TPTP Problem File: SCT169^3.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SCT169^3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 29, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : arrow_1000_thf_l29 [Bla11]
% Status : Theorem
% Rating : 1.00 v8.1.0, 0.75 v7.5.0, 1.00 v5.3.0
% Syntax : Number of formulae : 1387 ( 415 unt; 172 typ; 0 def)
% Number of atoms : 2840 (1182 equ; 22 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10020 ( 353 ~; 61 |; 238 &;8125 @)
% ( 305 <=>; 935 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 739 ( 739 >; 0 *; 0 +; 0 <<)
% Number of symbols : 169 ( 162 usr; 12 con; 0-3 aty)
% ( 0 !!; 3 ??; 0 @@+; 0 @@-)
% Number of variables : 3224 ( 82 ^;2936 !; 206 ?;3224 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:29:08
%------------------------------------------------------------------------------
%----Should-be-implicit typings (10)
thf(ty_ty_tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
arrow_1389031998le_alt: $tType ).
thf(ty_ty_tc__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_ty_tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_J,type,
list_A1402527603le_alt: $tType ).
thf(ty_ty_tc__List__Olist_Itc__List__Olist_Itc__Arrow____Order____Mirabelle____piyec,type,
list_l1230899348le_alt: $tType ).
thf(ty_ty_tc__List__Olist_Itc__List__Olist_Itc__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_ty_tc__List__Olist_Itc__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Option__Ooption_I_Eo_J,type,
option_o: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_J,type,
option1949451059le_alt: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Nat__Onat_J,type,
option_nat: $tType ).
%----Explicit typings (165)
thf(sy_c_All,type,
all: ( nat > $o ) > $o ).
thf(sy_c_Big__Operators_Olattice_OSup__fin_000tc__Nat__Onat,type,
big_Sup_fin_nat: ( nat > nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_Big__Operators_Olinorder__class_OMax_000tc__Nat__Onat,type,
big_linorder_Max_nat: ( nat > $o ) > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Arrow____Order____Mirabelle____piy,type,
minus_1594223470_alt_o: ( arrow_1389031998le_alt > $o ) > ( arrow_1389031998le_alt > $o ) > arrow_1389031998le_alt > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Obool_Obool__size,type,
bool_size: $o > nat ).
thf(sy_c_HOL_Oequal__class_Oequal_000tc__List__Olist_Itc__Arrow____Order____Mirabell,type,
equal_1919384209le_alt: list_A1402527603le_alt > list_A1402527603le_alt > $o ).
thf(sy_c_HOL_Oequal__class_Oequal_000tc__List__Olist_Itc__Nat__Onat_J,type,
equal_equal_list_nat: list_nat > list_nat > $o ).
thf(sy_c_If_000tc__List__Olist_I_Eo_J,type,
if_list_o: $o > list_o > list_o > list_o ).
thf(sy_c_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_,type,
if_lis613229059le_alt: $o > list_A1402527603le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_If_000tc__List__Olist_Itc__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_List_OListMem_000_Eo,type,
listMem_o: $o > list_o > $o ).
thf(sy_c_List_OListMem_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
listMe991720154le_alt: arrow_1389031998le_alt > list_A1402527603le_alt > $o ).
thf(sy_c_List_OListMem_000tc__Nat__Onat,type,
listMem_nat: nat > list_nat > $o ).
thf(sy_c_List_Oappend_000_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
append1092755459le_alt: list_A1402527603le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Oappend_000tc__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obutlast_000_Eo,type,
butlast_o: list_o > list_o ).
thf(sy_c_List_Obutlast_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
butlas1188620858le_alt: list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Obutlast_000tc__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Odistinct_000_Eo,type,
distinct_o: list_o > $o ).
thf(sy_c_List_Odistinct_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
distin1149997281le_alt: list_A1402527603le_alt > $o ).
thf(sy_c_List_Odistinct_000tc__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_OdropWhile_000_Eo,type,
dropWhile_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_OdropWhile_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
dropWh82971279le_alt: ( arrow_1389031998le_alt > $o ) > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_OdropWhile_000tc__Nat__Onat,type,
dropWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Odrop_000_Eo,type,
drop_o: nat > list_o > list_o ).
thf(sy_c_List_Odrop_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
drop_A112899118le_alt: nat > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Odrop_000tc__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldl_000tc__List__Olist_I_Eo_J_000_Eo,type,
foldl_list_o_o: ( list_o > $o > list_o ) > list_o > list_o > list_o ).
thf(sy_c_List_Ofoldl_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccgg,type,
foldl_2120284924le_alt: ( list_A1402527603le_alt > arrow_1389031998le_alt > list_A1402527603le_alt ) > list_A1402527603le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Ofoldl_000tc__List__Olist_Itc__Nat__Onat_J_000tc__Nat__Onat,type,
foldl_list_nat_nat: ( list_nat > nat > list_nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldl_000tc__Nat__Onat_000tc__Nat__Onat,type,
foldl_nat_nat: ( nat > nat > nat ) > nat > list_nat > nat ).
thf(sy_c_List_Ofoldr_000tc__Nat__Onat_000tc__Nat__Onat,type,
foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Ohd_000_Eo,type,
hd_o: list_o > $o ).
thf(sy_c_List_Ohd_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
hd_Arr731872705le_alt: list_A1402527603le_alt > arrow_1389031998le_alt ).
thf(sy_c_List_Ohd_000tc__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Oinsert_000_Eo,type,
insert_o: $o > list_o > list_o ).
thf(sy_c_List_Oinsert_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
insert886756100le_alt: arrow_1389031998le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olast_000_Eo,type,
last_o: list_o > $o ).
thf(sy_c_List_Olast_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
last_A2130988295le_alt: list_A1402527603le_alt > arrow_1389031998le_alt ).
thf(sy_c_List_Olast_000tc__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_000tc__Nat__Onat_000tc__Nat__Ona,type,
linord297828738at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_000tc__Nat__Onat_000tc__Nat__Onat,type,
linord1850739451at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_000tc__Nat__Onat_000tc__Nat__Onat,type,
linord852754176at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted_000tc__Nat__Onat,type,
linorder_sorted_nat: list_nat > $o ).
thf(sy_c_List_Olist_OCons_000_Eo,type,
cons_o: $o > list_o > list_o ).
thf(sy_c_List_Olist_OCons_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
cons_A1142416030le_alt: arrow_1389031998le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Olist_OCons_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piy,type,
cons_l2069870355le_alt: list_A1402527603le_alt > list_l1230899348le_alt > list_l1230899348le_alt ).
thf(sy_c_List_Olist_OCons_000tc__List__Olist_Itc__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_000tc__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_000_Eo,type,
nil_o: list_o ).
thf(sy_c_List_Olist_ONil_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
nil_Ar52383470le_alt: list_A1402527603le_alt ).
thf(sy_c_List_Olist_ONil_000tc__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Olist__case_000tc__List__Olist_I_Eo_J_000_Eo,type,
list_case_list_o_o: list_o > ( $o > list_o > list_o ) > list_o > list_o ).
thf(sy_c_List_Olist_Olist__case_000tc__List__Olist_Itc__Arrow____Order____Mirabelle_,type,
list_c1300280951le_alt: list_A1402527603le_alt > ( arrow_1389031998le_alt > list_A1402527603le_alt > list_A1402527603le_alt ) > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Olist_Olist__case_000tc__List__Olist_Itc__Nat__Onat_J_000tc__Nat__Onat,type,
list_c554461815at_nat: list_nat > ( nat > list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Olist__size_000_Eo,type,
list_size_o: ( $o > nat ) > list_o > nat ).
thf(sy_c_List_Olist_Olist__size_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oa,type,
list_s1163968880le_alt: ( arrow_1389031998le_alt > nat ) > list_A1402527603le_alt > nat ).
thf(sy_c_List_Olist_Olist__size_000tc__Nat__Onat,type,
list_size_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist__all_000_Eo,type,
list_all_o: ( $o > $o ) > list_o > $o ).
thf(sy_c_List_Olist__all_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
list_a1483787182le_alt: ( arrow_1389031998le_alt > $o ) > list_A1402527603le_alt > $o ).
thf(sy_c_List_Olist__all_000tc__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_000_Eo,type,
list_ex1_o: ( $o > $o ) > list_o > $o ).
thf(sy_c_List_Olist__ex1_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
list_e1152917937le_alt: ( arrow_1389031998le_alt > $o ) > list_A1402527603le_alt > $o ).
thf(sy_c_List_Olist__ex1_000tc__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olistrelp_000_Eo,type,
listrelp_o: ( $o > $o > $o ) > list_o > list_o > $o ).
thf(sy_c_List_Olistrelp_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
listre2126835016le_alt: ( arrow_1389031998le_alt > arrow_1389031998le_alt > $o ) > list_A1402527603le_alt > list_A1402527603le_alt > $o ).
thf(sy_c_List_Olistrelp_000tc__Nat__Onat,type,
listrelp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Omap_000tc__Nat__Onat_000tc__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__filter_000_Eo_000tc__Arrow____Order____Mirabelle____piyeccggrh__,type,
map_fi875972505le_alt: ( $o > option1949451059le_alt ) > list_o > list_A1402527603le_alt ).
thf(sy_c_List_Omap__filter_000_Eo_000tc__Nat__Onat,type,
map_filter_o_nat: ( $o > option_nat ) > list_o > list_nat ).
thf(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_00,type,
map_fi312952763_alt_o: ( arrow_1389031998le_alt > option_o ) > list_A1402527603le_alt > list_o ).
thf(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_00_001,type,
map_fi193489098le_alt: ( arrow_1389031998le_alt > option1949451059le_alt ) > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_00_002,type,
map_fi1378126230lt_nat: ( arrow_1389031998le_alt > option_nat ) > list_A1402527603le_alt > list_nat ).
thf(sy_c_List_Omap__filter_000tc__Nat__Onat_000_Eo,type,
map_filter_nat_o: ( nat > option_o ) > list_nat > list_o ).
thf(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____pi,type,
map_fi228886654le_alt: ( nat > option1949451059le_alt ) > list_nat > list_A1402527603le_alt ).
thf(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__Nat__Onat,type,
map_filter_nat_nat: ( nat > option_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_000_Eo_000tc__Nat__Onat,type,
maps_o_nat: ( $o > list_nat ) > list_o > list_nat ).
thf(sy_c_List_Omaps_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt_000tc__Ar,type,
maps_A2106577089le_alt: ( arrow_1389031998le_alt > list_A1402527603le_alt ) > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Omaps_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____piyeccggr,type,
maps_n2041629429le_alt: ( nat > list_A1402527603le_alt ) > list_nat > list_A1402527603le_alt ).
thf(sy_c_List_Omember_000_Eo,type,
member_o_1: list_o > $o > $o ).
thf(sy_c_List_Omember_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
member271157507le_alt: list_A1402527603le_alt > arrow_1389031998le_alt > $o ).
thf(sy_c_List_Omember_000tc__Nat__Onat,type,
member_nat_1: list_nat > nat > $o ).
thf(sy_c_List_Onth_000_Eo,type,
nth_o: list_o > nat > $o ).
thf(sy_c_List_Onth_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
nth_Ar1039551983le_alt: list_A1402527603le_alt > nat > arrow_1389031998le_alt ).
thf(sy_c_List_Onth_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh,type,
nth_li1391746178le_alt: list_l1230899348le_alt > nat > list_A1402527603le_alt ).
thf(sy_c_List_Onth_000tc__List__Olist_Itc__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_000tc__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onull_000_Eo,type,
null_o: list_o > $o ).
thf(sy_c_List_Onull_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
null_A287154422le_alt: list_A1402527603le_alt > $o ).
thf(sy_c_List_Onull_000tc__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Oremove1_000_Eo,type,
remove1_o: $o > list_o > list_o ).
thf(sy_c_List_Oremove1_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
remove356532612le_alt: arrow_1389031998le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Oremove1_000tc__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oreplicate_000_Eo,type,
replicate_o: nat > $o > list_o ).
thf(sy_c_List_Oreplicate_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
replic277728168le_alt: nat > arrow_1389031998le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Oreplicate_000tc__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Orev_000_Eo,type,
rev_o: list_o > list_o ).
