) ) ).
tff(fact_134_last__ConsL,axiom,
! [X_66: nat,Xs_89: list_nat] :
( ( Xs_89 = nil_nat )
=> ( last_nat(cons_nat(X_66,Xs_89)) = X_66 ) ) ).
tff(fact_135_last__ConsR,axiom,
! [X_65: arrow_931160158le_alt,Xs_88: list_A1748581779le_alt] :
( ( Xs_88 != nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(cons_A684544190le_alt(X_65,Xs_88)) = last_A1673116455le_alt(Xs_88) ) ) ).
tff(fact_136_last__ConsR,axiom,
! [X_65: nat,Xs_88: list_nat] :
( ( Xs_88 != nil_nat )
=> ( last_nat(cons_nat(X_65,Xs_88)) = last_nat(Xs_88) ) ) ).
tff(fact_137_last_Osimps,axiom,
! [X_64: arrow_931160158le_alt,Xs_87: list_A1748581779le_alt] :
( ( ( Xs_87 = nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(cons_A684544190le_alt(X_64,Xs_87)) = X_64 ) )
& ( ( Xs_87 != nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(cons_A684544190le_alt(X_64,Xs_87)) = last_A1673116455le_alt(Xs_87) ) ) ) ).
tff(fact_138_last_Osimps,axiom,
! [X_64: nat,Xs_87: list_nat] :
( ( ( Xs_87 = nil_nat )
=> ( last_nat(cons_nat(X_64,Xs_87)) = X_64 ) )
& ( ( Xs_87 != nil_nat )
=> ( last_nat(cons_nat(X_64,Xs_87)) = last_nat(Xs_87) ) ) ) ).
tff(fact_139_last__append,axiom,
! [Xs_86: list_A1748581779le_alt,Ys_20: list_A1748581779le_alt] :
( ( ( Ys_20 = nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(append634883619le_alt(Xs_86,Ys_20)) = last_A1673116455le_alt(Xs_86) ) )
& ( ( Ys_20 != nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(append634883619le_alt(Xs_86,Ys_20)) = last_A1673116455le_alt(Ys_20) ) ) ) ).
tff(fact_140_last__append,axiom,
! [Xs_86: list_nat,Ys_20: list_nat] :
( ( ( Ys_20 = nil_nat )
=> ( last_nat(append_nat(Xs_86,Ys_20)) = last_nat(Xs_86) ) )
& ( ( Ys_20 != nil_nat )
=> ( last_nat(append_nat(Xs_86,Ys_20)) = last_nat(Ys_20) ) ) ) ).
tff(fact_141_last__appendR,axiom,
! [Xs_85: list_A1748581779le_alt,Ys_19: list_A1748581779le_alt] :
( ( Ys_19 != nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(append634883619le_alt(Xs_85,Ys_19)) = last_A1673116455le_alt(Ys_19) ) ) ).
tff(fact_142_last__appendR,axiom,
! [Xs_85: list_nat,Ys_19: list_nat] :
( ( Ys_19 != nil_nat )
=> ( last_nat(append_nat(Xs_85,Ys_19)) = last_nat(Ys_19) ) ) ).
tff(fact_143_last__appendL,axiom,
! [Xs_84: list_A1748581779le_alt,Ys_18: list_A1748581779le_alt] :
( ( Ys_18 = nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(append634883619le_alt(Xs_84,Ys_18)) = last_A1673116455le_alt(Xs_84) ) ) ).
tff(fact_144_last__appendL,axiom,
! [Xs_84: list_nat,Ys_18: list_nat] :
( ( Ys_18 = nil_nat )
=> ( last_nat(append_nat(Xs_84,Ys_18)) = last_nat(Xs_84) ) ) ).
tff(fact_145_rev__eq__Cons__iff,axiom,
! [Xs_22: list_A1748581779le_alt,Y_9: arrow_931160158le_alt,Ys_4: list_A1748581779le_alt] :
( ( rev_Ar1562208110le_alt(Xs_22) = cons_A684544190le_alt(Y_9,Ys_4) )
<=> ( Xs_22 = append634883619le_alt(rev_Ar1562208110le_alt(Ys_4),cons_A684544190le_alt(Y_9,nil_Ar1741995278le_alt)) ) ) ).
tff(fact_146_rev__eq__Cons__iff,axiom,
! [Xs_22: list_nat,Y_9: nat,Ys_4: list_nat] :
( ( rev_nat(Xs_22) = cons_nat(Y_9,Ys_4) )
<=> ( Xs_22 = append_nat(rev_nat(Ys_4),cons_nat(Y_9,nil_nat)) ) ) ).
tff(fact_147_last__rev,axiom,
! [Xs_83: list_A1748581779le_alt] :
( ( Xs_83 != nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(rev_Ar1562208110le_alt(Xs_83)) = hd_Arr274000865le_alt(Xs_83) ) ) ).
tff(fact_148_last__rev,axiom,
! [Xs_83: list_nat] :
( ( Xs_83 != nil_nat )
=> ( last_nat(rev_nat(Xs_83)) = hd_nat(Xs_83) ) ) ).
tff(fact_149_hd__rev,axiom,
! [Xs_82: list_A1748581779le_alt] :
( ( Xs_82 != nil_Ar1741995278le_alt )
=> ( hd_Arr274000865le_alt(rev_Ar1562208110le_alt(Xs_82)) = last_A1673116455le_alt(Xs_82) ) ) ).
tff(fact_150_hd__rev,axiom,
! [Xs_82: list_nat] :
( ( Xs_82 != nil_nat )
=> ( hd_nat(rev_nat(Xs_82)) = last_nat(Xs_82) ) ) ).
tff(fact_151_equal,axiom,
equal_117954737le_alt = fequal414630996le_alt ).
tff(fact_152_equal__refl,axiom,
! [X_63: list_A1748581779le_alt] : hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(equal_117954737le_alt,X_63),X_63)) ).
tff(fact_153_equal__eq,axiom,
! [X_9: list_A1748581779le_alt,Y_9: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(equal_117954737le_alt,X_9),Y_9))
<=> ( X_9 = Y_9 ) ) ).
tff(fact_154_eq__equal,axiom,
fequal414630996le_alt = equal_117954737le_alt ).
tff(fact_155_hd__append,axiom,
! [Ys_17: list_A1748581779le_alt,Xs_81: list_A1748581779le_alt] :
( ( ( Xs_81 = nil_Ar1741995278le_alt )
=> ( hd_Arr274000865le_alt(append634883619le_alt(Xs_81,Ys_17)) = hd_Arr274000865le_alt(Ys_17) ) )
& ( ( Xs_81 != nil_Ar1741995278le_alt )
=> ( hd_Arr274000865le_alt(append634883619le_alt(Xs_81,Ys_17)) = hd_Arr274000865le_alt(Xs_81) ) ) ) ).
tff(fact_156_hd__append,axiom,
! [Ys_17: list_nat,Xs_81: list_nat] :
( ( ( Xs_81 = nil_nat )
=> ( hd_nat(append_nat(Xs_81,Ys_17)) = hd_nat(Ys_17) ) )
& ( ( Xs_81 != nil_nat )
=> ( hd_nat(append_nat(Xs_81,Ys_17)) = hd_nat(Xs_81) ) ) ) ).
tff(fact_157_hd__append2,axiom,
! [Ys_16: list_A1748581779le_alt,Xs_80: list_A1748581779le_alt] :
( ( Xs_80 != nil_Ar1741995278le_alt )
=> ( hd_Arr274000865le_alt(append634883619le_alt(Xs_80,Ys_16)) = hd_Arr274000865le_alt(Xs_80) ) ) ).
tff(fact_158_hd__append2,axiom,
! [Ys_16: list_nat,Xs_80: list_nat] :
( ( Xs_80 != nil_nat )
=> ( hd_nat(append_nat(Xs_80,Ys_16)) = hd_nat(Xs_80) ) ) ).
tff(fact_159_rotate1__hd__tl,axiom,
! [Xs_79: list_A1748581779le_alt] :
( ( Xs_79 != nil_Ar1741995278le_alt )
=> ( rotate791150427le_alt(Xs_79) = append634883619le_alt(tl_Arr921252325le_alt(Xs_79),cons_A684544190le_alt(hd_Arr274000865le_alt(Xs_79),nil_Ar1741995278le_alt)) ) ) ).
tff(fact_160_rotate1__hd__tl,axiom,
! [Xs_79: list_nat] :
( ( Xs_79 != nil_nat )
=> ( rotate1_nat(Xs_79) = append_nat(tl_nat(Xs_79),cons_nat(hd_nat(Xs_79),nil_nat)) ) ) ).
tff(fact_161_tl_Osimps_I2_J,axiom,
! [X_62: arrow_931160158le_alt,Xs_78: list_A1748581779le_alt] : tl_Arr921252325le_alt(cons_A684544190le_alt(X_62,Xs_78)) = Xs_78 ).
tff(fact_162_tl_Osimps_I1_J,axiom,
tl_Arr921252325le_alt(nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_163_tl_Osimps_I1_J,axiom,
tl_nat(nil_nat) = nil_nat ).
tff(fact_164_hd_Osimps,axiom,
! [X_61: arrow_931160158le_alt,Xs_77: list_A1748581779le_alt] : hd_Arr274000865le_alt(cons_A684544190le_alt(X_61,Xs_77)) = X_61 ).
tff(fact_165_distinct__tl,axiom,
! [Xs_76: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_76))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,tl_Arr921252325le_alt(Xs_76))) ) ).
tff(fact_166_sorted__tl,axiom,
! [Xs_75: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_75))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,tl_nat(Xs_75))) ) ).
tff(fact_167_tl__append2,axiom,
! [Ys_15: list_A1748581779le_alt,Xs_74: list_A1748581779le_alt] :
( ( Xs_74 != nil_Ar1741995278le_alt )
=> ( tl_Arr921252325le_alt(append634883619le_alt(Xs_74,Ys_15)) = append634883619le_alt(tl_Arr921252325le_alt(Xs_74),Ys_15) ) ) ).
tff(fact_168_tl__append2,axiom,
! [Ys_15: list_nat,Xs_74: list_nat] :
( ( Xs_74 != nil_nat )
=> ( tl_nat(append_nat(Xs_74,Ys_15)) = append_nat(tl_nat(Xs_74),Ys_15) ) ) ).
tff(fact_169_hd__dropWhile,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( ( dropWh1772583087le_alt(P_1,Xs_22) != nil_Ar1741995278le_alt )
=> ~ hBOOL(hAPP_A674791462t_bool(P_1,hd_Arr274000865le_alt(dropWh1772583087le_alt(P_1,Xs_22)))) ) ).
tff(fact_170_hd__dropWhile,axiom,
! [P_1: fun_nat_bool,Xs_22: list_nat] :
( ( dropWhile_nat(P_1,Xs_22) != nil_nat )
=> ~ hBOOL(hAPP_nat_bool(P_1,hd_nat(dropWhile_nat(P_1,Xs_22)))) ) ).
tff(fact_171_member__rec_I2_J,axiom,
! [Y_28: arrow_931160158le_alt] : ~ hBOOL(hAPP_A674791462t_bool(member1960769315le_alt(nil_Ar1741995278le_alt),Y_28)) ).
tff(fact_172_member__rec_I2_J,axiom,
! [Y_28: nat] : ~ hBOOL(hAPP_nat_bool(member_nat_1(nil_nat),Y_28)) ).
tff(fact_173_list__ex1__simps_I1_J,axiom,
! [P_1: fun_Ar1091019988t_bool] : ~ hBOOL(hAPP_l713721587t_bool(list_e695046097le_alt(P_1),nil_Ar1741995278le_alt)) ).
tff(fact_174_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_175_elem,axiom,
! [X_60: arrow_931160158le_alt,Xs_73: list_A1748581779le_alt] : hBOOL(hAPP_l713721587t_bool(listMe533848314le_alt(X_60),cons_A684544190le_alt(X_60,Xs_73))) ).
tff(fact_176_member__rec_I1_J,axiom,
! [X_9: arrow_931160158le_alt,Xs_22: list_A1748581779le_alt,Y_9: arrow_931160158le_alt] :
( hBOOL(hAPP_A674791462t_bool(member1960769315le_alt(cons_A684544190le_alt(X_9,Xs_22)),Y_9))
<=> ( ( X_9 = Y_9 )
| hBOOL(hAPP_A674791462t_bool(member1960769315le_alt(Xs_22),Y_9)) ) ) ).
tff(fact_177_insert,axiom,
! [Y_27: arrow_931160158le_alt,X_59: arrow_931160158le_alt,Xs_72: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(listMe533848314le_alt(X_59),Xs_72))
=> hBOOL(hAPP_l713721587t_bool(listMe533848314le_alt(X_59),cons_A684544190le_alt(Y_27,Xs_72))) ) ).
tff(fact_178_hd__replicate,axiom,
! [X_58: arrow_931160158le_alt,N_42: nat] :
( ( N_42 != zero_zero_nat )
=> ( hd_Arr274000865le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_42),X_58)) = X_58 ) ) ).
tff(fact_179_last__replicate,axiom,
! [X_57: arrow_931160158le_alt,N_41: nat] :
( ( N_41 != zero_zero_nat )
=> ( last_A1673116455le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_41),X_57)) = X_57 ) ) ).
tff(fact_180_replicate__eq__replicate,axiom,
! [M_1: nat,X_9: arrow_931160158le_alt,N_12: nat,Y_9: arrow_931160158le_alt] :
( ( hAPP_A1866301020le_alt(replic1967339976le_alt(M_1),X_9) = hAPP_A1866301020le_alt(replic1967339976le_alt(N_12),Y_9) )
<=> ( ( M_1 = N_12 )
& ( ( M_1 != zero_zero_nat )
=> ( X_9 = Y_9 ) ) ) ) ).
tff(fact_181_dropWhile_Osimps_I2_J,axiom,
! [Xs_22: list_A1748581779le_alt,P_1: fun_Ar1091019988t_bool,X_9: arrow_931160158le_alt] :
( ( hBOOL(hAPP_A674791462t_bool(P_1,X_9))
=> ( dropWh1772583087le_alt(P_1,cons_A684544190le_alt(X_9,Xs_22)) = dropWh1772583087le_alt(P_1,Xs_22) ) )
& ( ~ hBOOL(hAPP_A674791462t_bool(P_1,X_9))
=> ( dropWh1772583087le_alt(P_1,cons_A684544190le_alt(X_9,Xs_22)) = cons_A684544190le_alt(X_9,Xs_22) ) ) ) ).
tff(fact_182_dropWhile_Osimps_I1_J,axiom,
! [P_1: fun_Ar1091019988t_bool] : dropWh1772583087le_alt(P_1,nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_183_dropWhile_Osimps_I1_J,axiom,
! [P_1: fun_nat_bool] : dropWhile_nat(P_1,nil_nat) = nil_nat ).
tff(fact_184_distinct__dropWhile,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,dropWh1772583087le_alt(P_1,Xs_22))) ) ).
tff(fact_185_sorted__dropWhile,axiom,
! [P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_22))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,dropWhile_nat(P_1,Xs_22))) ) ).
tff(fact_186_replicate__0,axiom,
! [X_56: arrow_931160158le_alt] : hAPP_A1866301020le_alt(replic1967339976le_alt(zero_zero_nat),X_56) = nil_Ar1741995278le_alt ).
tff(fact_187_replicate__0,axiom,
! [X_56: nat] : hAPP_nat_list_nat(replicate_nat(zero_zero_nat),X_56) = nil_nat ).