thf(sy_c_List_Orev_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
rev_Ar2020079950le_alt: list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Orev_000tc__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_000_Eo,type,
rotate1_o: list_o > list_o ).
thf(sy_c_List_Orotate1_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
rotate1249022267le_alt: list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Orotate1_000tc__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Oset_000_Eo,type,
set_o: list_o > $o > $o ).
thf(sy_c_List_Oset_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
set_Ar1491127311le_alt: list_A1402527603le_alt > arrow_1389031998le_alt > $o ).
thf(sy_c_List_Oset_000tc__Nat__Onat,type,
set_nat: list_nat > nat > $o ).
thf(sy_c_List_Osplice_000_Eo,type,
splice_o: list_o > list_o > list_o ).
thf(sy_c_List_Osplice_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
splice287087809le_alt: list_A1402527603le_alt > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Osplice_000tc__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Osublist_000_Eo,type,
sublist_o: list_o > ( nat > $o ) > list_o ).
thf(sy_c_List_Osublist_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
sublis814142963le_alt: list_A1402527603le_alt > ( nat > $o ) > list_A1402527603le_alt ).
thf(sy_c_List_Osublist_000tc__Nat__Onat,type,
sublist_nat: list_nat > ( nat > $o ) > list_nat ).
thf(sy_c_List_OtakeWhile_000_Eo,type,
takeWhile_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_OtakeWhile_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
takeWh462480871le_alt: ( arrow_1389031998le_alt > $o ) > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_OtakeWhile_000tc__Nat__Onat,type,
takeWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Otake_000_Eo,type,
take_o: nat > list_o > list_o ).
thf(sy_c_List_Otake_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
take_A1527720662le_alt: nat > list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Otake_000tc__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otl_000_Eo,type,
tl_o: list_o > list_o ).
thf(sy_c_List_Otl_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
tl_Arr1379124165le_alt: list_A1402527603le_alt > list_A1402527603le_alt ).
thf(sy_c_List_Otl_000tc__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Onat__case_000_Eo,type,
nat_case_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Onat__case_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
nat_ca317420658le_alt: arrow_1389031998le_alt > ( nat > arrow_1389031998le_alt ) > nat > arrow_1389031998le_alt ).
thf(sy_c_Nat_Onat_Onat__case_000tc__List__Olist_I_Eo_J,type,
nat_case_list_o: list_o > ( nat > list_o ) > nat > list_o ).
thf(sy_c_Nat_Onat_Onat__case_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____,type,
nat_ca1434653759le_alt: list_A1402527603le_alt > ( nat > list_A1402527603le_alt ) > nat > list_A1402527603le_alt ).
thf(sy_c_Nat_Onat_Onat__case_000tc__List__Olist_Itc__Nat__Onat_J,type,
nat_case_list_nat: list_nat > ( nat > list_nat ) > nat > list_nat ).
thf(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
nat_case_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Arrow____Order____Mirabelle_,type,
size_s1146069981le_alt: list_A1402527603le_alt > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Nat__Onat,type,
size_size_nat: nat > nat ).
thf(sy_c_Option_Ooption_Ooption__case_000tc__List__Olist_I_Eo_J_000_Eo,type,
option_case_list_o_o: list_o > ( $o > list_o ) > option_o > list_o ).
thf(sy_c_Option_Ooption_Ooption__case_000tc__List__Olist_Itc__Arrow____Order____Mira,type,
option1388650770le_alt: list_A1402527603le_alt > ( arrow_1389031998le_alt > list_A1402527603le_alt ) > option1949451059le_alt > list_A1402527603le_alt ).
thf(sy_c_Option_Ooption_Ooption__case_000tc__List__Olist_Itc__Nat__Onat_J_000tc__Nat,type,
option306705938at_nat: list_nat > ( nat > list_nat ) > option_nat > list_nat ).
thf(sy_c_Orderings_Oord_Omax_000tc__Nat__Onat,type,
max_nat: ( nat > nat > $o ) > nat > nat > nat ).
thf(sy_c_Orderings_Oord_Omin_000tc__Nat__Onat,type,
min_nat: ( nat > nat > $o ) > nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Arrow____Order____Mirabelle____piy,type,
ord_le1573998569_alt_o: ( arrow_1389031998le_alt > $o ) > ( arrow_1389031998le_alt > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Arrow____Order____Mirabelle___,type,
ord_le1423491573_alt_o: ( arrow_1389031998le_alt > $o ) > ( arrow_1389031998le_alt > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
collec1655747795le_alt: ( arrow_1389031998le_alt > $o ) > arrow_1389031998le_alt > $o ).
thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh__O,type,
fequal68576820le_alt: list_A1402527603le_alt > list_A1402527603le_alt > $o ).
thf(sy_c_fequal_000tc__List__Olist_Itc__Nat__Onat_J,type,
fequal_list_nat: list_nat > list_nat > $o ).
thf(sy_c_member_000_Eo,type,
member_o: $o > ( $o > $o ) > $o ).
thf(sy_c_member_000tc__Arrow____Order____Mirabelle____piyeccggrh__Oalt,type,
member998036369le_alt: arrow_1389031998le_alt > ( arrow_1389031998le_alt > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_v_a,type,
a: arrow_1389031998le_alt ).
thf(sy_v_b,type,
b: arrow_1389031998le_alt ).
%----Relevant facts (1200)
thf(fact_0_alt3,axiom,
? [A_101: arrow_1389031998le_alt,B_1: arrow_1389031998le_alt,C: arrow_1389031998le_alt] : ( distin1149997281le_alt @ ( cons_A1142416030le_alt @ A_101 @ ( cons_A1142416030le_alt @ B_1 @ ( cons_A1142416030le_alt @ C @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_1_distinct_Osimps_I1_J,axiom,
distin1149997281le_alt @ nil_Ar52383470le_alt ).
thf(fact_2_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
thf(fact_3_list_Osimps_I2_J,axiom,
! [A_130: $o,List_7: list_o] :
( nil_o
!= ( cons_o @ A_130 @ List_7 ) ) ).
thf(fact_4_list_Osimps_I2_J,axiom,
! [A_130: arrow_1389031998le_alt,List_7: list_A1402527603le_alt] :
( nil_Ar52383470le_alt
!= ( cons_A1142416030le_alt @ A_130 @ List_7 ) ) ).
thf(fact_5_list_Osimps_I2_J,axiom,
! [A_130: nat,List_7: list_nat] :
( nil_nat
!= ( cons_nat @ A_130 @ List_7 ) ) ).
thf(fact_6_list_Osimps_I3_J,axiom,
! [A_129: $o,List_6: list_o] :
( ( cons_o @ A_129 @ List_6 )
!= nil_o ) ).
thf(fact_7_list_Osimps_I3_J,axiom,
! [A_129: arrow_1389031998le_alt,List_6: list_A1402527603le_alt] :
( ( cons_A1142416030le_alt @ A_129 @ List_6 )
!= nil_Ar52383470le_alt ) ).
thf(fact_8_list_Osimps_I3_J,axiom,
! [A_129: nat,List_6: list_nat] :
( ( cons_nat @ A_129 @ List_6 )
!= nil_nat ) ).
thf(fact_9_neq__Nil__conv,axiom,
! [Xs_268: list_o] :
( ( Xs_268 != nil_o )
<=> ? [Y_40: $o,Ys_7: list_o] :
( Xs_268
= ( cons_o @ Y_40 @ Ys_7 ) ) ) ).
thf(fact_10_neq__Nil__conv,axiom,
! [Xs_268: list_A1402527603le_alt] :
( ( Xs_268 != nil_Ar52383470le_alt )
<=> ? [Y_40: arrow_1389031998le_alt,Ys_7: list_A1402527603le_alt] :
( Xs_268
= ( cons_A1142416030le_alt @ Y_40 @ Ys_7 ) ) ) ).
thf(fact_11_neq__Nil__conv,axiom,
! [Xs_268: list_nat] :
( ( Xs_268 != nil_nat )
<=> ? [Y_40: nat,Ys_7: list_nat] :
( Xs_268
= ( cons_nat @ Y_40 @ Ys_7 ) ) ) ).
thf(fact_12_list_Oexhaust,axiom,
! [Y_110: list_o] :
( ( Y_110 != nil_o )
=> ~ ! [A_101: $o,List_5: list_o] :
( Y_110
!= ( cons_o @ A_101 @ List_5 ) ) ) ).
thf(fact_13_list_Oexhaust,axiom,
! [Y_110: list_A1402527603le_alt] :
( ( Y_110 != nil_Ar52383470le_alt )
=> ~ ! [A_101: arrow_1389031998le_alt,List_5: list_A1402527603le_alt] :
( Y_110
!= ( cons_A1142416030le_alt @ A_101 @ List_5 ) ) ) ).
thf(fact_14_list_Oexhaust,axiom,
! [Y_110: list_nat] :
( ( Y_110 != nil_nat )
=> ~ ! [A_101: nat,List_5: list_nat] :
( Y_110
!= ( cons_nat @ A_101 @ List_5 ) ) ) ).
thf(fact_15_not__Cons__self,axiom,
! [Xs_267: list_o,X_237: $o] :
( Xs_267
!= ( cons_o @ X_237 @ Xs_267 ) ) ).
thf(fact_16_not__Cons__self,axiom,
! [Xs_267: list_A1402527603le_alt,X_237: arrow_1389031998le_alt] :
( Xs_267
!= ( cons_A1142416030le_alt @ X_237 @ Xs_267 ) ) ).
thf(fact_17_not__Cons__self,axiom,
! [Xs_267: list_nat,X_237: nat] :
( Xs_267
!= ( cons_nat @ X_237 @ Xs_267 ) ) ).
thf(fact_18_not__Cons__self2,axiom,
! [X_236: $o,Xs_266: list_o] :
( ( cons_o @ X_236 @ Xs_266 )
!= Xs_266 ) ).
thf(fact_19_not__Cons__self2,axiom,
! [X_236: arrow_1389031998le_alt,Xs_266: list_A1402527603le_alt] :
( ( cons_A1142416030le_alt @ X_236 @ Xs_266 )
!= Xs_266 ) ).
thf(fact_20_not__Cons__self2,axiom,
! [X_236: nat,Xs_266: list_nat] :
( ( cons_nat @ X_236 @ Xs_266 )
!= Xs_266 ) ).
thf(fact_21_list_Oinject,axiom,
! [A_128: $o,List_4: list_o,A_127: $o,List_3: list_o] :
( ( ( cons_o @ A_128 @ List_4 )
= ( cons_o @ A_127 @ List_3 ) )
<=> ( ( A_128
<=> A_127 )
& ( List_4 = List_3 ) ) ) ).
thf(fact_22_list_Oinject,axiom,
! [A_128: arrow_1389031998le_alt,List_4: list_A1402527603le_alt,A_127: arrow_1389031998le_alt,List_3: list_A1402527603le_alt] :
( ( ( cons_A1142416030le_alt @ A_128 @ List_4 )
= ( cons_A1142416030le_alt @ A_127 @ List_3 ) )
<=> ( ( A_128 = A_127 )
& ( List_4 = List_3 ) ) ) ).
thf(fact_23_list_Oinject,axiom,
! [A_128: nat,List_4: list_nat,A_127: nat,List_3: list_nat] :
( ( ( cons_nat @ A_128 @ List_4 )
= ( cons_nat @ A_127 @ List_3 ) )
<=> ( ( A_128 = A_127 )
& ( List_4 = List_3 ) ) ) ).
thf(fact_24_splice_Osimps_I2_J,axiom,
! [V: $o,Va: list_o] :
( ( splice_o @ ( cons_o @ V @ Va ) @ nil_o )
= ( cons_o @ V @ Va ) ) ).
thf(fact_25_splice_Osimps_I2_J,axiom,
! [V: arrow_1389031998le_alt,Va: list_A1402527603le_alt] :
( ( splice287087809le_alt @ ( cons_A1142416030le_alt @ V @ Va ) @ nil_Ar52383470le_alt )
= ( cons_A1142416030le_alt @ V @ Va ) ) ).
thf(fact_26_splice_Osimps_I2_J,axiom,
! [V: nat,Va: list_nat] :
( ( splice_nat @ ( cons_nat @ V @ Va ) @ nil_nat )
= ( cons_nat @ V @ Va ) ) ).