tff(fact_188_empty__replicate,axiom,
! [N_12: nat,X_9: arrow_931160158le_alt] :
( ( nil_Ar1741995278le_alt = hAPP_A1866301020le_alt(replic1967339976le_alt(N_12),X_9) )
<=> ( N_12 = zero_zero_nat ) ) ).
tff(fact_189_empty__replicate,axiom,
! [N_12: nat,X_9: nat] :
( ( nil_nat = hAPP_nat_list_nat(replicate_nat(N_12),X_9) )
<=> ( N_12 = zero_zero_nat ) ) ).
tff(fact_190_replicate__empty,axiom,
! [N_12: nat,X_9: arrow_931160158le_alt] :
( ( hAPP_A1866301020le_alt(replic1967339976le_alt(N_12),X_9) = nil_Ar1741995278le_alt )
<=> ( N_12 = zero_zero_nat ) ) ).
tff(fact_191_replicate__empty,axiom,
! [N_12: nat,X_9: nat] :
( ( hAPP_nat_list_nat(replicate_nat(N_12),X_9) = nil_nat )
<=> ( N_12 = zero_zero_nat ) ) ).
tff(fact_192_sublist__singleton,axiom,
! [X_9: arrow_931160158le_alt,A: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublis356271123le_alt(cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt),A) = cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublis356271123le_alt(cons_A684544190le_alt(X_9,nil_Ar1741995278le_alt),A) = nil_Ar1741995278le_alt ) ) ) ).
tff(fact_193_sublist__singleton,axiom,
! [X_9: nat,A: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublist_nat(cons_nat(X_9,nil_nat),A) = cons_nat(X_9,nil_nat) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublist_nat(cons_nat(X_9,nil_nat),A) = nil_nat ) ) ) ).
tff(fact_194_dropWhile__eq__Cons__conv,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt,Y_9: arrow_931160158le_alt,Ys_4: list_A1748581779le_alt] :
( ( dropWh1772583087le_alt(P_1,Xs_22) = cons_A684544190le_alt(Y_9,Ys_4) )
<=> ( ( Xs_22 = append634883619le_alt(takeWh4609031le_alt(P_1,Xs_22),cons_A684544190le_alt(Y_9,Ys_4)) )
& ~ hBOOL(hAPP_A674791462t_bool(P_1,Y_9)) ) ) ).
tff(fact_195_list_Osize_I1_J,axiom,
! [Fa: fun_Ar518461033lt_nat] : list_s706097040le_alt(Fa,nil_Ar1741995278le_alt) = zero_zero_nat ).
tff(fact_196_list_Osize_I1_J,axiom,
! [Fa: fun_nat_nat] : list_size_nat(Fa,nil_nat) = zero_zero_nat ).
tff(fact_197_ListMem_Osimps,axiom,
! [A1: arrow_931160158le_alt,A2: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(listMe533848314le_alt(A1),A2))
<=> ( ? [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt] :
( ( A1 = X_52 )
& ( A2 = cons_A684544190le_alt(X_52,Xs_44) ) )
| ? [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt,Y_3: arrow_931160158le_alt] :
( ( A1 = X_52 )
& ( A2 = cons_A684544190le_alt(Y_3,Xs_44) )
& hBOOL(hAPP_l713721587t_bool(listMe533848314le_alt(X_52),Xs_44)) ) ) ) ).
tff(fact_198_zero__reorient,axiom,
! [X_9: int] :
( ( zero_zero_int = X_9 )
<=> ( X_9 = zero_zero_int ) ) ).
tff(fact_199_zero__reorient,axiom,
! [X_9: nat] :
( ( zero_zero_nat = X_9 )
<=> ( X_9 = zero_zero_nat ) ) ).
tff(fact_200_hd__conv__nth,axiom,
! [Xs_71: list_nat] :
( ( Xs_71 != nil_nat )
=> ( hd_nat(Xs_71) = hAPP_nat_nat(nth_nat(Xs_71),zero_zero_nat) ) ) ).
tff(fact_201_hd__conv__nth,axiom,
! [Xs_71: list_A1748581779le_alt] :
( ( Xs_71 != nil_Ar1741995278le_alt )
=> ( hd_Arr274000865le_alt(Xs_71) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_71),zero_zero_nat) ) ) ).
tff(fact_202_takeWhile_Osimps_I1_J,axiom,
! [P_1: fun_nat_bool] : takeWhile_nat(P_1,nil_nat) = nil_nat ).
tff(fact_203_takeWhile_Osimps_I1_J,axiom,
! [P_1: fun_Ar1091019988t_bool] : takeWh4609031le_alt(P_1,nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_204_distinct__takeWhile,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,takeWh4609031le_alt(P_1,Xs_22))) ) ).
tff(fact_205_sorted__takeWhile,axiom,
! [P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_22))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,takeWhile_nat(P_1,Xs_22))) ) ).
tff(fact_206_sublist__nil,axiom,
! [A: fun_nat_bool] : sublist_nat(nil_nat,A) = nil_nat ).
tff(fact_207_sublist__nil,axiom,
! [A: fun_nat_bool] : sublis356271123le_alt(nil_Ar1741995278le_alt,A) = nil_Ar1741995278le_alt ).
tff(fact_208_distinct__sublistI,axiom,
! [I_26: fun_nat_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,sublis356271123le_alt(Xs_22,I_26))) ) ).
tff(fact_209_nth__Cons__0,axiom,
! [X_55: nat,Xs_70: list_nat] : hAPP_nat_nat(nth_nat(cons_nat(X_55,Xs_70)),zero_zero_nat) = X_55 ).
tff(fact_210_nth__Cons__0,axiom,
! [X_55: arrow_931160158le_alt,Xs_70: list_A1748581779le_alt] : hAPP_n2017047337le_alt(nth_Ar581680143le_alt(cons_A684544190le_alt(X_55,Xs_70)),zero_zero_nat) = X_55 ).
tff(fact_211_takeWhile_Osimps_I2_J,axiom,
! [Xs_22: list_nat,P_1: fun_nat_bool,X_9: nat] :
( ( hBOOL(hAPP_nat_bool(P_1,X_9))
=> ( takeWhile_nat(P_1,cons_nat(X_9,Xs_22)) = cons_nat(X_9,takeWhile_nat(P_1,Xs_22)) ) )
& ( ~ hBOOL(hAPP_nat_bool(P_1,X_9))
=> ( takeWhile_nat(P_1,cons_nat(X_9,Xs_22)) = nil_nat ) ) ) ).
tff(fact_212_takeWhile_Osimps_I2_J,axiom,
! [Xs_22: list_A1748581779le_alt,P_1: fun_Ar1091019988t_bool,X_9: arrow_931160158le_alt] :
( ( hBOOL(hAPP_A674791462t_bool(P_1,X_9))
=> ( takeWh4609031le_alt(P_1,cons_A684544190le_alt(X_9,Xs_22)) = cons_A684544190le_alt(X_9,takeWh4609031le_alt(P_1,Xs_22)) ) )
& ( ~ hBOOL(hAPP_A674791462t_bool(P_1,X_9))
=> ( takeWh4609031le_alt(P_1,cons_A684544190le_alt(X_9,Xs_22)) = nil_Ar1741995278le_alt ) ) ) ).
tff(fact_213_takeWhile__tail,axiom,
! [Xs_22: list_A1748581779le_alt,L_1: list_A1748581779le_alt,P_1: fun_Ar1091019988t_bool,X_9: arrow_931160158le_alt] :
( ~ hBOOL(hAPP_A674791462t_bool(P_1,X_9))
=> ( takeWh4609031le_alt(P_1,append634883619le_alt(Xs_22,cons_A684544190le_alt(X_9,L_1))) = takeWh4609031le_alt(P_1,Xs_22) ) ) ).
tff(fact_214_takeWhile__dropWhile__id,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] : append634883619le_alt(takeWh4609031le_alt(P_1,Xs_22),dropWh1772583087le_alt(P_1,Xs_22)) = Xs_22 ).
tff(fact_215_nth__append__length,axiom,
! [Xs_69: list_nat,X_54: nat,Ys_14: list_nat] : hAPP_nat_nat(nth_nat(append_nat(Xs_69,cons_nat(X_54,Ys_14))),size_size_list_nat(Xs_69)) = X_54 ).
tff(fact_216_nth__append__length,axiom,
! [Xs_69: list_A1748581779le_alt,X_54: arrow_931160158le_alt,Ys_14: list_A1748581779le_alt] : hAPP_n2017047337le_alt(nth_Ar581680143le_alt(append634883619le_alt(Xs_69,cons_A684544190le_alt(X_54,Ys_14))),size_s1492124157le_alt(Xs_69)) = X_54 ).
tff(fact_217_foldr__max__sorted,axiom,
! [Y_9: int,Xs_22: list_int] :
( hBOOL(hAPP_list_int_bool(linorder_sorted_int,rev_int(Xs_22)))
=> ( ( ( Xs_22 = nil_int )
=> ( hAPP_int_int(foldr_int_int(ord_max_int,Xs_22),Y_9) = Y_9 ) )
& ( ( Xs_22 != nil_int )
=> ( hAPP_int_int(foldr_int_int(ord_max_int,Xs_22),Y_9) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,hAPP_nat_int(nth_int(Xs_22),zero_zero_nat)),Y_9) ) ) ) ) ).
tff(fact_218_foldr__max__sorted,axiom,
! [Y_9: nat,Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,rev_nat(Xs_22)))
=> ( ( ( Xs_22 = nil_nat )
=> ( hAPP_nat_nat(foldr_nat_nat(ord_max_nat,Xs_22),Y_9) = Y_9 ) )
& ( ( Xs_22 != nil_nat )
=> ( hAPP_nat_nat(foldr_nat_nat(ord_max_nat,Xs_22),Y_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(nth_nat(Xs_22),zero_zero_nat)),Y_9) ) ) ) ) ).
tff(fact_219_list__induct2_H,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_nat,P_1: fun_li1393995935t_bool] :
( hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,nil_Ar1741995278le_alt),nil_nat))
=> ( ! [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt] : hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,cons_A684544190le_alt(X_52,Xs_44)),nil_nat))
=> ( ! [Y_3: nat,Ys_5: list_nat] : hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,nil_Ar1741995278le_alt),cons_nat(Y_3,Ys_5)))
=> ( ! [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt,Y_3: nat,Ys_5: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,Xs_44),Ys_5))
=> hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,cons_A684544190le_alt(X_52,Xs_44)),cons_nat(Y_3,Ys_5))) )
=> hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,Xs_22),Ys_4)) ) ) ) ) ).
tff(fact_220_list__induct2_H,axiom,
! [Xs_22: list_nat,Ys_4: list_A1748581779le_alt,P_1: fun_li2096081351t_bool] :
( hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,nil_nat),nil_Ar1741995278le_alt))
=> ( ! [X_52: nat,Xs_44: list_nat] : hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,cons_nat(X_52,Xs_44)),nil_Ar1741995278le_alt))
=> ( ! [Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] : hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,nil_nat),cons_A684544190le_alt(Y_3,Ys_5)))
=> ( ! [X_52: nat,Xs_44: list_nat,Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,Xs_44),Ys_5))
=> hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,cons_nat(X_52,Xs_44)),cons_A684544190le_alt(Y_3,Ys_5))) )
=> hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,Xs_22),Ys_4)) ) ) ) ) ).
tff(fact_221_map__filter__simps_I2_J,axiom,
! [F: fun_Ar237091702on_nat] : map_fi747840374lt_nat(F,nil_Ar1741995278le_alt) = nil_nat ).
tff(fact_222_map__filter__simps_I2_J,axiom,
! [F: fun_na1357632374le_alt] : map_fi1918498462le_alt(F,nil_nat) = nil_Ar1741995278le_alt ).
tff(fact_223_listrelp_ONil,axiom,
! [R_1: fun_nat_fun_nat_bool] : hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(listrelp_nat(R_1),nil_nat),nil_nat)) ).
tff(fact_224_listrelp_ONil,axiom,
! [R_1: fun_Ar1414184139t_bool] : hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(listre1668963176le_alt(R_1),nil_Ar1741995278le_alt),nil_Ar1741995278le_alt)) ).
tff(fact_225_listrelp_OCons,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt,R_1: fun_Ar1414184139t_bool,X_9: arrow_931160158le_alt,Y_9: arrow_931160158le_alt] :
( hBOOL(hAPP_A674791462t_bool(hAPP_A1858212765t_bool(R_1,X_9),Y_9))
=> ( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(listre1668963176le_alt(R_1),Xs_22),Ys_4))
=> hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(listre1668963176le_alt(R_1),cons_A684544190le_alt(X_9,Xs_22)),cons_A684544190le_alt(Y_9,Ys_4))) ) ) ).
tff(fact_226_sublist__empty,axiom,
! [Xs_22: list_nat] : sublist_nat(Xs_22,bot_bot_fun_nat_bool) = nil_nat ).
tff(fact_227_sublist__empty,axiom,
! [Xs_22: list_A1748581779le_alt] : sublis356271123le_alt(Xs_22,bot_bot_fun_nat_bool) = nil_Ar1741995278le_alt ).
tff(fact_228_neq__if__length__neq,axiom,
! [Xs_68: list_A1748581779le_alt,Ys_13: list_A1748581779le_alt] :
( ( size_s1492124157le_alt(Xs_68) != size_s1492124157le_alt(Ys_13) )
=> ( Xs_68 != Ys_13 ) ) ).
tff(fact_229_length__rev,axiom,
! [Xs_67: list_nat] : size_size_list_nat(rev_nat(Xs_67)) = size_size_list_nat(Xs_67) ).
tff(fact_230_length__rev,axiom,
! [Xs_67: list_A1748581779le_alt] : size_s1492124157le_alt(rev_Ar1562208110le_alt(Xs_67)) = size_s1492124157le_alt(Xs_67) ).
tff(fact_231_length__replicate,axiom,
! [N_40: nat,X_53: arrow_931160158le_alt] : size_s1492124157le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_40),X_53)) = N_40 ).
tff(fact_232_length__rotate1,axiom,
! [Xs_66: list_A1748581779le_alt] : size_s1492124157le_alt(rotate791150427le_alt(Xs_66)) = size_s1492124157le_alt(Xs_66) ).
tff(fact_233_foldr_Osimps_I2_J,axiom,
! [F: fun_nat_fun_nat_nat,X_9: nat,Xs_22: list_nat,Aa: nat] : hAPP_nat_nat(foldr_nat_nat(F,cons_nat(X_9,Xs_22)),Aa) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,X_9),hAPP_nat_nat(foldr_nat_nat(F,Xs_22),Aa)) ).
tff(fact_234_foldr_Osimps_I1_J,axiom,
! [F: fun_nat_fun_nat_nat,Aa: nat] : hAPP_nat_nat(foldr_nat_nat(F,nil_nat),Aa) = Aa ).
tff(fact_235_foldr__append,axiom,
! [F: fun_nat_fun_nat_nat,Xs_22: list_nat,Ys_4: list_nat,Aa: nat] : hAPP_nat_nat(foldr_nat_nat(F,append_nat(Xs_22,Ys_4)),Aa) = hAPP_nat_nat(foldr_nat_nat(F,Xs_22),hAPP_nat_nat(foldr_nat_nat(F,Ys_4),Aa)) ).
tff(fact_236_length__0__conv,axiom,
! [Xs_22: list_nat] :
( ( size_size_list_nat(Xs_22) = zero_zero_nat )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_237_length__0__conv,axiom,
! [Xs_22: list_A1748581779le_alt] :
( ( size_s1492124157le_alt(Xs_22) = zero_zero_nat )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_238_list_Osize_I3_J,axiom,
size_size_list_nat(nil_nat) = zero_zero_nat ).