thf(fact_27_insert__Nil,axiom,
! [X_235: $o] :
( ( insert_o @ X_235 @ nil_o )
= ( cons_o @ X_235 @ nil_o ) ) ).
thf(fact_28_insert__Nil,axiom,
! [X_235: arrow_1389031998le_alt] :
( ( insert886756100le_alt @ X_235 @ nil_Ar52383470le_alt )
= ( cons_A1142416030le_alt @ X_235 @ nil_Ar52383470le_alt ) ) ).
thf(fact_29_insert__Nil,axiom,
! [X_235: nat] :
( ( insert_nat @ X_235 @ nil_nat )
= ( cons_nat @ X_235 @ nil_nat ) ) ).
thf(fact_30_list__nonempty__induct,axiom,
! [P_60: list_o > $o,Xs_265: list_o] :
( ( Xs_265 != nil_o )
=> ( ! [X_4: $o] : ( P_60 @ ( cons_o @ X_4 @ nil_o ) )
=> ( ! [X_4: $o,Xs_23: list_o] :
( ( Xs_23 != nil_o )
=> ( ( P_60 @ Xs_23 )
=> ( P_60 @ ( cons_o @ X_4 @ Xs_23 ) ) ) )
=> ( P_60 @ Xs_265 ) ) ) ) ).
thf(fact_31_list__nonempty__induct,axiom,
! [P_60: list_A1402527603le_alt > $o,Xs_265: list_A1402527603le_alt] :
( ( Xs_265 != nil_Ar52383470le_alt )
=> ( ! [X_4: arrow_1389031998le_alt] : ( P_60 @ ( cons_A1142416030le_alt @ X_4 @ nil_Ar52383470le_alt ) )
=> ( ! [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt] :
( ( Xs_23 != nil_Ar52383470le_alt )
=> ( ( P_60 @ Xs_23 )
=> ( P_60 @ ( cons_A1142416030le_alt @ X_4 @ Xs_23 ) ) ) )
=> ( P_60 @ Xs_265 ) ) ) ) ).
thf(fact_32_list__nonempty__induct,axiom,
! [P_60: list_nat > $o,Xs_265: list_nat] :
( ( Xs_265 != nil_nat )
=> ( ! [X_4: nat] : ( P_60 @ ( cons_nat @ X_4 @ nil_nat ) )
=> ( ! [X_4: nat,Xs_23: list_nat] :
( ( Xs_23 != nil_nat )
=> ( ( P_60 @ Xs_23 )
=> ( P_60 @ ( cons_nat @ X_4 @ Xs_23 ) ) ) )
=> ( P_60 @ Xs_265 ) ) ) ) ).
thf(fact_33_distinct__butlast,axiom,
! [Xs_264: list_A1402527603le_alt] :
( ( Xs_264 != nil_Ar52383470le_alt )
=> ( ( distin1149997281le_alt @ Xs_264 )
=> ( distin1149997281le_alt @ ( butlas1188620858le_alt @ Xs_264 ) ) ) ) ).
thf(fact_34_distinct__butlast,axiom,
! [Xs_264: list_nat] :
( ( Xs_264 != nil_nat )
=> ( ( distinct_nat @ Xs_264 )
=> ( distinct_nat @ ( butlast_nat @ Xs_264 ) ) ) ) ).
thf(fact_35_butlast_Osimps_I1_J,axiom,
( ( butlas1188620858le_alt @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_36_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
thf(fact_37_distinct__insert,axiom,
! [X_234: arrow_1389031998le_alt,Xs_263: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_263 )
=> ( distin1149997281le_alt @ ( insert886756100le_alt @ X_234 @ Xs_263 ) ) ) ).
thf(fact_38_distinct__insert,axiom,
! [X_234: nat,Xs_263: list_nat] :
( ( distinct_nat @ Xs_263 )
=> ( distinct_nat @ ( insert_nat @ X_234 @ Xs_263 ) ) ) ).
thf(fact_39_splice_Osimps_I3_J,axiom,
! [X_233: $o,Xs_262: list_o,Y_109: $o,Ys_64: list_o] :
( ( splice_o @ ( cons_o @ X_233 @ Xs_262 ) @ ( cons_o @ Y_109 @ Ys_64 ) )
= ( cons_o @ X_233 @ ( cons_o @ Y_109 @ ( splice_o @ Xs_262 @ Ys_64 ) ) ) ) ).
thf(fact_40_splice_Osimps_I3_J,axiom,
! [X_233: arrow_1389031998le_alt,Xs_262: list_A1402527603le_alt,Y_109: arrow_1389031998le_alt,Ys_64: list_A1402527603le_alt] :
( ( splice287087809le_alt @ ( cons_A1142416030le_alt @ X_233 @ Xs_262 ) @ ( cons_A1142416030le_alt @ Y_109 @ Ys_64 ) )
= ( cons_A1142416030le_alt @ X_233 @ ( cons_A1142416030le_alt @ Y_109 @ ( splice287087809le_alt @ Xs_262 @ Ys_64 ) ) ) ) ).
thf(fact_41_splice_Osimps_I3_J,axiom,
! [X_233: nat,Xs_262: list_nat,Y_109: nat,Ys_64: list_nat] :
( ( splice_nat @ ( cons_nat @ X_233 @ Xs_262 ) @ ( cons_nat @ Y_109 @ Ys_64 ) )
= ( cons_nat @ X_233 @ ( cons_nat @ Y_109 @ ( splice_nat @ Xs_262 @ Ys_64 ) ) ) ) ).
thf(fact_42_splice__Nil2,axiom,
! [Xs_261: list_A1402527603le_alt] :
( ( splice287087809le_alt @ Xs_261 @ nil_Ar52383470le_alt )
= Xs_261 ) ).
thf(fact_43_splice__Nil2,axiom,
! [Xs_261: list_nat] :
( ( splice_nat @ Xs_261 @ nil_nat )
= Xs_261 ) ).
thf(fact_44_splice_Osimps_I1_J,axiom,
! [Ys_63: list_A1402527603le_alt] :
( ( splice287087809le_alt @ nil_Ar52383470le_alt @ Ys_63 )
= Ys_63 ) ).
thf(fact_45_splice_Osimps_I1_J,axiom,
! [Ys_63: list_nat] :
( ( splice_nat @ nil_nat @ Ys_63 )
= Ys_63 ) ).
thf(fact_46_butlast_Osimps_I2_J,axiom,
! [X_232: $o,Xs_260: list_o] :
( ( ( Xs_260 = nil_o )
=> ( ( butlast_o @ ( cons_o @ X_232 @ Xs_260 ) )
= nil_o ) )
& ( ( Xs_260 != nil_o )
=> ( ( butlast_o @ ( cons_o @ X_232 @ Xs_260 ) )
= ( cons_o @ X_232 @ ( butlast_o @ Xs_260 ) ) ) ) ) ).
thf(fact_47_butlast_Osimps_I2_J,axiom,
! [X_232: arrow_1389031998le_alt,Xs_260: list_A1402527603le_alt] :
( ( ( Xs_260 = nil_Ar52383470le_alt )
=> ( ( butlas1188620858le_alt @ ( cons_A1142416030le_alt @ X_232 @ Xs_260 ) )
= nil_Ar52383470le_alt ) )
& ( ( Xs_260 != nil_Ar52383470le_alt )
=> ( ( butlas1188620858le_alt @ ( cons_A1142416030le_alt @ X_232 @ Xs_260 ) )
= ( cons_A1142416030le_alt @ X_232 @ ( butlas1188620858le_alt @ Xs_260 ) ) ) ) ) ).
thf(fact_48_butlast_Osimps_I2_J,axiom,
! [X_232: nat,Xs_260: list_nat] :
( ( ( Xs_260 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X_232 @ Xs_260 ) )
= nil_nat ) )
& ( ( Xs_260 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X_232 @ Xs_260 ) )
= ( cons_nat @ X_232 @ ( butlast_nat @ Xs_260 ) ) ) ) ) ).
thf(fact_49_list_Osimps_I4_J,axiom,
! [F1_3: list_A1402527603le_alt,F2_3: arrow_1389031998le_alt > list_A1402527603le_alt > list_A1402527603le_alt] :
( ( list_c1300280951le_alt @ F1_3 @ F2_3 @ nil_Ar52383470le_alt )
= F1_3 ) ).
thf(fact_50_list_Osimps_I4_J,axiom,
! [F1_3: list_nat,F2_3: nat > list_nat > list_nat] :
( ( list_c554461815at_nat @ F1_3 @ F2_3 @ nil_nat )
= F1_3 ) ).
thf(fact_51_list_Osimps_I5_J,axiom,
! [F1_2: list_A1402527603le_alt,F2_2: arrow_1389031998le_alt > list_A1402527603le_alt > list_A1402527603le_alt,A_126: arrow_1389031998le_alt,List_2: list_A1402527603le_alt] :
( ( list_c1300280951le_alt @ F1_2 @ F2_2 @ ( cons_A1142416030le_alt @ A_126 @ List_2 ) )
= ( F2_2 @ A_126 @ List_2 ) ) ).
thf(fact_52_list_Osimps_I5_J,axiom,
! [F1_2: list_nat,F2_2: nat > list_nat > list_nat,A_126: nat,List_2: list_nat] :
( ( list_c554461815at_nat @ F1_2 @ F2_2 @ ( cons_nat @ A_126 @ List_2 ) )
= ( F2_2 @ A_126 @ List_2 ) ) ).
thf(fact_53_eq__Nil__null,axiom,
! [Xs_259: list_A1402527603le_alt] :
( ( Xs_259 = nil_Ar52383470le_alt )
<=> ( null_A287154422le_alt @ Xs_259 ) ) ).
thf(fact_54_eq__Nil__null,axiom,
! [Xs_259: list_nat] :
( ( Xs_259 = nil_nat )
<=> ( null_nat @ Xs_259 ) ) ).
thf(fact_55_List_Onull__def,axiom,
! [Xs_258: list_A1402527603le_alt] :
( ( null_A287154422le_alt @ Xs_258 )
<=> ( Xs_258 = nil_Ar52383470le_alt ) ) ).
thf(fact_56_List_Onull__def,axiom,
! [Xs_258: list_nat] :
( ( null_nat @ Xs_258 )
<=> ( Xs_258 = nil_nat ) ) ).
thf(fact_57_null__rec_I2_J,axiom,
null_A287154422le_alt @ nil_Ar52383470le_alt ).
thf(fact_58_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
thf(fact_59_null__rec_I1_J,axiom,
! [X_231: $o,Xs_257: list_o] :
~ ( null_o @ ( cons_o @ X_231 @ Xs_257 ) ) ).
thf(fact_60_null__rec_I1_J,axiom,
! [X_231: arrow_1389031998le_alt,Xs_257: list_A1402527603le_alt] :
~ ( null_A287154422le_alt @ ( cons_A1142416030le_alt @ X_231 @ Xs_257 ) ) ).
thf(fact_61_null__rec_I1_J,axiom,
! [X_231: nat,Xs_257: list_nat] :
~ ( null_nat @ ( cons_nat @ X_231 @ Xs_257 ) ) ).
thf(fact_62_butlast__snoc,axiom,
! [Xs_256: list_o,X_230: $o] :
( ( butlast_o @ ( append_o @ Xs_256 @ ( cons_o @ X_230 @ nil_o ) ) )
= Xs_256 ) ).
thf(fact_63_butlast__snoc,axiom,
! [Xs_256: list_A1402527603le_alt,X_230: arrow_1389031998le_alt] :
( ( butlas1188620858le_alt @ ( append1092755459le_alt @ Xs_256 @ ( cons_A1142416030le_alt @ X_230 @ nil_Ar52383470le_alt ) ) )
= Xs_256 ) ).
thf(fact_64_butlast__snoc,axiom,
! [Xs_256: list_nat,X_230: nat] :
( ( butlast_nat @ ( append_nat @ Xs_256 @ ( cons_nat @ X_230 @ nil_nat ) ) )
= Xs_256 ) ).
thf(fact_65_sorted__single,axiom,
! [X_229: nat] : ( linorder_sorted_nat @ ( cons_nat @ X_229 @ nil_nat ) ) ).
thf(fact_66_last__ConsL,axiom,
! [X_228: $o,Xs_255: list_o] :
( ( Xs_255 = nil_o )
=> ( ( last_o @ ( cons_o @ X_228 @ Xs_255 ) )
<=> X_228 ) ) ).
thf(fact_67_last__ConsL,axiom,
! [X_228: arrow_1389031998le_alt,Xs_255: list_A1402527603le_alt] :
( ( Xs_255 = nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( cons_A1142416030le_alt @ X_228 @ Xs_255 ) )
= X_228 ) ) ).
thf(fact_68_last__ConsL,axiom,
! [X_228: nat,Xs_255: list_nat] :
( ( Xs_255 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_228 @ Xs_255 ) )
= X_228 ) ) ).
thf(fact_69_sorted_ONil,axiom,
linorder_sorted_nat @ nil_nat ).
thf(fact_70_append__assoc,axiom,
! [Xs_254: list_A1402527603le_alt,Ys_62: list_A1402527603le_alt,Zs_14: list_A1402527603le_alt] :
( ( append1092755459le_alt @ ( append1092755459le_alt @ Xs_254 @ Ys_62 ) @ Zs_14 )
= ( append1092755459le_alt @ Xs_254 @ ( append1092755459le_alt @ Ys_62 @ Zs_14 ) ) ) ).
thf(fact_71_append__assoc,axiom,
! [Xs_254: list_nat,Ys_62: list_nat,Zs_14: list_nat] :
( ( append_nat @ ( append_nat @ Xs_254 @ Ys_62 ) @ Zs_14 )
= ( append_nat @ Xs_254 @ ( append_nat @ Ys_62 @ Zs_14 ) ) ) ).
thf(fact_72_append__eq__append__conv2,axiom,
! [Xs_253: list_A1402527603le_alt,Ys_61: list_A1402527603le_alt,Zs_13: list_A1402527603le_alt,Ts: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_253 @ Ys_61 )
= ( append1092755459le_alt @ Zs_13 @ Ts ) )
<=> ? [Us_1: list_A1402527603le_alt] :
( ( ( Xs_253
= ( append1092755459le_alt @ Zs_13 @ Us_1 ) )
& ( ( append1092755459le_alt @ Us_1 @ Ys_61 )
= Ts ) )
| ( ( ( append1092755459le_alt @ Xs_253 @ Us_1 )
= Zs_13 )
& ( Ys_61
= ( append1092755459le_alt @ Us_1 @ Ts ) ) ) ) ) ).
thf(fact_73_append__eq__append__conv2,axiom,
! [Xs_253: list_nat,Ys_61: list_nat,Zs_13: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs_253 @ Ys_61 )
= ( append_nat @ Zs_13 @ Ts ) )
<=> ? [Us_1: list_nat] :
( ( ( Xs_253
= ( append_nat @ Zs_13 @ Us_1 ) )
& ( ( append_nat @ Us_1 @ Ys_61 )
= Ts ) )
| ( ( ( append_nat @ Xs_253 @ Us_1 )
= Zs_13 )
& ( Ys_61
= ( append_nat @ Us_1 @ Ts ) ) ) ) ) ).
thf(fact_74_same__append__eq,axiom,
! [Xs_252: list_A1402527603le_alt,Ys_60: list_A1402527603le_alt,Zs_12: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_252 @ Ys_60 )
= ( append1092755459le_alt @ Xs_252 @ Zs_12 ) )
<=> ( Ys_60 = Zs_12 ) ) ).
thf(fact_75_same__append__eq,axiom,
! [Xs_252: list_nat,Ys_60: list_nat,Zs_12: list_nat] :
( ( ( append_nat @ Xs_252 @ Ys_60 )
= ( append_nat @ Xs_252 @ Zs_12 ) )
<=> ( Ys_60 = Zs_12 ) ) ).
thf(fact_76_append__same__eq,axiom,
! [Ys_59: list_A1402527603le_alt,Xs_251: list_A1402527603le_alt,Zs_11: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Ys_59 @ Xs_251 )
= ( append1092755459le_alt @ Zs_11 @ Xs_251 ) )
<=> ( Ys_59 = Zs_11 ) ) ).
thf(fact_77_append__same__eq,axiom,
! [Ys_59: list_nat,Xs_251: list_nat,Zs_11: list_nat] :
( ( ( append_nat @ Ys_59 @ Xs_251 )
= ( append_nat @ Zs_11 @ Xs_251 ) )
<=> ( Ys_59 = Zs_11 ) ) ).
thf(fact_78_append__eq__appendI,axiom,
! [Ys_58: list_A1402527603le_alt,Us: list_A1402527603le_alt,Xs_250: list_A1402527603le_alt,Xs1_1: list_A1402527603le_alt,Zs_10: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_250 @ Xs1_1 )
= Zs_10 )
=> ( ( Ys_58
= ( append1092755459le_alt @ Xs1_1 @ Us ) )
=> ( ( append1092755459le_alt @ Xs_250 @ Ys_58 )
= ( append1092755459le_alt @ Zs_10 @ Us ) ) ) ) ).
thf(fact_79_append__eq__appendI,axiom,
! [Ys_58: list_nat,Us: list_nat,Xs_250: list_nat,Xs1_1: list_nat,Zs_10: list_nat] :
( ( ( append_nat @ Xs_250 @ Xs1_1 )
= Zs_10 )
=> ( ( Ys_58
= ( append_nat @ Xs1_1 @ Us ) )
=> ( ( append_nat @ Xs_250 @ Ys_58 )
= ( append_nat @ Zs_10 @ Us ) ) ) ) ).
thf(fact_80_last__appendL,axiom,
! [Xs_249: list_A1402527603le_alt,Ys_57: list_A1402527603le_alt] :
( ( Ys_57 = nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( append1092755459le_alt @ Xs_249 @ Ys_57 ) )
= ( last_A2130988295le_alt @ Xs_249 ) ) ) ).
thf(fact_81_last__appendL,axiom,
! [Xs_249: list_nat,Ys_57: list_nat] :
( ( Ys_57 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_249 @ Ys_57 ) )
= ( last_nat @ Xs_249 ) ) ) ).
thf(fact_82_last__appendR,axiom,
! [Xs_248: list_A1402527603le_alt,Ys_56: list_A1402527603le_alt] :
( ( Ys_56 != nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( append1092755459le_alt @ Xs_248 @ Ys_56 ) )
= ( last_A2130988295le_alt @ Ys_56 ) ) ) ).
thf(fact_83_last__appendR,axiom,
! [Xs_248: list_nat,Ys_56: list_nat] :
( ( Ys_56 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_248 @ Ys_56 ) )
= ( last_nat @ Ys_56 ) ) ) ).
thf(fact_84_last__append,axiom,
! [Xs_247: list_A1402527603le_alt,Ys_55: list_A1402527603le_alt] :
( ( ( Ys_55 = nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( append1092755459le_alt @ Xs_247 @ Ys_55 ) )
= ( last_A2130988295le_alt @ Xs_247 ) ) )
& ( ( Ys_55 != nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( append1092755459le_alt @ Xs_247 @ Ys_55 ) )
= ( last_A2130988295le_alt @ Ys_55 ) ) ) ) ).
thf(fact_85_last__append,axiom,
! [Xs_247: list_nat,Ys_55: list_nat] :
( ( ( Ys_55 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_247 @ Ys_55 ) )
= ( last_nat @ Xs_247 ) ) )
& ( ( Ys_55 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_247 @ Ys_55 ) )
= ( last_nat @ Ys_55 ) ) ) ) ).
thf(fact_86_maps__simps_I1_J,axiom,
! [F_30: arrow_1389031998le_alt > list_A1402527603le_alt,X_227: arrow_1389031998le_alt,Xs_246: list_A1402527603le_alt] :
( ( maps_A2106577089le_alt @ F_30 @ ( cons_A1142416030le_alt @ X_227 @ Xs_246 ) )
= ( append1092755459le_alt @ ( F_30 @ X_227 ) @ ( maps_A2106577089le_alt @ F_30 @ Xs_246 ) ) ) ).
thf(fact_87_maps__simps_I1_J,axiom,
! [F_30: nat > list_A1402527603le_alt,X_227: nat,Xs_246: list_nat] :
( ( maps_n2041629429le_alt @ F_30 @ ( cons_nat @ X_227 @ Xs_246 ) )
= ( append1092755459le_alt @ ( F_30 @ X_227 ) @ ( maps_n2041629429le_alt @ F_30 @ Xs_246 ) ) ) ).
thf(fact_88_maps__simps_I1_J,axiom,
! [F_30: $o > list_nat,X_227: $o,Xs_246: list_o] :
( ( maps_o_nat @ F_30 @ ( cons_o @ X_227 @ Xs_246 ) )
= ( append_nat @ ( F_30 @ X_227 ) @ ( maps_o_nat @ F_30 @ Xs_246 ) ) ) ).
thf(fact_89_last__snoc,axiom,
! [Xs_245: list_o,X_226: $o] :
( ( last_o @ ( append_o @ Xs_245 @ ( cons_o @ X_226 @ nil_o ) ) )
<=> X_226 ) ).
thf(fact_90_last__snoc,axiom,
! [Xs_245: list_A1402527603le_alt,X_226: arrow_1389031998le_alt] :
( ( last_A2130988295le_alt @ ( append1092755459le_alt @ Xs_245 @ ( cons_A1142416030le_alt @ X_226 @ nil_Ar52383470le_alt ) ) )
= X_226 ) ).
thf(fact_91_last__snoc,axiom,
! [Xs_245: list_nat,X_226: nat] :
( ( last_nat @ ( append_nat @ Xs_245 @ ( cons_nat @ X_226 @ nil_nat ) ) )
= X_226 ) ).
thf(fact_92_append__Cons,axiom,
! [X_225: $o,Xs_244: list_o,Ys_54: list_o] :
( ( append_o @ ( cons_o @ X_225 @ Xs_244 ) @ Ys_54 )
= ( cons_o @ X_225 @ ( append_o @ Xs_244 @ Ys_54 ) ) ) ).
thf(fact_93_append__Cons,axiom,
! [X_225: arrow_1389031998le_alt,Xs_244: list_A1402527603le_alt,Ys_54: list_A1402527603le_alt] :
( ( append1092755459le_alt @ ( cons_A1142416030le_alt @ X_225 @ Xs_244 ) @ Ys_54 )
= ( cons_A1142416030le_alt @ X_225 @ ( append1092755459le_alt @ Xs_244 @ Ys_54 ) ) ) ).
thf(fact_94_append__Cons,axiom,
! [X_225: nat,Xs_244: list_nat,Ys_54: list_nat] :
( ( append_nat @ ( cons_nat @ X_225 @ Xs_244 ) @ Ys_54 )
= ( cons_nat @ X_225 @ ( append_nat @ Xs_244 @ Ys_54 ) ) ) ).
thf(fact_95_Cons__eq__appendI,axiom,
! [Xs_243: list_o,Zs_9: list_o,X_224: $o,Xs1: list_o,Ys_53: list_o] :
( ( ( cons_o @ X_224 @ Xs1 )
= Ys_53 )
=> ( ( Xs_243
= ( append_o @ Xs1 @ Zs_9 ) )
=> ( ( cons_o @ X_224 @ Xs_243 )
= ( append_o @ Ys_53 @ Zs_9 ) ) ) ) ).
thf(fact_96_Cons__eq__appendI,axiom,
! [Xs_243: list_A1402527603le_alt,Zs_9: list_A1402527603le_alt,X_224: arrow_1389031998le_alt,Xs1: list_A1402527603le_alt,Ys_53: list_A1402527603le_alt] :
( ( ( cons_A1142416030le_alt @ X_224 @ Xs1 )
= Ys_53 )
=> ( ( Xs_243
= ( append1092755459le_alt @ Xs1 @ Zs_9 ) )
=> ( ( cons_A1142416030le_alt @ X_224 @ Xs_243 )
= ( append1092755459le_alt @ Ys_53 @ Zs_9 ) ) ) ) ).
thf(fact_97_Cons__eq__appendI,axiom,
! [Xs_243: list_nat,Zs_9: list_nat,X_224: nat,Xs1: list_nat,Ys_53: list_nat] :
( ( ( cons_nat @ X_224 @ Xs1 )
= Ys_53 )
=> ( ( Xs_243
= ( append_nat @ Xs1 @ Zs_9 ) )
=> ( ( cons_nat @ X_224 @ Xs_243 )
= ( append_nat @ Ys_53 @ Zs_9 ) ) ) ) ).