tff(fact_239_list_Osize_I3_J,axiom,
size_s1492124157le_alt(nil_Ar1741995278le_alt) = zero_zero_nat ).
tff(fact_240_list__induct2,axiom,
! [P_1: fun_li1393995935t_bool,Xs_22: list_A1748581779le_alt,Ys_4: list_nat] :
( ( size_s1492124157le_alt(Xs_22) = size_size_list_nat(Ys_4) )
=> ( hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,nil_Ar1741995278le_alt),nil_nat))
=> ( ! [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt,Y_3: nat,Ys_5: list_nat] :
( ( size_s1492124157le_alt(Xs_44) = size_size_list_nat(Ys_5) )
=> ( hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,Xs_44),Ys_5))
=> hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,cons_A684544190le_alt(X_52,Xs_44)),cons_nat(Y_3,Ys_5))) ) )
=> hBOOL(hAPP_list_nat_bool(hAPP_l931213803t_bool(P_1,Xs_22),Ys_4)) ) ) ) ).
tff(fact_241_list__induct2,axiom,
! [P_1: fun_li2096081351t_bool,Xs_22: list_nat,Ys_4: list_A1748581779le_alt] :
( ( size_size_list_nat(Xs_22) = size_s1492124157le_alt(Ys_4) )
=> ( hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,nil_nat),nil_Ar1741995278le_alt))
=> ( ! [X_52: nat,Xs_44: list_nat,Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] :
( ( size_size_list_nat(Xs_44) = size_s1492124157le_alt(Ys_5) )
=> ( hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,Xs_44),Ys_5))
=> hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,cons_nat(X_52,Xs_44)),cons_A684544190le_alt(Y_3,Ys_5))) ) )
=> hBOOL(hAPP_l713721587t_bool(hAPP_l824545683t_bool(P_1,Xs_22),Ys_4)) ) ) ) ).
tff(fact_242_list__induct2,axiom,
! [P_1: fun_li954148211t_bool,Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( size_s1492124157le_alt(Xs_22) = size_s1492124157le_alt(Ys_4) )
=> ( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(P_1,nil_Ar1741995278le_alt),nil_Ar1741995278le_alt))
=> ( ! [X_52: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt,Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] :
( ( size_s1492124157le_alt(Xs_44) = size_s1492124157le_alt(Ys_5) )
=> ( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(P_1,Xs_44),Ys_5))
=> hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(P_1,cons_A684544190le_alt(X_52,Xs_44)),cons_A684544190le_alt(Y_3,Ys_5))) ) )
=> hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(P_1,Xs_22),Ys_4)) ) ) ) ).
tff(fact_243_emptyE,axiom,
! [Aa: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),bot_bot_fun_int_bool)) ).
tff(fact_244_emptyE,axiom,
! [Aa: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Aa),bot_bot_fun_nat_bool)) ).
tff(fact_245_bot__fun__def,axiom,
! [X_52: nat] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_52))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_246_bot__apply,axiom,
! [X_9: nat] :
( hBOOL(hAPP_nat_bool(bot_bot_fun_nat_bool,X_9))
<=> hBOOL(bot_bot_bool) ) ).
tff(fact_247_Ex__list__of__length,axiom,
! [N_39: nat] :
? [Xs_44: list_A1748581779le_alt] : size_s1492124157le_alt(Xs_44) = N_39 ).
tff(fact_248_listrelp_Osimps,axiom,
! [R_1: fun_nat_fun_nat_bool,A1: list_nat,A2: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(listrelp_nat(R_1),A1),A2))
<=> ( ( ( A1 = nil_nat )
& ( A2 = nil_nat ) )
| ? [X_52: nat,Y_3: nat,Xs_44: list_nat,Ys_5: list_nat] :
( ( A1 = cons_nat(X_52,Xs_44) )
& ( A2 = cons_nat(Y_3,Ys_5) )
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(R_1,X_52),Y_3))
& hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(listrelp_nat(R_1),Xs_44),Ys_5)) ) ) ) ).
tff(fact_249_listrelp_Osimps,axiom,
! [R_1: fun_Ar1414184139t_bool,A1: list_A1748581779le_alt,A2: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(listre1668963176le_alt(R_1),A1),A2))
<=> ( ( ( A1 = nil_Ar1741995278le_alt )
& ( A2 = nil_Ar1741995278le_alt ) )
| ? [X_52: arrow_931160158le_alt,Y_3: arrow_931160158le_alt,Xs_44: list_A1748581779le_alt,Ys_5: list_A1748581779le_alt] :
( ( A1 = cons_A684544190le_alt(X_52,Xs_44) )
& ( A2 = cons_A684544190le_alt(Y_3,Ys_5) )
& hBOOL(hAPP_A674791462t_bool(hAPP_A1858212765t_bool(R_1,X_52),Y_3))
& hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(listre1668963176le_alt(R_1),Xs_44),Ys_5)) ) ) ) ).
tff(fact_250_equals0D,axiom,
! [Aa: int,A: fun_int_bool] :
( ( A = bot_bot_fun_int_bool )
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),A)) ) ).
tff(fact_251_equals0D,axiom,
! [Aa: nat,A: fun_nat_bool] :
( ( A = bot_bot_fun_nat_bool )
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Aa),A)) ) ).
tff(fact_252_Collect__empty__eq,axiom,
! [P_1: fun_nat_bool] :
( ( collect_nat(P_1) = bot_bot_fun_nat_bool )
<=> ! [X_52: nat] : ~ hBOOL(hAPP_nat_bool(P_1,X_52)) ) ).
tff(fact_253_empty__iff,axiom,
! [C_1: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_1),bot_bot_fun_int_bool)) ).
tff(fact_254_empty__iff,axiom,
! [C_1: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),bot_bot_fun_nat_bool)) ).
tff(fact_255_empty__Collect__eq,axiom,
! [P_1: fun_nat_bool] :
( ( bot_bot_fun_nat_bool = collect_nat(P_1) )
<=> ! [X_52: nat] : ~ hBOOL(hAPP_nat_bool(P_1,X_52)) ) ).
tff(fact_256_ex__in__conv,axiom,
! [A: fun_int_bool] :
( ? [X_52: int] : hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_52),A))
<=> ( A != bot_bot_fun_int_bool ) ) ).
tff(fact_257_ex__in__conv,axiom,
! [A: fun_nat_bool] :
( ? [X_52: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_52),A))
<=> ( A != bot_bot_fun_nat_bool ) ) ).
tff(fact_258_all__not__in__conv,axiom,
! [A: fun_int_bool] :
( ! [X_52: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_52),A))
<=> ( A = bot_bot_fun_int_bool ) ) ).
tff(fact_259_all__not__in__conv,axiom,
! [A: fun_nat_bool] :
( ! [X_52: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_52),A))
<=> ( A = bot_bot_fun_nat_bool ) ) ).
tff(fact_260_empty__def,axiom,
bot_bot_fun_nat_bool = collect_nat(cOMBK_bool_nat(fFalse)) ).
tff(fact_261_equals0I,axiom,
! [A: fun_int_bool] :
( ! [Y_3: int] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_3),A))
=> ( A = bot_bot_fun_int_bool ) ) ).
tff(fact_262_equals0I,axiom,
! [A: fun_nat_bool] :
( ! [Y_3: nat] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_3),A))
=> ( A = bot_bot_fun_nat_bool ) ) ).
tff(fact_263_sorted__list__of__set__empty,axiom,
linord810813188et_nat(bot_bot_fun_nat_bool) = nil_nat ).
tff(fact_264_min__max_Osup__assoc,axiom,
! [X_51: int,Y_26: int,Z_7: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_51),Y_26)),Z_7) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_51),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,Y_26),Z_7)) ).
tff(fact_265_min__max_Osup__assoc,axiom,
! [X_51: nat,Y_26: nat,Z_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_51),Y_26)),Z_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_51),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_26),Z_7)) ).
tff(fact_266_min__max_Osup_Oassoc,axiom,
! [A_45: int,B_29: int,C_24: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_45),B_29)),C_24) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_45),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,B_29),C_24)) ).
tff(fact_267_min__max_Osup_Oassoc,axiom,
! [A_45: nat,B_29: nat,C_24: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_45),B_29)),C_24) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_45),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_29),C_24)) ).
tff(fact_268_min__max_Osup__left__commute,axiom,
! [X_50: int,Y_25: int,Z_6: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_50),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,Y_25),Z_6)) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,Y_25),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_50),Z_6)) ).
tff(fact_269_min__max_Osup__left__commute,axiom,
! [X_50: nat,Y_25: nat,Z_6: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_50),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_25),Z_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_25),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_50),Z_6)) ).
tff(fact_270_min__max_Osup_Oleft__commute,axiom,
! [B_28: int,A_44: int,C_23: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,B_28),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_44),C_23)) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_44),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,B_28),C_23)) ).
tff(fact_271_min__max_Osup_Oleft__commute,axiom,
! [B_28: nat,A_44: nat,C_23: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_44),C_23)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_44),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_28),C_23)) ).
tff(fact_272_min__max_Osup_Oidem,axiom,
! [A_43: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_43),A_43) = A_43 ).
tff(fact_273_min__max_Osup_Oidem,axiom,
! [A_43: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_43),A_43) = A_43 ).
tff(fact_274_min__max_Osup_Ocommute,axiom,
! [A_42: int,B_27: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_42),B_27) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,B_27),A_42) ).
tff(fact_275_min__max_Osup_Ocommute,axiom,
! [A_42: nat,B_27: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_42),B_27) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_27),A_42) ).
tff(fact_276_min__max_Osup__commute,axiom,
! [X_49: int,Y_24: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_49),Y_24) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,Y_24),X_49) ).
tff(fact_277_min__max_Osup__commute,axiom,
! [X_49: nat,Y_24: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_49),Y_24) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_24),X_49) ).
tff(fact_278_min__max_Osup_Oleft__idem,axiom,
! [A_41: int,B_26: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_41),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_41),B_26)) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_41),B_26) ).
tff(fact_279_min__max_Osup_Oleft__idem,axiom,
! [A_41: nat,B_26: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_41),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_41),B_26)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_41),B_26) ).
tff(fact_280_min__max_Osup__left__idem,axiom,
! [X_48: int,Y_23: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_48),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_48),Y_23)) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_48),Y_23) ).
tff(fact_281_min__max_Osup__left__idem,axiom,
! [X_48: nat,Y_23: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_48),Y_23)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_48),Y_23) ).
tff(fact_282_Max_Oidem,axiom,
! [X_47: int] : hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_47),X_47) = X_47 ).
tff(fact_283_Max_Oidem,axiom,
! [X_47: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_47),X_47) = X_47 ).
tff(fact_284_dropWhile__eq__drop,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] : dropWh1772583087le_alt(P_1,Xs_22) = drop_A1802510926le_alt(size_s1492124157le_alt(takeWh4609031le_alt(P_1,Xs_22)),Xs_22) ).
tff(fact_285_nth__length__takeWhile,axiom,
! [P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,size_size_list_nat(takeWhile_nat(P_1,Xs_22))),size_size_list_nat(Xs_22)))
=> ~ hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs_22),size_size_list_nat(takeWhile_nat(P_1,Xs_22))))) ) ).
tff(fact_286_nth__length__takeWhile,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,size_s1492124157le_alt(takeWh4609031le_alt(P_1,Xs_22))),size_s1492124157le_alt(Xs_22)))
=> ~ hBOOL(hAPP_A674791462t_bool(P_1,hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),size_s1492124157le_alt(takeWh4609031le_alt(P_1,Xs_22))))) ) ).
tff(fact_287_takeWhile__nth,axiom,
! [J_1: nat,P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),size_size_list_nat(takeWhile_nat(P_1,Xs_22))))
=> ( hAPP_nat_nat(nth_nat(takeWhile_nat(P_1,Xs_22)),J_1) = hAPP_nat_nat(nth_nat(Xs_22),J_1) ) ) ).
tff(fact_288_takeWhile__nth,axiom,
! [J_1: nat,P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),size_s1492124157le_alt(takeWh4609031le_alt(P_1,Xs_22))))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(takeWh4609031le_alt(P_1,Xs_22)),J_1) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),J_1) ) ) ).
tff(fact_289_list__update__length,axiom,
! [Xs_65: list_A1748581779le_alt,X_46: arrow_931160158le_alt,Ys_12: list_A1748581779le_alt,Y_22: arrow_931160158le_alt] : hAPP_A1866301020le_alt(list_u518279030le_alt(append634883619le_alt(Xs_65,cons_A684544190le_alt(X_46,Ys_12)),size_s1492124157le_alt(Xs_65)),Y_22) = append634883619le_alt(Xs_65,cons_A684544190le_alt(Y_22,Ys_12)) ).
tff(fact_290_max__0L,axiom,
! [N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N_5) = N_5 ).
tff(fact_291_less__zeroE,axiom,
! [N_5: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),zero_zero_nat)) ).
tff(fact_292_list__update__overwrite,axiom,
! [Xs_64: list_A1748581779le_alt,I_25: nat,X_45: arrow_931160158le_alt,Y_21: arrow_931160158le_alt] : hAPP_A1866301020le_alt(list_u518279030le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_64,I_25),X_45),I_25),Y_21) = hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_64,I_25),Y_21) ).
tff(fact_293_list__update__swap,axiom,
! [Xs_63: list_A1748581779le_alt,X_44: arrow_931160158le_alt,X_43: arrow_931160158le_alt,I_24: nat,I_23: nat] :
( ( I_24 != I_23 )
=> ( hAPP_A1866301020le_alt(list_u518279030le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_63,I_24),X_44),I_23),X_43) = hAPP_A1866301020le_alt(list_u518279030le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_63,I_23),X_43),I_24),X_44) ) ) ).
tff(fact_294_nat__less__cases,axiom,
! [P_1: fun_nat_fun_nat_bool,M_1: nat,N_12: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_12),M_1)) )
=> ( ( ( M_1 = N_12 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_12),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_12),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_12),M_1)) ) ) ) ).
tff(fact_295_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_296_less__not__refl2,axiom,
! [N_5: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),M))
=> ( M != N_5 ) ) ).
tff(fact_297_less__irrefl__nat,axiom,
! [N_5: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),N_5)) ).
tff(fact_298_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_299_nat__neq__iff,axiom,
! [M_1: nat,N_12: nat] :
( ( M_1 != N_12 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),M_1)) ) ) ).
tff(fact_300_less__not__refl,axiom,
! [N_5: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),N_5)) ).
tff(fact_301_gr0I,axiom,
! [N_5: nat] :
( ( N_5 != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5)) ) ).
tff(fact_302_gr__implies__not0,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> ( N_5 != zero_zero_nat ) ) ).
tff(fact_303_less__nat__zero__code,axiom,
! [N_5: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),zero_zero_nat)) ).
tff(fact_304_neq0__conv,axiom,
! [N_12: nat] :
( ( N_12 != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) ) ).
tff(fact_305_not__less0,axiom,
! [N_5: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),zero_zero_nat)) ).
tff(fact_306_order__less__irrefl,axiom,
! [X_42: int] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_42),X_42)) ).
tff(fact_307_order__less__irrefl,axiom,
! [X_9: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),X_9)) ).
tff(fact_308_order__less__irrefl,axiom,
! [X_42: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_42),X_42)) ).
tff(fact_309_linorder__neq__iff,axiom,
! [X_9: int,Y_9: int] :
( ( X_9 != Y_9 )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_9))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_9),X_9)) ) ) ).
tff(fact_310_linorder__neq__iff,axiom,
! [X_9: nat,Y_9: nat] :
( ( X_9 != Y_9 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_9))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_9),X_9)) ) ) ).