thf(fact_98_append__Nil,axiom,
! [Ys_52: list_A1402527603le_alt] :
( ( append1092755459le_alt @ nil_Ar52383470le_alt @ Ys_52 )
= Ys_52 ) ).
thf(fact_99_append__Nil,axiom,
! [Ys_52: list_nat] :
( ( append_nat @ nil_nat @ Ys_52 )
= Ys_52 ) ).
thf(fact_100_Nil__is__append__conv,axiom,
! [Xs_242: list_A1402527603le_alt,Ys_51: list_A1402527603le_alt] :
( ( nil_Ar52383470le_alt
= ( append1092755459le_alt @ Xs_242 @ Ys_51 ) )
<=> ( ( Xs_242 = nil_Ar52383470le_alt )
& ( Ys_51 = nil_Ar52383470le_alt ) ) ) ).
thf(fact_101_Nil__is__append__conv,axiom,
! [Xs_242: list_nat,Ys_51: list_nat] :
( ( nil_nat
= ( append_nat @ Xs_242 @ Ys_51 ) )
<=> ( ( Xs_242 = nil_nat )
& ( Ys_51 = nil_nat ) ) ) ).
thf(fact_102_append__Nil2,axiom,
! [Xs_241: list_A1402527603le_alt] :
( ( append1092755459le_alt @ Xs_241 @ nil_Ar52383470le_alt )
= Xs_241 ) ).
thf(fact_103_append__Nil2,axiom,
! [Xs_241: list_nat] :
( ( append_nat @ Xs_241 @ nil_nat )
= Xs_241 ) ).
thf(fact_104_self__append__conv,axiom,
! [Xs_240: list_A1402527603le_alt,Ys_50: list_A1402527603le_alt] :
( ( Xs_240
= ( append1092755459le_alt @ Xs_240 @ Ys_50 ) )
<=> ( Ys_50 = nil_Ar52383470le_alt ) ) ).
thf(fact_105_self__append__conv,axiom,
! [Xs_240: list_nat,Ys_50: list_nat] :
( ( Xs_240
= ( append_nat @ Xs_240 @ Ys_50 ) )
<=> ( Ys_50 = nil_nat ) ) ).
thf(fact_106_self__append__conv2,axiom,
! [Ys_49: list_A1402527603le_alt,Xs_239: list_A1402527603le_alt] :
( ( Ys_49
= ( append1092755459le_alt @ Xs_239 @ Ys_49 ) )
<=> ( Xs_239 = nil_Ar52383470le_alt ) ) ).
thf(fact_107_self__append__conv2,axiom,
! [Ys_49: list_nat,Xs_239: list_nat] :
( ( Ys_49
= ( append_nat @ Xs_239 @ Ys_49 ) )
<=> ( Xs_239 = nil_nat ) ) ).
thf(fact_108_append__is__Nil__conv,axiom,
! [Xs_238: list_A1402527603le_alt,Ys_48: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_238 @ Ys_48 )
= nil_Ar52383470le_alt )
<=> ( ( Xs_238 = nil_Ar52383470le_alt )
& ( Ys_48 = nil_Ar52383470le_alt ) ) ) ).
thf(fact_109_append__is__Nil__conv,axiom,
! [Xs_238: list_nat,Ys_48: list_nat] :
( ( ( append_nat @ Xs_238 @ Ys_48 )
= nil_nat )
<=> ( ( Xs_238 = nil_nat )
& ( Ys_48 = nil_nat ) ) ) ).
thf(fact_110_append__self__conv,axiom,
! [Xs_237: list_A1402527603le_alt,Ys_47: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_237 @ Ys_47 )
= Xs_237 )
<=> ( Ys_47 = nil_Ar52383470le_alt ) ) ).
thf(fact_111_append__self__conv,axiom,
! [Xs_237: list_nat,Ys_47: list_nat] :
( ( ( append_nat @ Xs_237 @ Ys_47 )
= Xs_237 )
<=> ( Ys_47 = nil_nat ) ) ).
thf(fact_112_append__self__conv2,axiom,
! [Xs_236: list_A1402527603le_alt,Ys_46: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_236 @ Ys_46 )
= Ys_46 )
<=> ( Xs_236 = nil_Ar52383470le_alt ) ) ).
thf(fact_113_append__self__conv2,axiom,
! [Xs_236: list_nat,Ys_46: list_nat] :
( ( ( append_nat @ Xs_236 @ Ys_46 )
= Ys_46 )
<=> ( Xs_236 = nil_nat ) ) ).
thf(fact_114_eq__Nil__appendI,axiom,
! [Xs_235: list_A1402527603le_alt,Ys_45: list_A1402527603le_alt] :
( ( Xs_235 = Ys_45 )
=> ( Xs_235
= ( append1092755459le_alt @ nil_Ar52383470le_alt @ Ys_45 ) ) ) ).
thf(fact_115_eq__Nil__appendI,axiom,
! [Xs_235: list_nat,Ys_45: list_nat] :
( ( Xs_235 = Ys_45 )
=> ( Xs_235
= ( append_nat @ nil_nat @ Ys_45 ) ) ) ).
thf(fact_116_snoc__eq__iff__butlast,axiom,
! [Xs_234: list_o,X_223: $o,Ys_44: list_o] :
( ( ( append_o @ Xs_234 @ ( cons_o @ X_223 @ nil_o ) )
= Ys_44 )
<=> ( ( Ys_44 != nil_o )
& ( ( butlast_o @ Ys_44 )
= Xs_234 )
& ( ( last_o @ Ys_44 )
<=> X_223 ) ) ) ).
thf(fact_117_snoc__eq__iff__butlast,axiom,
! [Xs_234: list_A1402527603le_alt,X_223: arrow_1389031998le_alt,Ys_44: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs_234 @ ( cons_A1142416030le_alt @ X_223 @ nil_Ar52383470le_alt ) )
= Ys_44 )
<=> ( ( Ys_44 != nil_Ar52383470le_alt )
& ( ( butlas1188620858le_alt @ Ys_44 )
= Xs_234 )
& ( ( last_A2130988295le_alt @ Ys_44 )
= X_223 ) ) ) ).
thf(fact_118_snoc__eq__iff__butlast,axiom,
! [Xs_234: list_nat,X_223: nat,Ys_44: list_nat] :
( ( ( append_nat @ Xs_234 @ ( cons_nat @ X_223 @ nil_nat ) )
= Ys_44 )
<=> ( ( Ys_44 != nil_nat )
& ( ( butlast_nat @ Ys_44 )
= Xs_234 )
& ( ( last_nat @ Ys_44 )
= X_223 ) ) ) ).
thf(fact_119_append__butlast__last__id,axiom,
! [Xs_233: list_o] :
( ( Xs_233 != nil_o )
=> ( ( append_o @ ( butlast_o @ Xs_233 ) @ ( cons_o @ ( last_o @ Xs_233 ) @ nil_o ) )
= Xs_233 ) ) ).
thf(fact_120_append__butlast__last__id,axiom,
! [Xs_233: list_A1402527603le_alt] :
( ( Xs_233 != nil_Ar52383470le_alt )
=> ( ( append1092755459le_alt @ ( butlas1188620858le_alt @ Xs_233 ) @ ( cons_A1142416030le_alt @ ( last_A2130988295le_alt @ Xs_233 ) @ nil_Ar52383470le_alt ) )
= Xs_233 ) ) ).
thf(fact_121_append__butlast__last__id,axiom,
! [Xs_233: list_nat] :
( ( Xs_233 != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs_233 ) @ ( cons_nat @ ( last_nat @ Xs_233 ) @ nil_nat ) )
= Xs_233 ) ) ).
thf(fact_122_append__eq__Cons__conv,axiom,
! [Ys_43: list_o,Zs_8: list_o,X_222: $o,Xs_232: list_o] :
( ( ( append_o @ Ys_43 @ Zs_8 )
= ( cons_o @ X_222 @ Xs_232 ) )
<=> ( ( ( Ys_43 = nil_o )
& ( Zs_8
= ( cons_o @ X_222 @ Xs_232 ) ) )
| ? [Ys_42: list_o] :
( ( Ys_43
= ( cons_o @ X_222 @ Ys_42 ) )
& ( ( append_o @ Ys_42 @ Zs_8 )
= Xs_232 ) ) ) ) ).
thf(fact_123_append__eq__Cons__conv,axiom,
! [Ys_43: list_A1402527603le_alt,Zs_8: list_A1402527603le_alt,X_222: arrow_1389031998le_alt,Xs_232: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Ys_43 @ Zs_8 )
= ( cons_A1142416030le_alt @ X_222 @ Xs_232 ) )
<=> ( ( ( Ys_43 = nil_Ar52383470le_alt )
& ( Zs_8
= ( cons_A1142416030le_alt @ X_222 @ Xs_232 ) ) )
| ? [Ys_42: list_A1402527603le_alt] :
( ( Ys_43
= ( cons_A1142416030le_alt @ X_222 @ Ys_42 ) )
& ( ( append1092755459le_alt @ Ys_42 @ Zs_8 )
= Xs_232 ) ) ) ) ).
thf(fact_124_append__eq__Cons__conv,axiom,
! [Ys_43: list_nat,Zs_8: list_nat,X_222: nat,Xs_232: list_nat] :
( ( ( append_nat @ Ys_43 @ Zs_8 )
= ( cons_nat @ X_222 @ Xs_232 ) )
<=> ( ( ( Ys_43 = nil_nat )
& ( Zs_8
= ( cons_nat @ X_222 @ Xs_232 ) ) )
| ? [Ys_42: list_nat] :
( ( Ys_43
= ( cons_nat @ X_222 @ Ys_42 ) )
& ( ( append_nat @ Ys_42 @ Zs_8 )
= Xs_232 ) ) ) ) ).
thf(fact_125_Cons__eq__append__conv,axiom,
! [X_221: $o,Xs_231: list_o,Ys_41: list_o,Zs_7: list_o] :
( ( ( cons_o @ X_221 @ Xs_231 )
= ( append_o @ Ys_41 @ Zs_7 ) )
<=> ( ( ( Ys_41 = nil_o )
& ( ( cons_o @ X_221 @ Xs_231 )
= Zs_7 ) )
| ? [Ys_42: list_o] :
( ( ( cons_o @ X_221 @ Ys_42 )
= Ys_41 )
& ( Xs_231
= ( append_o @ Ys_42 @ Zs_7 ) ) ) ) ) ).
thf(fact_126_Cons__eq__append__conv,axiom,
! [X_221: arrow_1389031998le_alt,Xs_231: list_A1402527603le_alt,Ys_41: list_A1402527603le_alt,Zs_7: list_A1402527603le_alt] :
( ( ( cons_A1142416030le_alt @ X_221 @ Xs_231 )
= ( append1092755459le_alt @ Ys_41 @ Zs_7 ) )
<=> ( ( ( Ys_41 = nil_Ar52383470le_alt )
& ( ( cons_A1142416030le_alt @ X_221 @ Xs_231 )
= Zs_7 ) )
| ? [Ys_42: list_A1402527603le_alt] :
( ( ( cons_A1142416030le_alt @ X_221 @ Ys_42 )
= Ys_41 )
& ( Xs_231
= ( append1092755459le_alt @ Ys_42 @ Zs_7 ) ) ) ) ) ).
thf(fact_127_Cons__eq__append__conv,axiom,
! [X_221: nat,Xs_231: list_nat,Ys_41: list_nat,Zs_7: list_nat] :
( ( ( cons_nat @ X_221 @ Xs_231 )
= ( append_nat @ Ys_41 @ Zs_7 ) )
<=> ( ( ( Ys_41 = nil_nat )
& ( ( cons_nat @ X_221 @ Xs_231 )
= Zs_7 ) )
| ? [Ys_42: list_nat] :
( ( ( cons_nat @ X_221 @ Ys_42 )
= Ys_41 )
& ( Xs_231
= ( append_nat @ Ys_42 @ Zs_7 ) ) ) ) ) ).
thf(fact_128_append1__eq__conv,axiom,
! [Xs_230: list_o,X_220: $o,Ys_40: list_o,Y_108: $o] :
( ( ( append_o @ Xs_230 @ ( cons_o @ X_220 @ nil_o ) )
= ( append_o @ Ys_40 @ ( cons_o @ Y_108 @ nil_o ) ) )
<=> ( ( Xs_230 = Ys_40 )
& ( X_220
<=> Y_108 ) ) ) ).
thf(fact_129_append1__eq__conv,axiom,
! [Xs_230: list_A1402527603le_alt,X_220: arrow_1389031998le_alt,Ys_40: list_A1402527603le_alt,Y_108: arrow_1389031998le_alt] :
( ( ( append1092755459le_alt @ Xs_230 @ ( cons_A1142416030le_alt @ X_220 @ nil_Ar52383470le_alt ) )
= ( append1092755459le_alt @ Ys_40 @ ( cons_A1142416030le_alt @ Y_108 @ nil_Ar52383470le_alt ) ) )
<=> ( ( Xs_230 = Ys_40 )
& ( X_220 = Y_108 ) ) ) ).
thf(fact_130_append1__eq__conv,axiom,
! [Xs_230: list_nat,X_220: nat,Ys_40: list_nat,Y_108: nat] :
( ( ( append_nat @ Xs_230 @ ( cons_nat @ X_220 @ nil_nat ) )
= ( append_nat @ Ys_40 @ ( cons_nat @ Y_108 @ nil_nat ) ) )
<=> ( ( Xs_230 = Ys_40 )
& ( X_220 = Y_108 ) ) ) ).
thf(fact_131_butlast__append,axiom,
! [Xs_229: list_A1402527603le_alt,Ys_39: list_A1402527603le_alt] :
( ( ( Ys_39 = nil_Ar52383470le_alt )
=> ( ( butlas1188620858le_alt @ ( append1092755459le_alt @ Xs_229 @ Ys_39 ) )
= ( butlas1188620858le_alt @ Xs_229 ) ) )
& ( ( Ys_39 != nil_Ar52383470le_alt )
=> ( ( butlas1188620858le_alt @ ( append1092755459le_alt @ Xs_229 @ Ys_39 ) )
= ( append1092755459le_alt @ Xs_229 @ ( butlas1188620858le_alt @ Ys_39 ) ) ) ) ) ).
thf(fact_132_butlast__append,axiom,
! [Xs_229: list_nat,Ys_39: list_nat] :
( ( ( Ys_39 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs_229 @ Ys_39 ) )
= ( butlast_nat @ Xs_229 ) ) )
& ( ( Ys_39 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs_229 @ Ys_39 ) )
= ( append_nat @ Xs_229 @ ( butlast_nat @ Ys_39 ) ) ) ) ) ).
thf(fact_133_sorted__butlast,axiom,
! [Xs_228: list_nat] :
( ( Xs_228 != nil_nat )
=> ( ( linorder_sorted_nat @ Xs_228 )
=> ( linorder_sorted_nat @ ( butlast_nat @ Xs_228 ) ) ) ) ).
thf(fact_134_last_Osimps,axiom,
! [X_219: $o,Xs_227: list_o] :
( ( last_o @ ( cons_o @ X_219 @ Xs_227 ) )
<=> ( ( ( Xs_227 = nil_o )
=> X_219 )
& ( ( Xs_227 != nil_o )
=> ( last_o @ Xs_227 ) ) ) ) ).
thf(fact_135_last_Osimps,axiom,
! [X_219: arrow_1389031998le_alt,Xs_227: list_A1402527603le_alt] :
( ( ( Xs_227 = nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( cons_A1142416030le_alt @ X_219 @ Xs_227 ) )
= X_219 ) )
& ( ( Xs_227 != nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( cons_A1142416030le_alt @ X_219 @ Xs_227 ) )
= ( last_A2130988295le_alt @ Xs_227 ) ) ) ) ).
thf(fact_136_last_Osimps,axiom,
! [X_219: nat,Xs_227: list_nat] :
( ( ( Xs_227 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_219 @ Xs_227 ) )
= X_219 ) )
& ( ( Xs_227 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_219 @ Xs_227 ) )
= ( last_nat @ Xs_227 ) ) ) ) ).
thf(fact_137_last__ConsR,axiom,
! [X_218: $o,Xs_226: list_o] :
( ( Xs_226 != nil_o )
=> ( ( last_o @ ( cons_o @ X_218 @ Xs_226 ) )
<=> ( last_o @ Xs_226 ) ) ) ).
thf(fact_138_last__ConsR,axiom,
! [X_218: arrow_1389031998le_alt,Xs_226: list_A1402527603le_alt] :
( ( Xs_226 != nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( cons_A1142416030le_alt @ X_218 @ Xs_226 ) )
= ( last_A2130988295le_alt @ Xs_226 ) ) ) ).
thf(fact_139_last__ConsR,axiom,
! [X_218: nat,Xs_226: list_nat] :
( ( Xs_226 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_218 @ Xs_226 ) )
= ( last_nat @ Xs_226 ) ) ) ).
thf(fact_140_rev__induct,axiom,
! [Xs_225: list_o,P_59: list_o > $o] :
( ( P_59 @ nil_o )
=> ( ! [X_4: $o,Xs_23: list_o] :
( ( P_59 @ Xs_23 )
=> ( P_59 @ ( append_o @ Xs_23 @ ( cons_o @ X_4 @ nil_o ) ) ) )
=> ( P_59 @ Xs_225 ) ) ) ).
thf(fact_141_rev__induct,axiom,
! [Xs_225: list_A1402527603le_alt,P_59: list_A1402527603le_alt > $o] :
( ( P_59 @ nil_Ar52383470le_alt )
=> ( ! [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt] :
( ( P_59 @ Xs_23 )
=> ( P_59 @ ( append1092755459le_alt @ Xs_23 @ ( cons_A1142416030le_alt @ X_4 @ nil_Ar52383470le_alt ) ) ) )
=> ( P_59 @ Xs_225 ) ) ) ).
thf(fact_142_rev__induct,axiom,
! [Xs_225: list_nat,P_59: list_nat > $o] :
( ( P_59 @ nil_nat )
=> ( ! [X_4: nat,Xs_23: list_nat] :
( ( P_59 @ Xs_23 )
=> ( P_59 @ ( append_nat @ Xs_23 @ ( cons_nat @ X_4 @ nil_nat ) ) ) )
=> ( P_59 @ Xs_225 ) ) ) ).
thf(fact_143_rev__cases,axiom,
! [Xs_224: list_o] :
( ( Xs_224 != nil_o )
=> ~ ! [Ys_7: list_o,Y_40: $o] :
( Xs_224
!= ( append_o @ Ys_7 @ ( cons_o @ Y_40 @ nil_o ) ) ) ) ).
thf(fact_144_rev__cases,axiom,
! [Xs_224: list_A1402527603le_alt] :
( ( Xs_224 != nil_Ar52383470le_alt )
=> ~ ! [Ys_7: list_A1402527603le_alt,Y_40: arrow_1389031998le_alt] :
( Xs_224
!= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ Y_40 @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_145_rev__cases,axiom,
! [Xs_224: list_nat] :
( ( Xs_224 != nil_nat )
=> ~ ! [Ys_7: list_nat,Y_40: nat] :
( Xs_224
!= ( append_nat @ Ys_7 @ ( cons_nat @ Y_40 @ nil_nat ) ) ) ) ).
thf(fact_146_equal__Nil__null,axiom,
! [Xs_223: list_A1402527603le_alt] :
( ( equal_1919384209le_alt @ Xs_223 @ nil_Ar52383470le_alt )
<=> ( null_A287154422le_alt @ Xs_223 ) ) ).
thf(fact_147_equal__Nil__null,axiom,
! [Xs_223: list_nat] :
( ( equal_equal_list_nat @ Xs_223 @ nil_nat )
<=> ( null_nat @ Xs_223 ) ) ).
thf(fact_148_rotate__simps,axiom,
! [X_217: $o,Xs_222: list_o] :
( ( ( rotate1249022267le_alt @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt )
& ( ( rotate1_o @ ( cons_o @ X_217 @ Xs_222 ) )
= ( append_o @ Xs_222 @ ( cons_o @ X_217 @ nil_o ) ) ) ) ).
thf(fact_149_rotate__simps,axiom,
! [X_217: $o,Xs_222: list_o] :
( ( ( rotate1_nat @ nil_nat )
= nil_nat )
& ( ( rotate1_o @ ( cons_o @ X_217 @ Xs_222 ) )
= ( append_o @ Xs_222 @ ( cons_o @ X_217 @ nil_o ) ) ) ) ).
thf(fact_150_rotate__simps,axiom,
! [X_217: arrow_1389031998le_alt,Xs_222: list_A1402527603le_alt] :
( ( ( rotate1_nat @ nil_nat )
= nil_nat )
& ( ( rotate1249022267le_alt @ ( cons_A1142416030le_alt @ X_217 @ Xs_222 ) )
= ( append1092755459le_alt @ Xs_222 @ ( cons_A1142416030le_alt @ X_217 @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_151_rotate__simps,axiom,
! [X_217: arrow_1389031998le_alt,Xs_222: list_A1402527603le_alt] :
( ( ( rotate1249022267le_alt @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt )
& ( ( rotate1249022267le_alt @ ( cons_A1142416030le_alt @ X_217 @ Xs_222 ) )
= ( append1092755459le_alt @ Xs_222 @ ( cons_A1142416030le_alt @ X_217 @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_152_rotate__simps,axiom,
! [X_217: nat,Xs_222: list_nat] :
( ( ( rotate1_nat @ nil_nat )
= nil_nat )
& ( ( rotate1_nat @ ( cons_nat @ X_217 @ Xs_222 ) )
= ( append_nat @ Xs_222 @ ( cons_nat @ X_217 @ nil_nat ) ) ) ) ).
thf(fact_153_rotate__simps,axiom,
! [X_217: nat,Xs_222: list_nat] :
( ( ( rotate1249022267le_alt @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt )
& ( ( rotate1_nat @ ( cons_nat @ X_217 @ Xs_222 ) )
= ( append_nat @ Xs_222 @ ( cons_nat @ X_217 @ nil_nat ) ) ) ) ).
thf(fact_154_not__distinct__decomp,axiom,
! [Ws: list_o] :
( ~ ( distinct_o @ Ws )
=> ? [Xs_23: list_o,Ys_7: list_o,Zs_5: list_o,Y_40: $o] :
( Ws
= ( append_o @ Xs_23 @ ( append_o @ ( cons_o @ Y_40 @ nil_o ) @ ( append_o @ Ys_7 @ ( append_o @ ( cons_o @ Y_40 @ nil_o ) @ Zs_5 ) ) ) ) ) ) ).
thf(fact_155_not__distinct__decomp,axiom,
! [Ws: list_A1402527603le_alt] :
( ~ ( distin1149997281le_alt @ Ws )
=> ? [Xs_23: list_A1402527603le_alt,Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt,Y_40: arrow_1389031998le_alt] :
( Ws
= ( append1092755459le_alt @ Xs_23 @ ( append1092755459le_alt @ ( cons_A1142416030le_alt @ Y_40 @ nil_Ar52383470le_alt ) @ ( append1092755459le_alt @ Ys_7 @ ( append1092755459le_alt @ ( cons_A1142416030le_alt @ Y_40 @ nil_Ar52383470le_alt ) @ Zs_5 ) ) ) ) ) ) ).
thf(fact_156_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs_23: list_nat,Ys_7: list_nat,Zs_5: list_nat,Y_40: nat] :
( Ws
= ( append_nat @ Xs_23 @ ( append_nat @ ( cons_nat @ Y_40 @ nil_nat ) @ ( append_nat @ Ys_7 @ ( append_nat @ ( cons_nat @ Y_40 @ nil_nat ) @ Zs_5 ) ) ) ) ) ) ).
thf(fact_157_replicate__append__same,axiom,
! [I_16: nat,X_216: $o] :
( ( append_o @ ( replicate_o @ I_16 @ X_216 ) @ ( cons_o @ X_216 @ nil_o ) )
= ( cons_o @ X_216 @ ( replicate_o @ I_16 @ X_216 ) ) ) ).