tff(fact_311_not__less__iff__gr__or__eq,axiom,
! [X_9: int,Y_9: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_9))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_9),X_9))
| ( X_9 = Y_9 ) ) ) ).
tff(fact_312_not__less__iff__gr__or__eq,axiom,
! [X_9: nat,Y_9: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_9))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_9),X_9))
| ( X_9 = Y_9 ) ) ) ).
tff(fact_313_linorder__less__linear,axiom,
! [X_41: int,Y_20: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_41),Y_20))
| ( X_41 = Y_20 )
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),X_41)) ) ).
tff(fact_314_linorder__less__linear,axiom,
! [X_41: nat,Y_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_41),Y_20))
| ( X_41 = Y_20 )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),X_41)) ) ).
tff(fact_315_linorder__antisym__conv3,axiom,
! [Y_9: int,X_9: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_9),X_9))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_9))
<=> ( X_9 = Y_9 ) ) ) ).
tff(fact_316_linorder__antisym__conv3,axiom,
! [Y_9: nat,X_9: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_9),X_9))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_9))
<=> ( X_9 = Y_9 ) ) ) ).
tff(fact_317_linorder__neqE,axiom,
! [X_40: int,Y_19: int] :
( ( X_40 != Y_19 )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_40),Y_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_19),X_40)) ) ) ).
tff(fact_318_linorder__neqE,axiom,
! [X_40: nat,Y_19: nat] :
( ( X_40 != Y_19 )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_40),Y_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_19),X_40)) ) ) ).
tff(fact_319_less__imp__neq,axiom,
! [X_39: int,Y_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_39),Y_18))
=> ( X_39 != Y_18 ) ) ).
tff(fact_320_less__imp__neq,axiom,
! [X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ( X_9 != Y_9 ) ) ).
tff(fact_321_less__imp__neq,axiom,
! [X_39: nat,Y_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_39),Y_18))
=> ( X_39 != Y_18 ) ) ).
tff(fact_322_order__less__not__sym,axiom,
! [X_38: int,Y_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_38),Y_17))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_17),X_38)) ) ).
tff(fact_323_order__less__not__sym,axiom,
! [X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_9),X_9)) ) ).
tff(fact_324_order__less__not__sym,axiom,
! [X_38: nat,Y_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_38),Y_17))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_17),X_38)) ) ).
tff(fact_325_order__less__imp__not__less,axiom,
! [X_37: int,Y_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Y_16))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_16),X_37)) ) ).
tff(fact_326_order__less__imp__not__less,axiom,
! [X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_9),X_9)) ) ).
tff(fact_327_order__less__imp__not__less,axiom,
! [X_37: nat,Y_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Y_16))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_16),X_37)) ) ).
tff(fact_328_order__less__imp__not__eq,axiom,
! [X_36: int,Y_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_36),Y_15))
=> ( X_36 != Y_15 ) ) ).
tff(fact_329_order__less__imp__not__eq,axiom,
! [X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ( X_9 != Y_9 ) ) ).
tff(fact_330_order__less__imp__not__eq,axiom,
! [X_36: nat,Y_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_36),Y_15))
=> ( X_36 != Y_15 ) ) ).
tff(fact_331_order__less__imp__not__eq2,axiom,
! [X_35: int,Y_14: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Y_14))
=> ( Y_14 != X_35 ) ) ).
tff(fact_332_order__less__imp__not__eq2,axiom,
! [X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ( Y_9 != X_9 ) ) ).
tff(fact_333_order__less__imp__not__eq2,axiom,
! [X_35: nat,Y_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Y_14))
=> ( Y_14 != X_35 ) ) ).
tff(fact_334_order__less__imp__triv,axiom,
! [P_1: bool,X_9: int,Y_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Y_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_9),X_9))
=> hBOOL(P_1) ) ) ).
tff(fact_335_order__less__imp__triv,axiom,
! [P_1: bool,X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_9),X_9))
=> hBOOL(P_1) ) ) ).
tff(fact_336_order__less__imp__triv,axiom,
! [P_1: bool,X_9: nat,Y_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Y_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_9),X_9))
=> hBOOL(P_1) ) ) ).
tff(fact_337_order__less__asym_H,axiom,
! [A_40: int,B_25: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_40),B_25))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_25),A_40)) ) ).
tff(fact_338_order__less__asym_H,axiom,
! [Aa: fun_nat_bool,Ba: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa)) ) ).
tff(fact_339_order__less__asym_H,axiom,
! [A_40: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_40),B_25))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_25),A_40)) ) ).
tff(fact_340_xt1_I9_J,axiom,
! [B_24: int,A_39: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_24),A_39))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_39),B_24)) ) ).
tff(fact_341_xt1_I9_J,axiom,
! [Ba: fun_nat_bool,Aa: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba)) ) ).
tff(fact_342_xt1_I9_J,axiom,
! [B_24: nat,A_39: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_24),A_39))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_39),B_24)) ) ).
tff(fact_343_ord__eq__less__trans,axiom,
! [C_22: int,A_38: int,B_23: int] :
( ( A_38 = B_23 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_23),C_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_38),C_22)) ) ) ).
tff(fact_344_ord__eq__less__trans,axiom,
! [C_1: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
( ( Aa = Ba )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),C_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),C_1)) ) ) ).
tff(fact_345_ord__eq__less__trans,axiom,
! [C_22: nat,A_38: nat,B_23: nat] :
( ( A_38 = B_23 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_23),C_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_38),C_22)) ) ) ).
tff(fact_346_xt1_I1_J,axiom,
! [C_21: int,A_37: int,B_22: int] :
( ( A_37 = B_22 )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_21),B_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_21),A_37)) ) ) ).
tff(fact_347_xt1_I1_J,axiom,
! [C_1: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
( ( Aa = Ba )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C_1),Ba))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C_1),Aa)) ) ) ).
tff(fact_348_xt1_I1_J,axiom,
! [C_21: nat,A_37: nat,B_22: nat] :
( ( A_37 = B_22 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_21),B_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_21),A_37)) ) ) ).
tff(fact_349_ord__less__eq__trans,axiom,
! [C_20: int,A_36: int,B_21: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),B_21))
=> ( ( B_21 = C_20 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),C_20)) ) ) ).
tff(fact_350_ord__less__eq__trans,axiom,
! [C_1: fun_nat_bool,Aa: fun_nat_bool,Ba: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),Ba))
=> ( ( Ba = C_1 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),C_1)) ) ) ).
tff(fact_351_ord__less__eq__trans,axiom,
! [C_20: nat,A_36: nat,B_21: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),B_21))
=> ( ( B_21 = C_20 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),C_20)) ) ) ).
tff(fact_352_xt1_I2_J,axiom,
! [C_19: int,B_20: int,A_35: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_20),A_35))
=> ( ( B_20 = C_19 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_19),A_35)) ) ) ).
tff(fact_353_xt1_I2_J,axiom,
! [C_1: fun_nat_bool,Ba: fun_nat_bool,Aa: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ba),Aa))
=> ( ( Ba = C_1 )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C_1),Aa)) ) ) ).
tff(fact_354_xt1_I2_J,axiom,
! [C_19: nat,B_20: nat,A_35: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_20),A_35))
=> ( ( B_20 = C_19 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_19),A_35)) ) ) ).
tff(fact_355_order__less__trans,axiom,
! [Z_5: int,X_34: int,Y_13: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_34),Y_13))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_13),Z_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_34),Z_5)) ) ) ).
tff(fact_356_order__less__trans,axiom,
! [Z_3: fun_nat_bool,X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_9),Z_3))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Z_3)) ) ) ).
tff(fact_357_order__less__trans,axiom,
! [Z_5: nat,X_34: nat,Y_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_34),Y_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_13),Z_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_34),Z_5)) ) ) ).
tff(fact_358_xt1_I10_J,axiom,
! [Z_4: int,Y_12: int,X_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_33))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_4),Y_12))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_4),X_33)) ) ) ).
tff(fact_359_xt1_I10_J,axiom,
! [Z_3: fun_nat_bool,Y_9: fun_nat_bool,X_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_9),X_9))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_3),Y_9))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_3),X_9)) ) ) ).
tff(fact_360_xt1_I10_J,axiom,
! [Z_4: nat,Y_12: nat,X_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_12),X_33))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_4),Y_12))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_4),X_33)) ) ) ).
tff(fact_361_order__less__asym,axiom,
! [X_32: int,Y_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),Y_11))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_11),X_32)) ) ).
tff(fact_362_order__less__asym,axiom,
! [X_9: fun_nat_bool,Y_9: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_9),Y_9))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Y_9),X_9)) ) ).
tff(fact_363_order__less__asym,axiom,
! [X_32: nat,Y_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_32),Y_11))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_11),X_32)) ) ).
tff(fact_364_linorder__cases,axiom,
! [X_31: int,Y_10: int] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_10))
=> ( ( X_31 != Y_10 )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_10),X_31)) ) ) ).
tff(fact_365_linorder__cases,axiom,
! [X_31: nat,Y_10: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_31),Y_10))
=> ( ( X_31 != Y_10 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_10),X_31)) ) ) ).
tff(fact_366_list__update__append1,axiom,
! [Ys_11: list_A1748581779le_alt,X_30: arrow_931160158le_alt,I_22: nat,Xs_62: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_22),size_s1492124157le_alt(Xs_62)))
=> ( hAPP_A1866301020le_alt(list_u518279030le_alt(append634883619le_alt(Xs_62,Ys_11),I_22),X_30) = append634883619le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_62,I_22),X_30),Ys_11) ) ) ).
tff(fact_367_nth__list__update,axiom,
! [X_29: arrow_931160158le_alt,J_5: nat,I_21: nat,Xs_61: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_21),size_s1492124157le_alt(Xs_61)))
=> ( ( ( I_21 = J_5 )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_61,I_21),X_29)),J_5) = X_29 ) )
& ( ( I_21 != J_5 )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_61,I_21),X_29)),J_5) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_61),J_5) ) ) ) ) ).
tff(fact_368_nth__list__update,axiom,
! [X_29: nat,J_5: nat,I_21: nat,Xs_61: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_21),size_size_list_nat(Xs_61)))
=> ( ( ( I_21 = J_5 )
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(list_update_nat(Xs_61,I_21),X_29)),J_5) = X_29 ) )
& ( ( I_21 != J_5 )
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(list_update_nat(Xs_61,I_21),X_29)),J_5) = hAPP_nat_nat(nth_nat(Xs_61),J_5) ) ) ) ) ).
tff(fact_369_list__update__same__conv,axiom,
! [X_9: arrow_931160158le_alt,I: nat,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),size_s1492124157le_alt(Xs_22)))
=> ( ( hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_22,I),X_9) = Xs_22 )
<=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),I) = X_9 ) ) ) ).
tff(fact_370_list__update__same__conv,axiom,
! [X_9: nat,I: nat,Xs_22: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),size_size_list_nat(Xs_22)))
=> ( ( hAPP_nat_list_nat(list_update_nat(Xs_22,I),X_9) = Xs_22 )
<=> ( hAPP_nat_nat(nth_nat(Xs_22),I) = X_9 ) ) ) ).
tff(fact_371_nth__list__update__eq,axiom,
! [X_28: arrow_931160158le_alt,I_20: nat,Xs_60: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_20),size_s1492124157le_alt(Xs_60)))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_60,I_20),X_28)),I_20) = X_28 ) ) ).
tff(fact_372_nth__list__update__eq,axiom,
! [X_28: nat,I_20: nat,Xs_60: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_20),size_size_list_nat(Xs_60)))
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(list_update_nat(Xs_60,I_20),X_28)),I_20) = X_28 ) ) ).
tff(fact_373_last__drop,axiom,
! [N_38: nat,Xs_59: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_38),size_s1492124157le_alt(Xs_59)))
=> ( last_A1673116455le_alt(drop_A1802510926le_alt(N_38,Xs_59)) = last_A1673116455le_alt(Xs_59) ) ) ).
tff(fact_374_not__less__bot,axiom,
! [Aa: bool] : ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_bool,Aa),bot_bot_bool)) ).
tff(fact_375_not__less__bot,axiom,
! [A_34: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_34),bot_bot_nat)) ).
tff(fact_376_not__less__bot,axiom,
! [Aa: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Aa),bot_bot_fun_nat_bool)) ).
tff(fact_377_bot__less,axiom,
! [Aa: bool] :
( ~ ( hBOOL(Aa)
<=> hBOOL(bot_bot_bool) )
<=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_bool,bot_bot_bool),Aa)) ) ).
tff(fact_378_bot__less,axiom,
! [Aa: nat] :
( ( Aa != bot_bot_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,bot_bot_nat),Aa)) ) ).
tff(fact_379_bot__less,axiom,
! [Aa: fun_nat_bool] :
( ( Aa != bot_bot_fun_nat_bool )
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,bot_bot_fun_nat_bool),Aa)) ) ).
tff(fact_380_drop__0,axiom,
! [Xs_58: list_A1748581779le_alt] : drop_A1802510926le_alt(zero_zero_nat,Xs_58) = Xs_58 ).
tff(fact_381_drop__Nil,axiom,
! [N_37: nat] : drop_nat(N_37,nil_nat) = nil_nat ).
tff(fact_382_drop__Nil,axiom,
! [N_37: nat] : drop_A1802510926le_alt(N_37,nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_383_less__max__iff__disj,axiom,
! [Z_3: int,X_9: int,Y_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_9),Y_9)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),X_9))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_3),Y_9)) ) ) ).
tff(fact_384_less__max__iff__disj,axiom,
! [Z_3: nat,X_9: nat,Y_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_9),Y_9)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),X_9))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_3),Y_9)) ) ) ).
tff(fact_385_max__less__iff__conj,axiom,
! [X_9: int,Y_9: int,Z_3: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_9),Y_9)),Z_3))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_9),Z_3))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_9),Z_3)) ) ) ).
tff(fact_386_max__less__iff__conj,axiom,
! [X_9: nat,Y_9: nat,Z_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_9),Y_9)),Z_3))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),Z_3))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_9),Z_3)) ) ) ).
tff(fact_387_min__max_Oless__supI1,axiom,
! [B_19: int,X_27: int,A_33: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_27),A_33))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_27),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_33),B_19))) ) ).
tff(fact_388_min__max_Oless__supI1,axiom,
! [B_19: nat,X_27: nat,A_33: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_27),A_33))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_33),B_19))) ) ).
tff(fact_389_min__max_Oless__supI2,axiom,
! [A_32: int,X_26: int,B_18: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_26),B_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_26),hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,A_32),B_18))) ) ).
tff(fact_390_min__max_Oless__supI2,axiom,
! [A_32: nat,X_26: nat,B_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_26),B_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_26),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_32),B_18))) ) ).
tff(fact_391_distinct__drop,axiom,
! [I_19: nat,Xs_57: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_57))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,drop_A1802510926le_alt(I_19,Xs_57))) ) ).
tff(fact_392_sorted__drop,axiom,
! [N_36: nat,Xs_56: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_56))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,drop_nat(N_36,Xs_56))) ) ).
tff(fact_393_list__update__nonempty,axiom,
! [Xs_22: list_nat,K_3: nat,X_9: nat] :
( ( hAPP_nat_list_nat(list_update_nat(Xs_22,K_3),X_9) = nil_nat )
<=> ( Xs_22 = nil_nat ) ) ).
tff(fact_394_list__update__nonempty,axiom,
! [Xs_22: list_A1748581779le_alt,K_3: nat,X_9: arrow_931160158le_alt] :
( ( hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_22,K_3),X_9) = nil_Ar1741995278le_alt )
<=> ( Xs_22 = nil_Ar1741995278le_alt ) ) ).