thf(fact_158_replicate__append__same,axiom,
! [I_16: nat,X_216: arrow_1389031998le_alt] :
( ( append1092755459le_alt @ ( replic277728168le_alt @ I_16 @ X_216 ) @ ( cons_A1142416030le_alt @ X_216 @ nil_Ar52383470le_alt ) )
= ( cons_A1142416030le_alt @ X_216 @ ( replic277728168le_alt @ I_16 @ X_216 ) ) ) ).
thf(fact_159_replicate__append__same,axiom,
! [I_16: nat,X_216: nat] :
( ( append_nat @ ( replicate_nat @ I_16 @ X_216 ) @ ( cons_nat @ X_216 @ nil_nat ) )
= ( cons_nat @ X_216 @ ( replicate_nat @ I_16 @ X_216 ) ) ) ).
thf(fact_160_rev_Osimps_I2_J,axiom,
! [X_215: $o,Xs_221: list_o] :
( ( rev_o @ ( cons_o @ X_215 @ Xs_221 ) )
= ( append_o @ ( rev_o @ Xs_221 ) @ ( cons_o @ X_215 @ nil_o ) ) ) ).
thf(fact_161_rev_Osimps_I2_J,axiom,
! [X_215: arrow_1389031998le_alt,Xs_221: list_A1402527603le_alt] :
( ( rev_Ar2020079950le_alt @ ( cons_A1142416030le_alt @ X_215 @ Xs_221 ) )
= ( append1092755459le_alt @ ( rev_Ar2020079950le_alt @ Xs_221 ) @ ( cons_A1142416030le_alt @ X_215 @ nil_Ar52383470le_alt ) ) ) ).
thf(fact_162_rev_Osimps_I2_J,axiom,
! [X_215: nat,Xs_221: list_nat] :
( ( rev_nat @ ( cons_nat @ X_215 @ Xs_221 ) )
= ( append_nat @ ( rev_nat @ Xs_221 ) @ ( cons_nat @ X_215 @ nil_nat ) ) ) ).
thf(fact_163_rev__eq__Cons__iff,axiom,
! [Xs_220: list_o,Y_107: $o,Ys_38: list_o] :
( ( ( rev_o @ Xs_220 )
= ( cons_o @ Y_107 @ Ys_38 ) )
<=> ( Xs_220
= ( append_o @ ( rev_o @ Ys_38 ) @ ( cons_o @ Y_107 @ nil_o ) ) ) ) ).
thf(fact_164_rev__eq__Cons__iff,axiom,
! [Xs_220: list_A1402527603le_alt,Y_107: arrow_1389031998le_alt,Ys_38: list_A1402527603le_alt] :
( ( ( rev_Ar2020079950le_alt @ Xs_220 )
= ( cons_A1142416030le_alt @ Y_107 @ Ys_38 ) )
<=> ( Xs_220
= ( append1092755459le_alt @ ( rev_Ar2020079950le_alt @ Ys_38 ) @ ( cons_A1142416030le_alt @ Y_107 @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_165_rev__eq__Cons__iff,axiom,
! [Xs_220: list_nat,Y_107: nat,Ys_38: list_nat] :
( ( ( rev_nat @ Xs_220 )
= ( cons_nat @ Y_107 @ Ys_38 ) )
<=> ( Xs_220
= ( append_nat @ ( rev_nat @ Ys_38 ) @ ( cons_nat @ Y_107 @ nil_nat ) ) ) ) ).
thf(fact_166_hd__append2,axiom,
! [Ys_37: list_A1402527603le_alt,Xs_219: list_A1402527603le_alt] :
( ( Xs_219 != nil_Ar52383470le_alt )
=> ( ( hd_Arr731872705le_alt @ ( append1092755459le_alt @ Xs_219 @ Ys_37 ) )
= ( hd_Arr731872705le_alt @ Xs_219 ) ) ) ).
thf(fact_167_hd__append2,axiom,
! [Ys_37: list_nat,Xs_219: list_nat] :
( ( Xs_219 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs_219 @ Ys_37 ) )
= ( hd_nat @ Xs_219 ) ) ) ).
thf(fact_168_hd__append,axiom,
! [Ys_36: list_A1402527603le_alt,Xs_218: list_A1402527603le_alt] :
( ( ( Xs_218 = nil_Ar52383470le_alt )
=> ( ( hd_Arr731872705le_alt @ ( append1092755459le_alt @ Xs_218 @ Ys_36 ) )
= ( hd_Arr731872705le_alt @ Ys_36 ) ) )
& ( ( Xs_218 != nil_Ar52383470le_alt )
=> ( ( hd_Arr731872705le_alt @ ( append1092755459le_alt @ Xs_218 @ Ys_36 ) )
= ( hd_Arr731872705le_alt @ Xs_218 ) ) ) ) ).
thf(fact_169_hd__append,axiom,
! [Ys_36: list_nat,Xs_218: list_nat] :
( ( ( Xs_218 = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs_218 @ Ys_36 ) )
= ( hd_nat @ Ys_36 ) ) )
& ( ( Xs_218 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs_218 @ Ys_36 ) )
= ( hd_nat @ Xs_218 ) ) ) ) ).
thf(fact_170_tl__append2,axiom,
! [Ys_35: list_A1402527603le_alt,Xs_217: list_A1402527603le_alt] :
( ( Xs_217 != nil_Ar52383470le_alt )
=> ( ( tl_Arr1379124165le_alt @ ( append1092755459le_alt @ Xs_217 @ Ys_35 ) )
= ( append1092755459le_alt @ ( tl_Arr1379124165le_alt @ Xs_217 ) @ Ys_35 ) ) ) ).
thf(fact_171_tl__append2,axiom,
! [Ys_35: list_nat,Xs_217: list_nat] :
( ( Xs_217 != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs_217 @ Ys_35 ) )
= ( append_nat @ ( tl_nat @ Xs_217 ) @ Ys_35 ) ) ) ).
thf(fact_172_rev__is__rev__conv,axiom,
! [Xs_216: list_nat,Ys_34: list_nat] :
( ( ( rev_nat @ Xs_216 )
= ( rev_nat @ Ys_34 ) )
<=> ( Xs_216 = Ys_34 ) ) ).
thf(fact_173_rev__is__rev__conv,axiom,
! [Xs_216: list_A1402527603le_alt,Ys_34: list_A1402527603le_alt] :
( ( ( rev_Ar2020079950le_alt @ Xs_216 )
= ( rev_Ar2020079950le_alt @ Ys_34 ) )
<=> ( Xs_216 = Ys_34 ) ) ).
thf(fact_174_rev__swap,axiom,
! [Xs_215: list_nat,Ys_33: list_nat] :
( ( ( rev_nat @ Xs_215 )
= Ys_33 )
<=> ( Xs_215
= ( rev_nat @ Ys_33 ) ) ) ).
thf(fact_175_rev__swap,axiom,
! [Xs_215: list_A1402527603le_alt,Ys_33: list_A1402527603le_alt] :
( ( ( rev_Ar2020079950le_alt @ Xs_215 )
= Ys_33 )
<=> ( Xs_215
= ( rev_Ar2020079950le_alt @ Ys_33 ) ) ) ).
thf(fact_176_mem__def,axiom,
! [X_214: arrow_1389031998le_alt,A_125: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ X_214 @ A_125 )
<=> ( A_125 @ X_214 ) ) ).
thf(fact_177_mem__def,axiom,
! [X_214: nat,A_125: nat > $o] :
( ( member_nat @ X_214 @ A_125 )
<=> ( A_125 @ X_214 ) ) ).
thf(fact_178_Collect__def,axiom,
! [P_58: nat > $o] :
( ( collect_nat @ P_58 )
= P_58 ) ).
thf(fact_179_rev__replicate,axiom,
! [N_68: nat,X_213: nat] :
( ( rev_nat @ ( replicate_nat @ N_68 @ X_213 ) )
= ( replicate_nat @ N_68 @ X_213 ) ) ).
thf(fact_180_rev__replicate,axiom,
! [N_68: nat,X_213: arrow_1389031998le_alt] :
( ( rev_Ar2020079950le_alt @ ( replic277728168le_alt @ N_68 @ X_213 ) )
= ( replic277728168le_alt @ N_68 @ X_213 ) ) ).
thf(fact_181_rev__rev__ident,axiom,
! [Xs_214: list_nat] :
( ( rev_nat @ ( rev_nat @ Xs_214 ) )
= Xs_214 ) ).
thf(fact_182_rev__rev__ident,axiom,
! [Xs_214: list_A1402527603le_alt] :
( ( rev_Ar2020079950le_alt @ ( rev_Ar2020079950le_alt @ Xs_214 ) )
= Xs_214 ) ).
thf(fact_183_equal__list__def,axiom,
! [X_212: list_nat,Y_106: list_nat] :
( ( equal_equal_list_nat @ X_212 @ Y_106 )
<=> ( X_212 = Y_106 ) ) ).
thf(fact_184_equal__list__def,axiom,
! [X_212: list_A1402527603le_alt,Y_106: list_A1402527603le_alt] :
( ( equal_1919384209le_alt @ X_212 @ Y_106 )
<=> ( X_212 = Y_106 ) ) ).
thf(fact_185_rev__is__Nil__conv,axiom,
! [Xs_213: list_A1402527603le_alt] :
( ( ( rev_Ar2020079950le_alt @ Xs_213 )
= nil_Ar52383470le_alt )
<=> ( Xs_213 = nil_Ar52383470le_alt ) ) ).
thf(fact_186_rev__is__Nil__conv,axiom,
! [Xs_213: list_nat] :
( ( ( rev_nat @ Xs_213 )
= nil_nat )
<=> ( Xs_213 = nil_nat ) ) ).
thf(fact_187_Nil__is__rev__conv,axiom,
! [Xs_212: list_A1402527603le_alt] :
( ( nil_Ar52383470le_alt
= ( rev_Ar2020079950le_alt @ Xs_212 ) )
<=> ( Xs_212 = nil_Ar52383470le_alt ) ) ).
thf(fact_188_Nil__is__rev__conv,axiom,
! [Xs_212: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs_212 ) )
<=> ( Xs_212 = nil_nat ) ) ).
thf(fact_189_rev_Osimps_I1_J,axiom,
( ( rev_Ar2020079950le_alt @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_190_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
thf(fact_191_rev__append,axiom,
! [Xs_211: list_A1402527603le_alt,Ys_32: list_A1402527603le_alt] :
( ( rev_Ar2020079950le_alt @ ( append1092755459le_alt @ Xs_211 @ Ys_32 ) )
= ( append1092755459le_alt @ ( rev_Ar2020079950le_alt @ Ys_32 ) @ ( rev_Ar2020079950le_alt @ Xs_211 ) ) ) ).
thf(fact_192_rev__append,axiom,
! [Xs_211: list_nat,Ys_32: list_nat] :
( ( rev_nat @ ( append_nat @ Xs_211 @ Ys_32 ) )
= ( append_nat @ ( rev_nat @ Ys_32 ) @ ( rev_nat @ Xs_211 ) ) ) ).
thf(fact_193_distinct__rev,axiom,
! [Xs_210: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ ( rev_Ar2020079950le_alt @ Xs_210 ) )
<=> ( distin1149997281le_alt @ Xs_210 ) ) ).
thf(fact_194_distinct__rev,axiom,
! [Xs_210: list_nat] :
( ( distinct_nat @ ( rev_nat @ Xs_210 ) )
<=> ( distinct_nat @ Xs_210 ) ) ).
thf(fact_195_last__rev,axiom,
! [Xs_209: list_A1402527603le_alt] :
( ( Xs_209 != nil_Ar52383470le_alt )
=> ( ( last_A2130988295le_alt @ ( rev_Ar2020079950le_alt @ Xs_209 ) )
= ( hd_Arr731872705le_alt @ Xs_209 ) ) ) ).
thf(fact_196_last__rev,axiom,
! [Xs_209: list_nat] :
( ( Xs_209 != nil_nat )
=> ( ( last_nat @ ( rev_nat @ Xs_209 ) )
= ( hd_nat @ Xs_209 ) ) ) ).
thf(fact_197_hd__rev,axiom,
! [Xs_208: list_A1402527603le_alt] :
( ( Xs_208 != nil_Ar52383470le_alt )
=> ( ( hd_Arr731872705le_alt @ ( rev_Ar2020079950le_alt @ Xs_208 ) )
= ( last_A2130988295le_alt @ Xs_208 ) ) ) ).