tff(fact_395_list__update_Osimps_I1_J,axiom,
! [I_18: nat,V_3: nat] : hAPP_nat_list_nat(list_update_nat(nil_nat,I_18),V_3) = nil_nat ).
tff(fact_396_list__update_Osimps_I1_J,axiom,
! [I_18: nat,V_3: arrow_931160158le_alt] : hAPP_A1866301020le_alt(list_u518279030le_alt(nil_Ar1741995278le_alt,I_18),V_3) = nil_Ar1741995278le_alt ).
tff(fact_397_list__update__code_I1_J,axiom,
! [I_17: nat,Y_8: nat] : hAPP_nat_list_nat(list_update_nat(nil_nat,I_17),Y_8) = nil_nat ).
tff(fact_398_list__update__code_I1_J,axiom,
! [I_17: nat,Y_8: arrow_931160158le_alt] : hAPP_A1866301020le_alt(list_u518279030le_alt(nil_Ar1741995278le_alt,I_17),Y_8) = nil_Ar1741995278le_alt ).
tff(fact_399_length__list__update,axiom,
! [Xs_55: list_A1748581779le_alt,I_16: nat,X_25: arrow_931160158le_alt] : size_s1492124157le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_55,I_16),X_25)) = size_s1492124157le_alt(Xs_55) ).
tff(fact_400_list__update__id,axiom,
! [Xs_54: list_A1748581779le_alt,I_15: nat] : hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_54,I_15),hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_54),I_15)) = Xs_54 ).
tff(fact_401_list__update__id,axiom,
! [Xs_54: list_nat,I_15: nat] : hAPP_nat_list_nat(list_update_nat(Xs_54,I_15),hAPP_nat_nat(nth_nat(Xs_54),I_15)) = Xs_54 ).
tff(fact_402_nth__list__update__neq,axiom,
! [Xs_53: list_A1748581779le_alt,X_24: arrow_931160158le_alt,I_14: nat,J_4: nat] :
( ( I_14 != J_4 )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_53,I_14),X_24)),J_4) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_53),J_4) ) ) ).
tff(fact_403_nth__list__update__neq,axiom,
! [Xs_53: list_nat,X_24: nat,I_14: nat,J_4: nat] :
( ( I_14 != J_4 )
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(list_update_nat(Xs_53,I_14),X_24)),J_4) = hAPP_nat_nat(nth_nat(Xs_53),J_4) ) ) ).
tff(fact_404_drop__butlast,axiom,
! [N_35: nat,Xs_52: list_A1748581779le_alt] : drop_A1802510926le_alt(N_35,butlas730749018le_alt(Xs_52)) = butlas730749018le_alt(drop_A1802510926le_alt(N_35,Xs_52)) ).
tff(fact_405_butlast__drop,axiom,
! [N_34: nat,Xs_51: list_A1748581779le_alt] : butlas730749018le_alt(drop_A1802510926le_alt(N_34,Xs_51)) = drop_A1802510926le_alt(N_34,butlas730749018le_alt(Xs_51)) ).
tff(fact_406_tl__drop,axiom,
! [N_33: nat,Xs_50: list_A1748581779le_alt] : tl_Arr921252325le_alt(drop_A1802510926le_alt(N_33,Xs_50)) = drop_A1802510926le_alt(N_33,tl_Arr921252325le_alt(Xs_50)) ).
tff(fact_407_drop__tl,axiom,
! [N_32: nat,Xs_49: list_A1748581779le_alt] : drop_A1802510926le_alt(N_32,tl_Arr921252325le_alt(Xs_49)) = tl_Arr921252325le_alt(drop_A1802510926le_alt(N_32,Xs_49)) ).
tff(fact_408_hd__drop__conv__nth,axiom,
! [N_31: nat,Xs_48: list_nat] :
( ( Xs_48 != nil_nat )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_31),size_size_list_nat(Xs_48)))
=> ( hd_nat(drop_nat(N_31,Xs_48)) = hAPP_nat_nat(nth_nat(Xs_48),N_31) ) ) ) ).
tff(fact_409_hd__drop__conv__nth,axiom,
! [N_31: nat,Xs_48: list_A1748581779le_alt] :
( ( Xs_48 != nil_Ar1741995278le_alt )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_31),size_s1492124157le_alt(Xs_48)))
=> ( hd_Arr274000865le_alt(drop_A1802510926le_alt(N_31,Xs_48)) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_48),N_31) ) ) ) ).
tff(fact_410_nth__via__drop,axiom,
! [N_30: nat,Xs_47: list_nat,Y_7: nat,Ys_10: list_nat] :
( ( drop_nat(N_30,Xs_47) = cons_nat(Y_7,Ys_10) )
=> ( hAPP_nat_nat(nth_nat(Xs_47),N_30) = Y_7 ) ) ).
tff(fact_411_nth__via__drop,axiom,
! [N_30: nat,Xs_47: list_A1748581779le_alt,Y_7: arrow_931160158le_alt,Ys_10: list_A1748581779le_alt] :
( ( drop_A1802510926le_alt(N_30,Xs_47) = cons_A684544190le_alt(Y_7,Ys_10) )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_47),N_30) = Y_7 ) ) ).
tff(fact_412_list__eq__iff__nth__eq,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( ( Xs_22 = Ys_4 )
<=> ( ( size_s1492124157le_alt(Xs_22) = size_s1492124157le_alt(Ys_4) )
& ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_s1492124157le_alt(Xs_22)))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),I_5) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Ys_4),I_5) ) ) ) ) ).
tff(fact_413_list__eq__iff__nth__eq,axiom,
! [Xs_22: list_nat,Ys_4: list_nat] :
( ( Xs_22 = Ys_4 )
<=> ( ( size_size_list_nat(Xs_22) = size_size_list_nat(Ys_4) )
& ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_size_list_nat(Xs_22)))
=> ( hAPP_nat_nat(nth_nat(Xs_22),I_5) = hAPP_nat_nat(nth_nat(Ys_4),I_5) ) ) ) ) ).
tff(fact_414_list__update__code_I2_J,axiom,
! [X_23: arrow_931160158le_alt,Xs_46: list_A1748581779le_alt,Y_6: arrow_931160158le_alt] : hAPP_A1866301020le_alt(list_u518279030le_alt(cons_A684544190le_alt(X_23,Xs_46),zero_zero_nat),Y_6) = cons_A684544190le_alt(Y_6,Xs_46) ).
tff(fact_415_nth__replicate,axiom,
! [X_22: nat,I_13: nat,N_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_13),N_29))
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(replicate_nat(N_29),X_22)),I_13) = X_22 ) ) ).
tff(fact_416_nth__replicate,axiom,
! [X_22: arrow_931160158le_alt,I_13: nat,N_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_13),N_29))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_29),X_22)),I_13) = X_22 ) ) ).
tff(fact_417_length__greater__0__conv,axiom,
! [Xs_22: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs_22)))
<=> ( Xs_22 != nil_nat ) ) ).
tff(fact_418_length__greater__0__conv,axiom,
! [Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),size_s1492124157le_alt(Xs_22)))
<=> ( Xs_22 != nil_Ar1741995278le_alt ) ) ).
tff(fact_419_distinct__conv__nth,axiom,
! [Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_22))
<=> ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_size_list_nat(Xs_22)))
=> ! [J_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_3),size_size_list_nat(Xs_22)))
=> ( ( I_5 != J_3 )
=> ( hAPP_nat_nat(nth_nat(Xs_22),I_5) != hAPP_nat_nat(nth_nat(Xs_22),J_3) ) ) ) ) ) ).
tff(fact_420_distinct__conv__nth,axiom,
! [Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22))
<=> ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_s1492124157le_alt(Xs_22)))
=> ! [J_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_3),size_s1492124157le_alt(Xs_22)))
=> ( ( I_5 != J_3 )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),I_5) != hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),J_3) ) ) ) ) ) ).
tff(fact_421_nth__eq__iff__index__eq,axiom,
! [J_1: nat,I: nat,Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),size_size_list_nat(Xs_22)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),size_size_list_nat(Xs_22)))
=> ( ( hAPP_nat_nat(nth_nat(Xs_22),I) = hAPP_nat_nat(nth_nat(Xs_22),J_1) )
<=> ( I = J_1 ) ) ) ) ) ).
tff(fact_422_nth__eq__iff__index__eq,axiom,
! [J_1: nat,I: nat,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),size_s1492124157le_alt(Xs_22)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),size_s1492124157le_alt(Xs_22)))
=> ( ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),I) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),J_1) )
<=> ( I = J_1 ) ) ) ) ) ).
tff(fact_423_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
tff(fact_424_max__0R,axiom,
! [N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_5),zero_zero_nat) = N_5 ).
tff(fact_425_nth__equalityI,axiom,
! [Xs_45: list_A1748581779le_alt,Ys_9: list_A1748581779le_alt] :
( ( size_s1492124157le_alt(Xs_45) = size_s1492124157le_alt(Ys_9) )
=> ( ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_s1492124157le_alt(Xs_45)))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_45),I_5) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Ys_9),I_5) ) )
=> ( Xs_45 = Ys_9 ) ) ) ).
tff(fact_426_nth__equalityI,axiom,
! [Xs_45: list_nat,Ys_9: list_nat] :
( ( size_size_list_nat(Xs_45) = size_size_list_nat(Ys_9) )
=> ( ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_size_list_nat(Xs_45)))
=> ( hAPP_nat_nat(nth_nat(Xs_45),I_5) = hAPP_nat_nat(nth_nat(Ys_9),I_5) ) )
=> ( Xs_45 = Ys_9 ) ) ) ).
tff(fact_427_Skolem__list__nth,axiom,
! [P_1: fun_na81669751t_bool,K_3: nat] :
( ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),K_3))
=> ? [X1: arrow_931160158le_alt] : hBOOL(hAPP_A674791462t_bool(hAPP_n127720521t_bool(P_1,I_5),X1)) )
<=> ? [Xs_44: list_A1748581779le_alt] :
( ( size_s1492124157le_alt(Xs_44) = K_3 )
& ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),K_3))
=> hBOOL(hAPP_A674791462t_bool(hAPP_n127720521t_bool(P_1,I_5),hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_44),I_5))) ) ) ) ).
tff(fact_428_Skolem__list__nth,axiom,
! [P_1: fun_nat_fun_nat_bool,K_3: nat] :
( ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),K_3))
=> ? [X1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,I_5),X1)) )
<=> ? [Xs_44: list_nat] :
( ( size_size_list_nat(Xs_44) = K_3 )
& ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),K_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,I_5),hAPP_nat_nat(nth_nat(Xs_44),I_5))) ) ) ) ).
tff(fact_429_dropWhile__nth,axiom,
! [J_1: nat,P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),size_size_list_nat(dropWhile_nat(P_1,Xs_22))))
=> ( hAPP_nat_nat(nth_nat(dropWhile_nat(P_1,Xs_22)),J_1) = hAPP_nat_nat(nth_nat(Xs_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),size_size_list_nat(takeWhile_nat(P_1,Xs_22)))) ) ) ).
tff(fact_430_dropWhile__nth,axiom,
! [J_1: nat,P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),size_s1492124157le_alt(dropWh1772583087le_alt(P_1,Xs_22))))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(dropWh1772583087le_alt(P_1,Xs_22)),J_1) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),size_s1492124157le_alt(takeWh4609031le_alt(P_1,Xs_22)))) ) ) ).
tff(fact_431_nth__drop_H,axiom,
! [I_12: nat,Xs_43: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_12),size_size_list_nat(Xs_43)))
=> ( cons_nat(hAPP_nat_nat(nth_nat(Xs_43),I_12),drop_nat(hAPP_nat_nat(suc,I_12),Xs_43)) = drop_nat(I_12,Xs_43) ) ) ).
tff(fact_432_nth__drop_H,axiom,
! [I_12: nat,Xs_43: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_12),size_s1492124157le_alt(Xs_43)))
=> ( cons_A684544190le_alt(hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_43),I_12),drop_A1802510926le_alt(hAPP_nat_nat(suc,I_12),Xs_43)) = drop_A1802510926le_alt(I_12,Xs_43) ) ) ).
tff(fact_433_list__ex__length,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),Xs_22))
<=> ? [N_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_7),size_s1492124157le_alt(Xs_22)))
& hBOOL(hAPP_A674791462t_bool(P_1,hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),N_7))) ) ) ).
tff(fact_434_list__ex__length,axiom,
! [P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_ex_nat(P_1),Xs_22))
<=> ? [N_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_7),size_size_list_nat(Xs_22)))
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs_22),N_7))) ) ) ).
tff(fact_435_lessI,axiom,
! [N_5: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),hAPP_nat_nat(suc,N_5))) ).
tff(fact_436_Suc__mono,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_5))) ) ).
tff(fact_437_zero__less__Suc,axiom,
! [N_5: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N_5))) ).
tff(fact_438_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
tff(fact_439_nat__add__right__cancel,axiom,
! [M_1: nat,K_3: nat,N_12: 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_12),K_3) )
<=> ( M_1 = N_12 ) ) ).
tff(fact_440_nat__add__left__cancel,axiom,
! [K_3: nat,M_1: nat,N_12: 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_12) )
<=> ( M_1 = N_12 ) ) ).
tff(fact_441_nat__add__assoc,axiom,
! [M: nat,N_5: nat,K_4: 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_5)),K_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_5),K_4)) ).
tff(fact_442_nat_Oinject,axiom,
! [Nat_4: nat,Nat_3: nat] :
( ( hAPP_nat_nat(suc,Nat_4) = hAPP_nat_nat(suc,Nat_3) )
<=> ( Nat_4 = Nat_3 ) ) ).
tff(fact_443_add__Suc__shift,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N_5)) ).
tff(fact_444_add__Suc,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N_5) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_5)) ).
tff(fact_445_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_1: 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_1)) = 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_1)) ).
tff(fact_446_add__Suc__right,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N_5)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_5)) ).
tff(fact_447_nat__add__commute,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_5),M) ).
tff(fact_448_less__iff__Suc__add,axiom,
! [M_1: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12))
<=> ? [K_6: nat] : N_12 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_6)) ) ).
tff(fact_449_less__add__Suc2,axiom,
! [I_3: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I_3)))) ).
tff(fact_450_less__add__Suc1,axiom,
! [I_3: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_3),M)))) ).
tff(fact_451_Suc__n__not__n,axiom,
! [N_5: nat] : hAPP_nat_nat(suc,N_5) != N_5 ).
tff(fact_452_n__not__Suc__n,axiom,
! [N_5: nat] : N_5 != hAPP_nat_nat(suc,N_5) ).
tff(fact_453_add__is__1,axiom,
! [M_1: nat,N_12: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_12) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_12 = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( N_12 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_454_one__is__add,axiom,
! [M_1: nat,N_12: nat] :
( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_12) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_12 = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( N_12 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_455_add__right__imp__eq,axiom,
! [B_17: nat,A_31: nat,C_18: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_17),A_31) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_18),A_31) )
=> ( B_17 = C_18 ) ) ).
tff(fact_456_add__right__imp__eq,axiom,
! [B_17: int,A_31: int,C_18: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_17),A_31) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_18),A_31) )
=> ( B_17 = C_18 ) ) ).
tff(fact_457_add__imp__eq,axiom,
! [A_30: nat,B_16: nat,C_17: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),B_16) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),C_17) )
=> ( B_16 = C_17 ) ) ).
tff(fact_458_add__imp__eq,axiom,
! [A_30: int,B_16: int,C_17: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),B_16) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),C_17) )
=> ( B_16 = C_17 ) ) ).
tff(fact_459_add__left__imp__eq,axiom,
! [A_29: nat,B_15: nat,C_16: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_29),B_15) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_29),C_16) )
=> ( B_15 = C_16 ) ) ).
tff(fact_460_add__left__imp__eq,axiom,
! [A_29: int,B_15: int,C_16: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_29),B_15) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_29),C_16) )
=> ( B_15 = C_16 ) ) ).
tff(fact_461_add__right__cancel,axiom,
! [Ba: nat,Aa: nat,C_1: 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_1),Aa) )
<=> ( Ba = C_1 ) ) ).
tff(fact_462_add__right__cancel,axiom,
! [Ba: int,Aa: int,C_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),Aa) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),Aa) )
<=> ( Ba = C_1 ) ) ).
tff(fact_463_add__left__cancel,axiom,
! [Aa: nat,Ba: nat,C_1: 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_1) )
<=> ( Ba = C_1 ) ) ).
tff(fact_464_add__left__cancel,axiom,
! [Aa: int,Ba: int,C_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),C_1) )
<=> ( Ba = C_1 ) ) ).
tff(fact_465_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_28: nat,B_14: nat,C_15: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_28),B_14)),C_15) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_28),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_14),C_15)) ).
tff(fact_466_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_28: int,B_14: int,C_15: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_28),B_14)),C_15) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_28),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_14),C_15)) ).
tff(fact_467_add_Ocomm__neutral,axiom,
! [A_27: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_27),zero_zero_nat) = A_27 ).
tff(fact_468_add_Ocomm__neutral,axiom,
! [A_27: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_27),zero_zero_int) = A_27 ).
tff(fact_469_add__0__right,axiom,
! [A_26: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_26),zero_zero_nat) = A_26 ).
tff(fact_470_add__0__right,axiom,
! [A_26: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_26),zero_zero_int) = A_26 ).
tff(fact_471_double__zero__sym,axiom,
! [Aa: int] :
( ( zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa) )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_472_add__0,axiom,
! [A_25: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_25) = A_25 ).
tff(fact_473_add__0,axiom,
! [A_25: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_25) = A_25 ).
tff(fact_474_add__0__left,axiom,
! [A_24: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_24) = A_24 ).
tff(fact_475_add__0__left,axiom,
! [A_24: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_24) = A_24 ).
tff(fact_476_add__less__imp__less__left,axiom,
! [C_14: nat,A_23: nat,B_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_14),A_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_14),B_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_23),B_13)) ) ).
tff(fact_477_add__less__imp__less__left,axiom,
! [C_14: int,A_23: int,B_13: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_14),A_23)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_14),B_13)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_23),B_13)) ) ).
tff(fact_478_add__less__imp__less__right,axiom,
! [A_22: nat,C_13: nat,B_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_22),C_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_12),C_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_22),B_12)) ) ).
tff(fact_479_add__less__imp__less__right,axiom,
! [A_22: int,C_13: int,B_12: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_22),C_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_12),C_13)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_22),B_12)) ) ).
tff(fact_480_add__strict__mono,axiom,
! [C_12: nat,D_5: nat,A_21: nat,B_11: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_21),B_11))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_12),D_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_21),C_12)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_11),D_5))) ) ) ).
tff(fact_481_add__strict__mono,axiom,
! [C_12: int,D_5: int,A_21: int,B_11: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_21),B_11))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_12),D_5))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_21),C_12)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_11),D_5))) ) ) ).
tff(fact_482_add__strict__left__mono,axiom,
! [C_11: nat,A_20: nat,B_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_20),B_10))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_11),A_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_11),B_10))) ) ).
tff(fact_483_add__strict__left__mono,axiom,
! [C_11: int,A_20: int,B_10: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_20),B_10))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_11),A_20)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_11),B_10))) ) ).
tff(fact_484_add__strict__right__mono,axiom,
! [C_10: nat,A_19: nat,B_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_19),B_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_19),C_10)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_9),C_10))) ) ).
tff(fact_485_add__strict__right__mono,axiom,
! [C_10: int,A_19: int,B_9: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_19),B_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_19),C_10)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_9),C_10))) ) ).
tff(fact_486_add__less__cancel__left,axiom,
! [C_1: 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_1),Aa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_1),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_487_add__less__cancel__left,axiom,
! [C_1: int,Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),Aa)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),Ba)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
tff(fact_488_add__less__cancel__right,axiom,
! [Aa: nat,C_1: 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_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),C_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_489_add__less__cancel__right,axiom,
! [Aa: int,C_1: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),C_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba)) ) ).
tff(fact_490_Zero__not__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_491_nat_Osimps_I2_J,axiom,
! [Nat_2: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat_2) ).
tff(fact_492_Suc__not__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_493_nat_Osimps_I3_J,axiom,
! [Nat_1: nat] : hAPP_nat_nat(suc,Nat_1) != zero_zero_nat ).
tff(fact_494_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_495_Suc__neq__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_496_not__less__eq,axiom,
! [M_1: nat,N_12: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),hAPP_nat_nat(suc,M_1))) ) ).
tff(fact_497_less__Suc__eq,axiom,
! [M_1: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_12)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12))
| ( M_1 = N_12 ) ) ) ).
tff(fact_498_Suc__less__eq,axiom,
! [M_1: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_12)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12)) ) ).
tff(fact_499_not__less__less__Suc__eq,axiom,
! [N_12: nat,M_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),hAPP_nat_nat(suc,M_1)))
<=> ( N_12 = M_1 ) ) ) ).
tff(fact_500_less__antisym,axiom,
! [N_5: nat,M: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_5),hAPP_nat_nat(suc,M)))
=> ( M = N_5 ) ) ) ).
tff(fact_501_less__SucI,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_5))) ) ).
tff(fact_502_Suc__lessI,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> ( ( hAPP_nat_nat(suc,M) != N_5 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_5)) ) ) ).
tff(fact_503_less__trans__Suc,axiom,
! [K_4: nat,I_3: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_3)),K_4)) ) ) ).
tff(fact_504_less__SucE,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N_5)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> ( M = N_5 ) ) ) ).
tff(fact_505_Suc__lessD,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5)) ) ).
tff(fact_506_Suc__less__SucD,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_5)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5)) ) ).
tff(fact_507_not__psubset__empty,axiom,
! [A: fun_nat_bool] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),bot_bot_fun_nat_bool)) ).
tff(fact_508_plus__nat_Oadd__0,axiom,
! [N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N_5) = N_5 ).
tff(fact_509_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_510_add__is__0,axiom,
! [M_1: nat,N_12: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_12) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
& ( N_12 = zero_zero_nat ) ) ) ).
tff(fact_511_add__eq__self__zero,axiom,
! [M: nat,N_5: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_5) = M )
=> ( N_5 = zero_zero_nat ) ) ).
tff(fact_512_not__add__less1,axiom,
! [I_3: 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_3),J)),I_3)) ).
tff(fact_513_not__add__less2,axiom,
! [J: nat,I_3: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I_3)),I_3)) ).
tff(fact_514_nat__add__left__cancel__less,axiom,
! [K_3: nat,M_1: nat,N_12: 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_12)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12)) ) ).
tff(fact_515_trans__less__add1,axiom,
! [M: nat,I_3: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
tff(fact_516_trans__less__add2,axiom,
! [M: nat,I_3: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
tff(fact_517_add__less__mono1,axiom,
! [K_4: nat,I_3: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_3),K_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_4))) ) ).
tff(fact_518_add__less__mono,axiom,
! [K_4: nat,L: nat,I_3: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_4),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_3),K_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).
tff(fact_519_less__add__eq__less,axiom,
! [M: nat,N_5: nat,K_4: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_4),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_4),N_5) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5)) ) ) ).
tff(fact_520_add__lessD1,axiom,
! [I_3: nat,J: nat,K_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_3),J)),K_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),K_4)) ) ).
tff(fact_521_max__add__distrib__left,axiom,
! [X_21: nat,Y_5: nat,Z_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_21),Y_5)),Z_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_21),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_5),Z_2)) ).
tff(fact_522_max__add__distrib__left,axiom,
! [X_21: int,Y_5: int,Z_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_21),Y_5)),Z_2) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_21),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_5),Z_2)) ).
tff(fact_523_drop__drop,axiom,
! [N_28: nat,M_4: nat,Xs_42: list_A1748581779le_alt] : drop_A1802510926le_alt(N_28,drop_A1802510926le_alt(M_4,Xs_42)) = drop_A1802510926le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_28),M_4),Xs_42) ).
tff(fact_524_max__Suc__Suc,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_5)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N_5)) ).
tff(fact_525_list_Osize_I4_J,axiom,
! [A_18: arrow_931160158le_alt,List_1: list_A1748581779le_alt] : size_s1492124157le_alt(cons_A684544190le_alt(A_18,List_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_s1492124157le_alt(List_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_526_list_Osize_I2_J,axiom,
! [Fa: fun_Ar518461033lt_nat,Aa: arrow_931160158le_alt,List: list_A1748581779le_alt] : list_s706097040le_alt(Fa,cons_A684544190le_alt(Aa,List)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_A846389249lt_nat(Fa,Aa)),list_s706097040le_alt(Fa,List))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_527_list__ex__simps_I1_J,axiom,
! [P_1: fun_Ar1091019988t_bool,X_9: arrow_931160158le_alt,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),cons_A684544190le_alt(X_9,Xs_22)))
<=> ( hBOOL(hAPP_A674791462t_bool(P_1,X_9))
| hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),Xs_22)) ) ) ).
tff(fact_528_list__ex__simps_I2_J,axiom,
! [P_1: fun_nat_bool] : ~ hBOOL(hAPP_list_nat_bool(list_ex_nat(P_1),nil_nat)) ).
tff(fact_529_list__ex__simps_I2_J,axiom,
! [P_1: fun_Ar1091019988t_bool] : ~ hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),nil_Ar1741995278le_alt)) ).
tff(fact_530_list__ex__append,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),append634883619le_alt(Xs_22,Ys_4)))
<=> ( hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),Xs_22))
| hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),Ys_4)) ) ) ).
tff(fact_531_zero__less__double__add__iff__zero__less__single__add,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Aa)) ) ).
tff(fact_532_double__add__less__zero__iff__single__add__less__zero,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).
tff(fact_533_add__pos__pos,axiom,
! [B_8: nat,A_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_17))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_8))
=> 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_17),B_8))) ) ) ).
tff(fact_534_add__pos__pos,axiom,
! [B_8: int,A_17: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_17),B_8))) ) ) ).
tff(fact_535_add__neg__neg,axiom,
! [B_7: nat,A_16: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_16),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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_16),B_7)),zero_zero_nat)) ) ) ).
tff(fact_536_add__neg__neg,axiom,
! [B_7: int,A_16: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_16),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),zero_zero_int))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_16),B_7)),zero_zero_int)) ) ) ).
tff(fact_537_list__ex__rev,axiom,
! [P_1: fun_nat_bool,Xs_22: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_ex_nat(P_1),rev_nat(Xs_22)))
<=> hBOOL(hAPP_list_nat_bool(list_ex_nat(P_1),Xs_22)) ) ).
tff(fact_538_list__ex__rev,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),rev_Ar1562208110le_alt(Xs_22)))
<=> hBOOL(hAPP_l713721587t_bool(list_e1708603724le_alt(P_1),Xs_22)) ) ).
tff(fact_539_gr0__conv__Suc,axiom,
! [N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
<=> ? [M_2: nat] : N_12 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_540_less__Suc0,axiom,
! [N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),hAPP_nat_nat(suc,zero_zero_nat)))
<=> ( N_12 = zero_zero_nat ) ) ).
tff(fact_541_less__Suc__eq__0__disj,axiom,
! [M_1: nat,N_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_12)))
<=> ( ( M_1 = zero_zero_nat )
| ? [J_3: nat] :
( ( M_1 = hAPP_nat_nat(suc,J_3) )
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_3),N_12)) ) ) ) ).
tff(fact_542_add__gr__0,axiom,
! [M_1: nat,N_12: 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_12)))
<=> ( 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_12)) ) ) ).
tff(fact_543_nth__Cons__Suc,axiom,
! [X_20: nat,Xs_41: list_nat,N_27: nat] : hAPP_nat_nat(nth_nat(cons_nat(X_20,Xs_41)),hAPP_nat_nat(suc,N_27)) = hAPP_nat_nat(nth_nat(Xs_41),N_27) ).
tff(fact_544_nth__Cons__Suc,axiom,
! [X_20: arrow_931160158le_alt,Xs_41: list_A1748581779le_alt,N_27: nat] : hAPP_n2017047337le_alt(nth_Ar581680143le_alt(cons_A684544190le_alt(X_20,Xs_41)),hAPP_nat_nat(suc,N_27)) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_41),N_27) ).
tff(fact_545_length__append,axiom,
! [Xs_40: list_A1748581779le_alt,Ys_8: list_A1748581779le_alt] : size_s1492124157le_alt(append634883619le_alt(Xs_40,Ys_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_s1492124157le_alt(Xs_40)),size_s1492124157le_alt(Ys_8)) ).
tff(fact_546_drop__Suc__Cons,axiom,
! [N_26: nat,X_19: arrow_931160158le_alt,Xs_39: list_A1748581779le_alt] : drop_A1802510926le_alt(hAPP_nat_nat(suc,N_26),cons_A684544190le_alt(X_19,Xs_39)) = drop_A1802510926le_alt(N_26,Xs_39) ).
tff(fact_547_list__update__code_I3_J,axiom,
! [X_18: arrow_931160158le_alt,Xs_38: list_A1748581779le_alt,I_11: nat,Y_4: arrow_931160158le_alt] : hAPP_A1866301020le_alt(list_u518279030le_alt(cons_A684544190le_alt(X_18,Xs_38),hAPP_nat_nat(suc,I_11)),Y_4) = cons_A684544190le_alt(X_18,hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_38,I_11),Y_4)) ).
tff(fact_548_replicate__Suc,axiom,
! [N_25: nat,X_17: arrow_931160158le_alt] : hAPP_A1866301020le_alt(replic1967339976le_alt(hAPP_nat_nat(suc,N_25)),X_17) = cons_A684544190le_alt(X_17,hAPP_A1866301020le_alt(replic1967339976le_alt(N_25),X_17)) ).
tff(fact_549_replicate__add,axiom,
! [N_24: nat,M_3: nat,X_16: arrow_931160158le_alt] : hAPP_A1866301020le_alt(replic1967339976le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_24),M_3)),X_16) = append634883619le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_24),X_16),hAPP_A1866301020le_alt(replic1967339976le_alt(M_3),X_16)) ).
tff(fact_550_drop__Suc,axiom,
! [N_23: nat,Xs_37: list_A1748581779le_alt] : drop_A1802510926le_alt(hAPP_nat_nat(suc,N_23),Xs_37) = drop_A1802510926le_alt(N_23,tl_Arr921252325le_alt(Xs_37)) ).
tff(fact_551_length__splice,axiom,
! [Xs_36: list_A1748581779le_alt,Ys_7: list_A1748581779le_alt] : size_s1492124157le_alt(splice1976699617le_alt(Xs_36,Ys_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_s1492124157le_alt(Xs_36)),size_s1492124157le_alt(Ys_7)) ).
tff(fact_552_nth__append__length__plus,axiom,
! [Xs_35: list_nat,Ys_6: list_nat,N_22: nat] : hAPP_nat_nat(nth_nat(append_nat(Xs_35,Ys_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs_35)),N_22)) = hAPP_nat_nat(nth_nat(Ys_6),N_22) ).
tff(fact_553_nth__append__length__plus,axiom,
! [Xs_35: list_A1748581779le_alt,Ys_6: list_A1748581779le_alt,N_22: nat] : hAPP_n2017047337le_alt(nth_Ar581680143le_alt(append634883619le_alt(Xs_35,Ys_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,size_s1492124157le_alt(Xs_35)),N_22)) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Ys_6),N_22) ).
tff(fact_554_less__imp__Suc__add,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> ? [K_6: nat] : N_5 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_6)) ) ).
tff(fact_555_length__Suc__conv,axiom,
! [Xs_22: list_A1748581779le_alt,N_12: nat] :
( ( size_s1492124157le_alt(Xs_22) = hAPP_nat_nat(suc,N_12) )
<=> ? [Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] :
( ( Xs_22 = cons_A684544190le_alt(Y_3,Ys_5) )
& ( size_s1492124157le_alt(Ys_5) = N_12 ) ) ) ).
tff(fact_556_Suc__length__conv,axiom,
! [N_12: nat,Xs_22: list_A1748581779le_alt] :
( ( hAPP_nat_nat(suc,N_12) = size_s1492124157le_alt(Xs_22) )
<=> ? [Y_3: arrow_931160158le_alt,Ys_5: list_A1748581779le_alt] :
( ( Xs_22 = cons_A684544190le_alt(Y_3,Ys_5) )
& ( size_s1492124157le_alt(Ys_5) = N_12 ) ) ) ).
tff(fact_557_less__imp__add__positive,axiom,
! [I_3: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J))
=> ? [K_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_6))
& ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_3),K_6) = J ) ) ) ).
tff(fact_558_psubsetD,axiom,
! [C_1: int,A: fun_int_bool,B: fun_int_bool] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A),B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,C_1),B)) ) ) ).
tff(fact_559_psubsetD,axiom,
! [C_1: 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_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B)) ) ) ).
tff(fact_560_gr0__implies__Suc,axiom,
! [N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> ? [M_2: nat] : N_5 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_561_lift__Suc__mono__less,axiom,
! [N_12: nat,N_21: nat,F: fun_nat_nat] :
( ! [N_7: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_7)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_7))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_12)),hAPP_nat_nat(F,N_21))) ) ) ).
tff(fact_562_lift__Suc__mono__less__iff,axiom,
! [N_12: nat,M_1: nat,F: fun_nat_nat] :
( ! [N_7: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_7)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_7))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_12)),hAPP_nat_nat(F,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),M_1)) ) ) ).
tff(fact_563_even__less__0__iff,axiom,
! [Aa: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ).
tff(fact_564_pos__add__strict,axiom,
! [B_6: int,C_9: int,A_15: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_6),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_6),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_15),C_9))) ) ) ).
tff(fact_565_pos__add__strict,axiom,
! [B_6: nat,C_9: nat,A_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_6),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_15),C_9))) ) ) ).
tff(fact_566_double__eq__0__iff,axiom,
! [Aa: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aa),Aa) = zero_zero_int )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_567_lessE,axiom,
! [I_3: nat,K_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),K_4))
=> ( ( K_4 != hAPP_nat_nat(suc,I_3) )
=> ~ ! [J_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J_3))
=> ( K_4 != hAPP_nat_nat(suc,J_3) ) ) ) ) ).
tff(fact_568_Suc__lessE,axiom,
! [I_3: nat,K_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I_3)),K_4))
=> ~ ! [J_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_3),J_3))
=> ( K_4 != hAPP_nat_nat(suc,J_3) ) ) ) ).
tff(fact_569_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] : Y != hAPP_nat_nat(suc,Nat) ) ).
tff(fact_570_not0__implies__Suc,axiom,
! [N_5: nat] :
( ( N_5 != zero_zero_nat )
=> ? [M_2: nat] : N_5 = hAPP_nat_nat(suc,M_2) ) ).
tff(fact_571_nat__induct,axiom,
! [N_12: nat,P_1: fun_nat_bool] :
( hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( ! [N_7: nat] :
( hBOOL(hAPP_nat_bool(P_1,N_7))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_7))) )
=> hBOOL(hAPP_nat_bool(P_1,N_12)) ) ) ).
tff(fact_572_zero__induct,axiom,
! [P_1: fun_nat_bool,K_3: nat] :
( hBOOL(hAPP_nat_bool(P_1,K_3))
=> ( ! [N_7: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_7)))
=> hBOOL(hAPP_nat_bool(P_1,N_7)) )
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ).
tff(fact_573_take__hd__drop,axiom,
! [N_20: nat,Xs_34: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_20),size_s1492124157le_alt(Xs_34)))
=> ( append634883619le_alt(take_A1069848822le_alt(N_20,Xs_34),cons_A684544190le_alt(hd_Arr274000865le_alt(drop_A1802510926le_alt(N_20,Xs_34)),nil_Ar1741995278le_alt)) = take_A1069848822le_alt(hAPP_nat_nat(suc,N_20),Xs_34) ) ) ).
tff(fact_574_take__tl,axiom,
! [N_19: nat,Xs_33: list_A1748581779le_alt] : take_A1069848822le_alt(N_19,tl_Arr921252325le_alt(Xs_33)) = tl_Arr921252325le_alt(take_A1069848822le_alt(hAPP_nat_nat(suc,N_19),Xs_33)) ).
tff(fact_575_take__Nil,axiom,
! [N_18: nat] : take_A1069848822le_alt(N_18,nil_Ar1741995278le_alt) = nil_Ar1741995278le_alt ).
tff(fact_576_distinct__take,axiom,
! [I_10: nat,Xs_32: list_A1748581779le_alt] :
( hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,Xs_32))
=> hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,take_A1069848822le_alt(I_10,Xs_32))) ) ).
tff(fact_577_takeWhile__eq__take,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt] : takeWh4609031le_alt(P_1,Xs_22) = take_A1069848822le_alt(size_s1492124157le_alt(takeWh4609031le_alt(P_1,Xs_22)),Xs_22) ).
tff(fact_578_append__take__drop__id,axiom,
! [N_17: nat,Xs_31: list_A1748581779le_alt] : append634883619le_alt(take_A1069848822le_alt(N_17,Xs_31),drop_A1802510926le_alt(N_17,Xs_31)) = Xs_31 ).
tff(fact_579_take__Suc__Cons,axiom,
! [N_16: nat,X_15: arrow_931160158le_alt,Xs_30: list_A1748581779le_alt] : take_A1069848822le_alt(hAPP_nat_nat(suc,N_16),cons_A684544190le_alt(X_15,Xs_30)) = cons_A684544190le_alt(X_15,take_A1069848822le_alt(N_16,Xs_30)) ).
tff(fact_580_take__eq__Nil,axiom,
! [N_12: nat,Xs_22: list_A1748581779le_alt] :
( ( take_A1069848822le_alt(N_12,Xs_22) = nil_Ar1741995278le_alt )
<=> ( ( N_12 = zero_zero_nat )
| ( Xs_22 = nil_Ar1741995278le_alt ) ) ) ).
tff(fact_581_take__0,axiom,
! [Xs_29: list_A1748581779le_alt] : take_A1069848822le_alt(zero_zero_nat,Xs_29) = nil_Ar1741995278le_alt ).
tff(fact_582_append__eq__conv__conj,axiom,
! [Xs_22: list_A1748581779le_alt,Ys_4: list_A1748581779le_alt,Zs: list_A1748581779le_alt] :
( ( append634883619le_alt(Xs_22,Ys_4) = Zs )
<=> ( ( Xs_22 = take_A1069848822le_alt(size_s1492124157le_alt(Xs_22),Zs) )
& ( Ys_4 = drop_A1802510926le_alt(size_s1492124157le_alt(Xs_22),Zs) ) ) ) ).
tff(fact_583_take__add,axiom,
! [I_9: nat,J_2: nat,Xs_28: list_A1748581779le_alt] : take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_9),J_2),Xs_28) = append634883619le_alt(take_A1069848822le_alt(I_9,Xs_28),take_A1069848822le_alt(J_2,drop_A1802510926le_alt(I_9,Xs_28))) ).
tff(fact_584_take__butlast,axiom,
! [N_15: nat,Xs_27: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_15),size_s1492124157le_alt(Xs_27)))
=> ( take_A1069848822le_alt(N_15,butlas730749018le_alt(Xs_27)) = take_A1069848822le_alt(N_15,Xs_27) ) ) ).
tff(fact_585_take__Suc,axiom,
! [N_14: nat,Xs_26: list_A1748581779le_alt] :
( ( Xs_26 != nil_Ar1741995278le_alt )
=> ( take_A1069848822le_alt(hAPP_nat_nat(suc,N_14),Xs_26) = cons_A684544190le_alt(hd_Arr274000865le_alt(Xs_26),take_A1069848822le_alt(N_14,tl_Arr921252325le_alt(Xs_26))) ) ) ).
tff(fact_586_take__Suc__conv__app__nth,axiom,
! [I_8: nat,Xs_25: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_8),size_s1492124157le_alt(Xs_25)))
=> ( take_A1069848822le_alt(hAPP_nat_nat(suc,I_8),Xs_25) = append634883619le_alt(take_A1069848822le_alt(I_8,Xs_25),cons_A684544190le_alt(hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_25),I_8),nil_Ar1741995278le_alt)) ) ) ).
tff(fact_587_id__take__nth__drop,axiom,
! [I_7: nat,Xs_24: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_7),size_s1492124157le_alt(Xs_24)))
=> ( Xs_24 = append634883619le_alt(take_A1069848822le_alt(I_7,Xs_24),cons_A684544190le_alt(hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_24),I_7),drop_A1802510926le_alt(hAPP_nat_nat(suc,I_7),Xs_24))) ) ) ).
tff(fact_588_upd__conv__take__nth__drop,axiom,
! [A_14: arrow_931160158le_alt,I_6: nat,Xs_23: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_6),size_s1492124157le_alt(Xs_23)))
=> ( hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_23,I_6),A_14) = append634883619le_alt(take_A1069848822le_alt(I_6,Xs_23),cons_A684544190le_alt(A_14,drop_A1802510926le_alt(hAPP_nat_nat(suc,I_6),Xs_23))) ) ) ).
tff(fact_589_takeWhile__eq__take__P__nth,axiom,
! [P_1: fun_Ar1091019988t_bool,Xs_22: list_A1748581779le_alt,N_12: nat] :
( ! [I_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),N_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_5),size_s1492124157le_alt(Xs_22)))
=> hBOOL(hAPP_A674791462t_bool(P_1,hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),I_5))) ) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),size_s1492124157le_alt(Xs_22)))
=> ~ hBOOL(hAPP_A674791462t_bool(P_1,hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_22),N_12))) )
=> ( takeWh4609031le_alt(P_1,Xs_22) = take_A1069848822le_alt(N_12,Xs_22) ) ) ) ).
tff(fact_590_termination__basic__simps_I1_J,axiom,
! [Z_1: 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_1))) ) ).
tff(fact_591_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).
tff(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_13: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_13) = A_13 ).
tff(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_13: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_13) = A_13 ).
tff(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_12: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_12),zero_zero_int) = A_12 ).
tff(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_12: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),zero_zero_nat) = A_12 ).
tff(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_11: int,B_5: int,C_8: int,D_4: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),B_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_8),D_4)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_11),C_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_5),D_4)) ).
tff(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_11: nat,B_5: nat,C_8: nat,D_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),B_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_8),D_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),C_8)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_5),D_4)) ).
tff(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_10: int,B_4: int,C_7: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_10),B_4)),C_7) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_10),C_7)),B_4) ).
tff(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_10: nat,B_4: nat,C_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_10),B_4)),C_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_10),C_7)),B_4) ).
tff(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_9: int,B_3: int,C_6: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_9),B_3)),C_6) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_9),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_3),C_6)) ).
tff(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_9: nat,B_3: nat,C_6: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_9),B_3)),C_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_3),C_6)) ).
tff(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_8: int,C_5: int,D_3: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_8),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_5),D_3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_8),C_5)),D_3) ).
tff(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_8: nat,C_5: nat,D_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_5),D_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_8),C_5)),D_3) ).
tff(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_7: int,C_4: int,D_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),D_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),D_2)) ).
tff(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_7: nat,C_4: nat,D_2: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),D_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_7),D_2)) ).
tff(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_6: int,C_3: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_6),C_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_3),A_6) ).
tff(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_6: nat,C_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_6),C_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_3),A_6) ).
tff(fact_608_add__0__iff,axiom,
! [Ba: int,Aa: int] :
( ( Ba = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ba),Aa) )
<=> ( Aa = zero_zero_int ) ) ).
tff(fact_609_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_610_rev__nth,axiom,
! [N_13: nat,Xs_21: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_13),size_s1492124157le_alt(Xs_21)))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(rev_Ar1562208110le_alt(Xs_21)),N_13) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_21)),hAPP_nat_nat(suc,N_13))) ) ) ).
tff(fact_611_drop__replicate,axiom,
! [I_4: nat,K_5: nat,X_14: arrow_931160158le_alt] : drop_A1802510926le_alt(I_4,hAPP_A1866301020le_alt(replic1967339976le_alt(K_5),X_14)) = hAPP_A1866301020le_alt(replic1967339976le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_5),I_4)),X_14) ).
tff(fact_612_diff__cancel2,axiom,
! [M: nat,K_4: nat,N_5: 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_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_5),K_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5) ).
tff(fact_613_diff__cancel,axiom,
! [K_4: nat,M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_4),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_4),N_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5) ).
tff(fact_614_diff__diff__left,axiom,
! [I_3: nat,J: nat,K_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_3),J)),K_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_4)) ).
tff(fact_615_diff__add__inverse,axiom,
! [N_5: 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_5),M)),N_5) = M ).
tff(fact_616_diff__add__inverse2,axiom,
! [M: nat,N_5: 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_5)),N_5) = M ).
tff(fact_617_diff__eq__diff__less,axiom,
! [Aa: int,Ba: int,C_1: int,D_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),D_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),D_1)) ) ) ).
tff(fact_618_add__diff__cancel,axiom,
! [A_5: int,B_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_5),B_2)),B_2) = A_5 ).
tff(fact_619_diff__add__cancel,axiom,
! [A_4: int,B_1: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_4),B_1)),B_1) = A_4 ).
tff(fact_620_less__imp__diff__less,axiom,
! [N_5: nat,J: nat,K_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N_5)),K_4)) ) ).
tff(fact_621_diff__less__mono2,axiom,
! [L: nat,M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> ( 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_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).
tff(fact_622_less__iff__diff__less__0,axiom,
! [Aa: int,Ba: int] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),Ba))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),zero_zero_int)) ) ).
tff(fact_623_diff__less,axiom,
! [M: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> ( 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_5)),M)) ) ) ).
tff(fact_624_zero__less__diff,axiom,
! [N_12: 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_12),M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_12)) ) ).
tff(fact_625_diff__less__Suc,axiom,
! [M: nat,N_5: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5)),hAPP_nat_nat(suc,M))) ).
tff(fact_626_diff__add__0,axiom,
! [N_5: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_5),M)) = zero_zero_nat ).
tff(fact_627_less__diff__conv,axiom,
! [I: nat,J_1: nat,K_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),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),K_3)),J_1)) ) ).
tff(fact_628_add__diff__inverse,axiom,
! [M: nat,N_5: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_5))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5)) = M ) ) ).
tff(fact_629_diff__commute,axiom,
! [I_3: nat,J: nat,K_4: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_3),J)),K_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_3),K_4)),J) ).
tff(fact_630_max__diff__distrib__left,axiom,
! [X_13: int,Y_2: int,Z: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,X_13),Y_2)),Z) = hAPP_int_int(hAPP_int_fun_int_int(ord_max_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_13),Z)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),Z)) ).
tff(fact_631_diff__eq__diff__eq,axiom,
! [Aa: int,Ba: int,C_1: int,D_1: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),D_1) )
=> ( ( Aa = Ba )
<=> ( C_1 = D_1 ) ) ) ).
tff(fact_632_diff__0__right,axiom,
! [A_3: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),zero_zero_int) = A_3 ).
tff(fact_633_diff__self,axiom,
! [A_2: int] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),A_2) = zero_zero_int ).
tff(fact_634_eq__iff__diff__eq__0,axiom,
! [Aa: int,Ba: int] :
( ( Aa = Ba )
<=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = zero_zero_int ) ) ).
tff(fact_635_right__minus__eq,axiom,
! [Aa: int,Ba: int] :
( ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba) = zero_zero_int )
<=> ( Aa = Ba ) ) ).
tff(fact_636_diffs0__imp__equal,axiom,
! [M: nat,N_5: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5) = zero_zero_nat )
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_5),M) = zero_zero_nat )
=> ( M = N_5 ) ) ) ).
tff(fact_637_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_638_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_639_diff__0__eq__0,axiom,
! [N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N_5) = zero_zero_nat ).
tff(fact_640_diff__Suc__Suc,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5) ).
tff(fact_641_Suc__diff__diff,axiom,
! [M: nat,N_5: nat,K_4: 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_5)),hAPP_nat_nat(suc,K_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N_5)),K_4) ).
tff(fact_642_nat__minus__add__max,axiom,
! [N_5: 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_5),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_5),M) ).
tff(fact_643_diff__Suc__less,axiom,
! [I_3: nat,N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_5),hAPP_nat_nat(suc,I_3))),N_5)) ) ).
tff(fact_644_Suc__pred,axiom,
! [N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_5),hAPP_nat_nat(suc,zero_zero_nat))) = N_5 ) ) ).
tff(fact_645_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_646_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_647_take__append,axiom,
! [N_11: nat,Xs_20: list_A1748581779le_alt,Ys_3: list_A1748581779le_alt] : take_A1069848822le_alt(N_11,append634883619le_alt(Xs_20,Ys_3)) = append634883619le_alt(take_A1069848822le_alt(N_11,Xs_20),take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),size_s1492124157le_alt(Xs_20)),Ys_3)) ).
tff(fact_648_drop__append,axiom,
! [N_10: nat,Xs_19: list_A1748581779le_alt,Ys_2: list_A1748581779le_alt] : drop_A1802510926le_alt(N_10,append634883619le_alt(Xs_19,Ys_2)) = append634883619le_alt(drop_A1802510926le_alt(N_10,Xs_19),drop_A1802510926le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),size_s1492124157le_alt(Xs_19)),Ys_2)) ).
tff(fact_649_nth__append,axiom,
! [Ys_1: list_A1748581779le_alt,N_9: nat,Xs_18: list_A1748581779le_alt] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_9),size_s1492124157le_alt(Xs_18)))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(append634883619le_alt(Xs_18,Ys_1)),N_9) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_18),N_9) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_9),size_s1492124157le_alt(Xs_18)))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(append634883619le_alt(Xs_18,Ys_1)),N_9) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Ys_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_9),size_s1492124157le_alt(Xs_18))) ) ) ) ).
tff(fact_650_list__update__append,axiom,
! [Ys: list_A1748581779le_alt,X_12: arrow_931160158le_alt,N_8: nat,Xs_17: list_A1748581779le_alt] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),size_s1492124157le_alt(Xs_17)))
=> ( hAPP_A1866301020le_alt(list_u518279030le_alt(append634883619le_alt(Xs_17,Ys),N_8),X_12) = append634883619le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_17,N_8),X_12),Ys) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),size_s1492124157le_alt(Xs_17)))
=> ( hAPP_A1866301020le_alt(list_u518279030le_alt(append634883619le_alt(Xs_17,Ys),N_8),X_12) = append634883619le_alt(Xs_17,hAPP_A1866301020le_alt(list_u518279030le_alt(Ys,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_8),size_s1492124157le_alt(Xs_17))),X_12)) ) ) ) ).
tff(fact_651_rev__drop,axiom,
! [I_2: nat,Xs_16: list_A1748581779le_alt] : rev_Ar1562208110le_alt(drop_A1802510926le_alt(I_2,Xs_16)) = take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_16)),I_2),rev_Ar1562208110le_alt(Xs_16)) ).
tff(fact_652_rev__take,axiom,
! [I_1: nat,Xs_15: list_A1748581779le_alt] : rev_Ar1562208110le_alt(take_A1069848822le_alt(I_1,Xs_15)) = drop_A1802510926le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_15)),I_1),rev_Ar1562208110le_alt(Xs_15)) ).
tff(fact_653_zero__induct__lemma,axiom,
! [I: nat,P_1: fun_nat_bool,K_3: nat] :
( hBOOL(hAPP_nat_bool(P_1,K_3))
=> ( ! [N_7: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_7)))
=> hBOOL(hAPP_nat_bool(P_1,N_7)) )
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_3),I))) ) ) ).
tff(fact_654_last__list__update,axiom,
! [X_11: arrow_931160158le_alt,K_2: nat,Xs_14: list_A1748581779le_alt] :
( ( Xs_14 != nil_Ar1741995278le_alt )
=> ( ( ( K_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_14)),one_one_nat) )
=> ( last_A1673116455le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_14,K_2),X_11)) = X_11 ) )
& ( ( K_2 != hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_14)),one_one_nat) )
=> ( last_A1673116455le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_14,K_2),X_11)) = last_A1673116455le_alt(Xs_14) ) ) ) ) ).
tff(fact_655_empty__Diff,axiom,
! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,bot_bot_fun_nat_bool),A) = bot_bot_fun_nat_bool ).
tff(fact_656_Diff__empty,axiom,
! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),bot_bot_fun_nat_bool) = A ).
tff(fact_657_Diff__cancel,axiom,
! [A: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),A) = bot_bot_fun_nat_bool ).
tff(fact_658_diff__Suc__1,axiom,
! [N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_5)),one_one_nat) = N_5 ).
tff(fact_659_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N_5: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_5) ).
tff(fact_660_Suc__eq__plus1__left,axiom,
! [N_5: nat] : hAPP_nat_nat(suc,N_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N_5) ).
tff(fact_661_Suc__eq__plus1,axiom,
! [N_5: nat] : hAPP_nat_nat(suc,N_5) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_5),one_one_nat) ).
tff(fact_662_drop__1__Cons,axiom,
! [X_10: arrow_931160158le_alt,Xs_13: list_A1748581779le_alt] : drop_A1802510926le_alt(one_one_nat,cons_A684544190le_alt(X_10,Xs_13)) = Xs_13 ).
tff(fact_663_one__reorient,axiom,
! [X_9: nat] :
( ( one_one_nat = X_9 )
<=> ( X_9 = one_one_nat ) ) ).
tff(fact_664_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
tff(fact_665_less__add__one,axiom,
! [A_1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),one_one_nat))) ).
tff(fact_666_less__add__one,axiom,
! [A_1: int] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),one_one_int))) ).
tff(fact_667_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
tff(fact_668_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
tff(fact_669_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
tff(fact_670_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
tff(fact_671_zero__less__two,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).
tff(fact_672_zero__less__two,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).
tff(fact_673_take__1__Cons,axiom,
! [X_8: arrow_931160158le_alt,Xs_12: list_A1748581779le_alt] : take_A1069848822le_alt(one_one_nat,cons_A684544190le_alt(X_8,Xs_12)) = cons_A684544190le_alt(X_8,nil_Ar1741995278le_alt) ).
tff(fact_674_length__butlast,axiom,
! [Xs_11: list_A1748581779le_alt] : size_s1492124157le_alt(butlas730749018le_alt(Xs_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_11)),one_one_nat) ).
tff(fact_675_length__tl,axiom,
! [Xs_10: list_A1748581779le_alt] : size_s1492124157le_alt(tl_Arr921252325le_alt(Xs_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_10)),one_one_nat) ).
tff(fact_676_tl__take,axiom,
! [N_6: nat,Xs_9: list_A1748581779le_alt] : tl_Arr921252325le_alt(take_A1069848822le_alt(N_6,Xs_9)) = take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_6),one_one_nat),tl_Arr921252325le_alt(Xs_9)) ).
tff(fact_677_Suc__pred_H,axiom,
! [N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> ( N_5 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_5),one_one_nat)) ) ) ).
tff(fact_678_Suc__diff__1,axiom,
! [N_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_5),one_one_nat)) = N_5 ) ) ).
tff(fact_679_add__eq__if,axiom,
! [N_5: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_5) = N_5 ) )
& ( ( M != zero_zero_nat )
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N_5) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N_5)) ) ) ) ).
tff(fact_680_nth__Cons_H,axiom,
! [X_7: arrow_931160158le_alt,Xs_8: list_A1748581779le_alt,N_4: nat] :
( ( ( N_4 = zero_zero_nat )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(cons_A684544190le_alt(X_7,Xs_8)),N_4) = X_7 ) )
& ( ( N_4 != zero_zero_nat )
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(cons_A684544190le_alt(X_7,Xs_8)),N_4) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_4),one_one_nat)) ) ) ) ).
tff(fact_681_drop__Cons_H,axiom,
! [X_6: arrow_931160158le_alt,Xs_7: list_A1748581779le_alt,N_3: nat] :
( ( ( N_3 = zero_zero_nat )
=> ( drop_A1802510926le_alt(N_3,cons_A684544190le_alt(X_6,Xs_7)) = cons_A684544190le_alt(X_6,Xs_7) ) )
& ( ( N_3 != zero_zero_nat )
=> ( drop_A1802510926le_alt(N_3,cons_A684544190le_alt(X_6,Xs_7)) = drop_A1802510926le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_3),one_one_nat),Xs_7) ) ) ) ).
tff(fact_682_butlast__conv__take,axiom,
! [Xs_6: list_A1748581779le_alt] : butlas730749018le_alt(Xs_6) = take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_6)),one_one_nat),Xs_6) ).
tff(fact_683_butlast__list__update,axiom,
! [X_5: arrow_931160158le_alt,K_1: nat,Xs_5: list_A1748581779le_alt] :
( ( ( K_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_5)),one_one_nat) )
=> ( butlas730749018le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_5,K_1),X_5)) = butlas730749018le_alt(Xs_5) ) )
& ( ( K_1 != hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_5)),one_one_nat) )
=> ( butlas730749018le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_5,K_1),X_5)) = hAPP_A1866301020le_alt(list_u518279030le_alt(butlas730749018le_alt(Xs_5),K_1),X_5) ) ) ) ).
tff(fact_684_tl__replicate,axiom,
! [X_4: arrow_931160158le_alt,N_2: nat] :
( ( N_2 != zero_zero_nat )
=> ( tl_Arr921252325le_alt(hAPP_A1866301020le_alt(replic1967339976le_alt(N_2),X_4)) = hAPP_A1866301020le_alt(replic1967339976le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),one_one_nat)),X_4) ) ) ).
tff(fact_685_nth__Cons__pos,axiom,
! [X_3: arrow_931160158le_alt,Xs_4: list_A1748581779le_alt,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hAPP_n2017047337le_alt(nth_Ar581680143le_alt(cons_A684544190le_alt(X_3,Xs_4)),N_1) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)) ) ) ).
tff(fact_686_take__Cons_H,axiom,
! [X_2: arrow_931160158le_alt,Xs_3: list_A1748581779le_alt,N: nat] :
( ( ( N = zero_zero_nat )
=> ( take_A1069848822le_alt(N,cons_A684544190le_alt(X_2,Xs_3)) = nil_Ar1741995278le_alt ) )
& ( ( N != zero_zero_nat )
=> ( take_A1069848822le_alt(N,cons_A684544190le_alt(X_2,Xs_3)) = cons_A684544190le_alt(X_2,take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),Xs_3)) ) ) ) ).
tff(fact_687_rev__update,axiom,
! [Y_1: arrow_931160158le_alt,K: nat,Xs_2: list_A1748581779le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),size_s1492124157le_alt(Xs_2)))
=> ( rev_Ar1562208110le_alt(hAPP_A1866301020le_alt(list_u518279030le_alt(Xs_2,K),Y_1)) = hAPP_A1866301020le_alt(list_u518279030le_alt(rev_Ar1562208110le_alt(Xs_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_2)),K)),one_one_nat)),Y_1) ) ) ).
tff(fact_688_last__conv__nth,axiom,
! [Xs_1: list_A1748581779le_alt] :
( ( Xs_1 != nil_Ar1741995278le_alt )
=> ( last_A1673116455le_alt(Xs_1) = hAPP_n2017047337le_alt(nth_Ar581680143le_alt(Xs_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,size_s1492124157le_alt(Xs_1)),one_one_nat)) ) ) ).
tff(fact_689_take__Cons__number__of,axiom,
! [X_1: arrow_931160158le_alt,Xs: list_A1748581779le_alt,V_2: int] :
( ( ( number_number_of_nat(V_2) = zero_zero_nat )
=> ( take_A1069848822le_alt(number_number_of_nat(V_2),cons_A684544190le_alt(X_1,Xs)) = nil_Ar1741995278le_alt ) )
& ( ( number_number_of_nat(V_2) != zero_zero_nat )
=> ( take_A1069848822le_alt(number_number_of_nat(V_2),cons_A684544190le_alt(X_1,Xs)) = cons_A684544190le_alt(X_1,take_A1069848822le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,number_number_of_nat(V_2)),one_one_nat),Xs)) ) ) ) ).
tff(fact_690_Ints__odd__less__0,axiom,
! [Aa: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Aa)),Aa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Aa),zero_zero_int)) ) ) ).
tff(fact_691_DiffE,axiom,
! [C_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B)) ) ) ).
tff(fact_692_DiffI,axiom,
! [B: fun_nat_bool,C_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).
tff(fact_693_Ints__diff,axiom,
! [Ba: int,Aa: int] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Aa),ring_1_Ints_int))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ba),ring_1_Ints_int))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Aa),Ba)),ring_1_Ints_int)) ) ) ).
tff(fact_694_number__of__diff,axiom,
! [V_1: int,W_1: int] : number_number_of_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_1),W_1)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,number_number_of_int(V_1)),number_number_of_int(W_1)) ).
tff(fact_695_add__number__of__diff1,axiom,
! [V: int,W: int,C_2: int] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,number_number_of_int(V)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,number_number_of_int(W)),C_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,number_number_of_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),W))),C_2) ).
tff(fact_696_DiffD2,axiom,
! [C_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B)) ) ).
tff(fact_697_DiffD1,axiom,
! [C_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A)) ) ).
tff(fact_698_Diff__iff,axiom,
! [C_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B)) ) ) ).
tff(fact_699_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,cOMBC_226598744l_bool(member_nat,A)),cOMBB_bool_bool_nat(fNot,cOMBC_226598744l_bool(member_nat,B)))) ).
%----Helper facts (14)
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_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_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P: bool,Q: nat] : hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P ).
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_fequal_1_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ct,axiom,
! [X: list_A1748581779le_alt,Y: list_A1748581779le_alt] :
( ~ hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(fequal414630996le_alt,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ct,axiom,
! [X: list_A1748581779le_alt,Y: list_A1748581779le_alt] :
( ( X != Y )
| hBOOL(hAPP_l713721587t_bool(hAPP_l1173894719t_bool(fequal414630996le_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(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)) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
a != b ).
tff(conj_1,conjecture,
? [C: arrow_931160158le_alt] : hBOOL(hAPP_l713721587t_bool(distin692125441le_alt,cons_A684544190le_alt(a,cons_A684544190le_alt(b,cons_A684544190le_alt(C,nil_Ar1741995278le_alt))))) ).
%------------------------------------------------------------------------------