thf(fact_198_hd__rev,axiom,
! [Xs_208: list_nat] :
( ( Xs_208 != nil_nat )
=> ( ( hd_nat @ ( rev_nat @ Xs_208 ) )
= ( last_nat @ Xs_208 ) ) ) ).
thf(fact_199_append__replicate__commute,axiom,
! [N_67: nat,X_211: arrow_1389031998le_alt,K_9: nat] :
( ( append1092755459le_alt @ ( replic277728168le_alt @ N_67 @ X_211 ) @ ( replic277728168le_alt @ K_9 @ X_211 ) )
= ( append1092755459le_alt @ ( replic277728168le_alt @ K_9 @ X_211 ) @ ( replic277728168le_alt @ N_67 @ X_211 ) ) ) ).
thf(fact_200_append__replicate__commute,axiom,
! [N_67: nat,X_211: nat,K_9: nat] :
( ( append_nat @ ( replicate_nat @ N_67 @ X_211 ) @ ( replicate_nat @ K_9 @ X_211 ) )
= ( append_nat @ ( replicate_nat @ K_9 @ X_211 ) @ ( replicate_nat @ N_67 @ X_211 ) ) ) ).
thf(fact_201_rotate1__hd__tl,axiom,
! [Xs_207: list_o] :
( ( Xs_207 != nil_o )
=> ( ( rotate1_o @ Xs_207 )
= ( append_o @ ( tl_o @ Xs_207 ) @ ( cons_o @ ( hd_o @ Xs_207 ) @ nil_o ) ) ) ) ).
thf(fact_202_rotate1__hd__tl,axiom,
! [Xs_207: list_A1402527603le_alt] :
( ( Xs_207 != nil_Ar52383470le_alt )
=> ( ( rotate1249022267le_alt @ Xs_207 )
= ( append1092755459le_alt @ ( tl_Arr1379124165le_alt @ Xs_207 ) @ ( cons_A1142416030le_alt @ ( hd_Arr731872705le_alt @ Xs_207 ) @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_203_rotate1__hd__tl,axiom,
! [Xs_207: list_nat] :
( ( Xs_207 != nil_nat )
=> ( ( rotate1_nat @ Xs_207 )
= ( append_nat @ ( tl_nat @ Xs_207 ) @ ( cons_nat @ ( hd_nat @ Xs_207 ) @ nil_nat ) ) ) ) ).
thf(fact_204_tl_Osimps_I2_J,axiom,
! [X_210: $o,Xs_206: list_o] :
( ( tl_o @ ( cons_o @ X_210 @ Xs_206 ) )
= Xs_206 ) ).
thf(fact_205_tl_Osimps_I2_J,axiom,
! [X_210: arrow_1389031998le_alt,Xs_206: list_A1402527603le_alt] :
( ( tl_Arr1379124165le_alt @ ( cons_A1142416030le_alt @ X_210 @ Xs_206 ) )
= Xs_206 ) ).
thf(fact_206_tl_Osimps_I2_J,axiom,
! [X_210: nat,Xs_206: list_nat] :
( ( tl_nat @ ( cons_nat @ X_210 @ Xs_206 ) )
= Xs_206 ) ).
thf(fact_207_tl_Osimps_I1_J,axiom,
( ( tl_Arr1379124165le_alt @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_208_tl_Osimps_I1_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
thf(fact_209_hd_Osimps,axiom,
! [X_209: $o,Xs_205: list_o] :
( ( hd_o @ ( cons_o @ X_209 @ Xs_205 ) )
<=> X_209 ) ).
thf(fact_210_hd_Osimps,axiom,
! [X_209: arrow_1389031998le_alt,Xs_205: list_A1402527603le_alt] :
( ( hd_Arr731872705le_alt @ ( cons_A1142416030le_alt @ X_209 @ Xs_205 ) )
= X_209 ) ).
thf(fact_211_hd_Osimps,axiom,
! [X_209: nat,Xs_205: list_nat] :
( ( hd_nat @ ( cons_nat @ X_209 @ Xs_205 ) )
= X_209 ) ).
thf(fact_212_distinct__tl,axiom,
! [Xs_204: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_204 )
=> ( distin1149997281le_alt @ ( tl_Arr1379124165le_alt @ Xs_204 ) ) ) ).
thf(fact_213_distinct__tl,axiom,
! [Xs_204: list_nat] :
( ( distinct_nat @ Xs_204 )
=> ( distinct_nat @ ( tl_nat @ Xs_204 ) ) ) ).
thf(fact_214_sorted__tl,axiom,
! [Xs_203: list_nat] :
( ( linorder_sorted_nat @ Xs_203 )
=> ( linorder_sorted_nat @ ( tl_nat @ Xs_203 ) ) ) ).
thf(fact_215_rotate1__is__Nil__conv,axiom,
! [Xs_202: list_A1402527603le_alt] :
( ( ( rotate1249022267le_alt @ Xs_202 )
= nil_Ar52383470le_alt )
<=> ( Xs_202 = nil_Ar52383470le_alt ) ) ).
thf(fact_216_rotate1__is__Nil__conv,axiom,
! [Xs_202: list_nat] :
( ( ( rotate1_nat @ Xs_202 )
= nil_nat )
<=> ( Xs_202 = nil_nat ) ) ).
thf(fact_217_distinct1__rotate,axiom,
! [Xs_201: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ ( rotate1249022267le_alt @ Xs_201 ) )
<=> ( distin1149997281le_alt @ Xs_201 ) ) ).
thf(fact_218_distinct1__rotate,axiom,
! [Xs_201: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs_201 ) )
<=> ( distinct_nat @ Xs_201 ) ) ).
thf(fact_219_singleton__rev__conv,axiom,
! [X_208: $o,Xs_200: list_o] :
( ( ( cons_o @ X_208 @ nil_o )
= ( rev_o @ Xs_200 ) )
<=> ( Xs_200
= ( cons_o @ X_208 @ nil_o ) ) ) ).
thf(fact_220_singleton__rev__conv,axiom,
! [X_208: arrow_1389031998le_alt,Xs_200: list_A1402527603le_alt] :
( ( ( cons_A1142416030le_alt @ X_208 @ nil_Ar52383470le_alt )
= ( rev_Ar2020079950le_alt @ Xs_200 ) )
<=> ( Xs_200
= ( cons_A1142416030le_alt @ X_208 @ nil_Ar52383470le_alt ) ) ) ).
thf(fact_221_singleton__rev__conv,axiom,
! [X_208: nat,Xs_200: list_nat] :
( ( ( cons_nat @ X_208 @ nil_nat )
= ( rev_nat @ Xs_200 ) )
<=> ( Xs_200
= ( cons_nat @ X_208 @ nil_nat ) ) ) ).
thf(fact_222_rev__singleton__conv,axiom,
! [Xs_199: list_o,X_207: $o] :
( ( ( rev_o @ Xs_199 )
= ( cons_o @ X_207 @ nil_o ) )
<=> ( Xs_199
= ( cons_o @ X_207 @ nil_o ) ) ) ).
thf(fact_223_rev__singleton__conv,axiom,
! [Xs_199: list_A1402527603le_alt,X_207: arrow_1389031998le_alt] :
( ( ( rev_Ar2020079950le_alt @ Xs_199 )
= ( cons_A1142416030le_alt @ X_207 @ nil_Ar52383470le_alt ) )
<=> ( Xs_199
= ( cons_A1142416030le_alt @ X_207 @ nil_Ar52383470le_alt ) ) ) ).
thf(fact_224_rev__singleton__conv,axiom,
! [Xs_199: list_nat,X_207: nat] :
( ( ( rev_nat @ Xs_199 )
= ( cons_nat @ X_207 @ nil_nat ) )
<=> ( Xs_199
= ( cons_nat @ X_207 @ nil_nat ) ) ) ).
thf(fact_225_replicate__app__Cons__same,axiom,
! [N_66: nat,X_206: $o,Xs_198: list_o] :
( ( append_o @ ( replicate_o @ N_66 @ X_206 ) @ ( cons_o @ X_206 @ Xs_198 ) )
= ( cons_o @ X_206 @ ( append_o @ ( replicate_o @ N_66 @ X_206 ) @ Xs_198 ) ) ) ).
thf(fact_226_replicate__app__Cons__same,axiom,
! [N_66: nat,X_206: arrow_1389031998le_alt,Xs_198: list_A1402527603le_alt] :
( ( append1092755459le_alt @ ( replic277728168le_alt @ N_66 @ X_206 ) @ ( cons_A1142416030le_alt @ X_206 @ Xs_198 ) )
= ( cons_A1142416030le_alt @ X_206 @ ( append1092755459le_alt @ ( replic277728168le_alt @ N_66 @ X_206 ) @ Xs_198 ) ) ) ).
thf(fact_227_replicate__app__Cons__same,axiom,
! [N_66: nat,X_206: nat,Xs_198: list_nat] :
( ( append_nat @ ( replicate_nat @ N_66 @ X_206 ) @ ( cons_nat @ X_206 @ Xs_198 ) )
= ( cons_nat @ X_206 @ ( append_nat @ ( replicate_nat @ N_66 @ X_206 ) @ Xs_198 ) ) ) ).
thf(fact_228_equal,axiom,
equal_equal_list_nat = fequal_list_nat ).
thf(fact_229_equal,axiom,
equal_1919384209le_alt = fequal68576820le_alt ).
thf(fact_230_equal__refl,axiom,
! [X_205: list_nat] : ( equal_equal_list_nat @ X_205 @ X_205 ) ).
thf(fact_231_equal__refl,axiom,
! [X_205: list_A1402527603le_alt] : ( equal_1919384209le_alt @ X_205 @ X_205 ) ).
thf(fact_232_equal__eq,axiom,
! [X_204: list_nat,Y_105: list_nat] :
( ( equal_equal_list_nat @ X_204 @ Y_105 )
<=> ( X_204 = Y_105 ) ) ).
thf(fact_233_equal__eq,axiom,
! [X_204: list_A1402527603le_alt,Y_105: list_A1402527603le_alt] :
( ( equal_1919384209le_alt @ X_204 @ Y_105 )
<=> ( X_204 = Y_105 ) ) ).
thf(fact_234_eq__equal,axiom,
fequal_list_nat = equal_equal_list_nat ).
thf(fact_235_eq__equal,axiom,
fequal68576820le_alt = equal_1919384209le_alt ).
thf(fact_236_hd__dropWhile,axiom,
! [P_57: arrow_1389031998le_alt > $o,Xs_197: list_A1402527603le_alt] :
( ( ( dropWh82971279le_alt @ P_57 @ Xs_197 )
!= nil_Ar52383470le_alt )
=> ~ ( P_57 @ ( hd_Arr731872705le_alt @ ( dropWh82971279le_alt @ P_57 @ Xs_197 ) ) ) ) ).
thf(fact_237_hd__dropWhile,axiom,
! [P_57: nat > $o,Xs_197: list_nat] :
( ( ( dropWhile_nat @ P_57 @ Xs_197 )
!= nil_nat )
=> ~ ( P_57 @ ( hd_nat @ ( dropWhile_nat @ P_57 @ Xs_197 ) ) ) ) ).
thf(fact_238_list__ex1__simps_I1_J,axiom,
! [P_56: arrow_1389031998le_alt > $o] :
~ ( list_e1152917937le_alt @ P_56 @ nil_Ar52383470le_alt ) ).
thf(fact_239_list__ex1__simps_I1_J,axiom,
! [P_56: nat > $o] :
~ ( list_ex1_nat @ P_56 @ nil_nat ) ).
thf(fact_240_member__rec_I2_J,axiom,
! [Y_104: arrow_1389031998le_alt] :
~ ( member271157507le_alt @ nil_Ar52383470le_alt @ Y_104 ) ).
thf(fact_241_member__rec_I2_J,axiom,
! [Y_104: nat] :
~ ( member_nat_1 @ nil_nat @ Y_104 ) ).
thf(fact_242_insert,axiom,
! [Y_103: $o,X_203: $o,Xs_196: list_o] :
( ( listMem_o @ X_203 @ Xs_196 )
=> ( listMem_o @ X_203 @ ( cons_o @ Y_103 @ Xs_196 ) ) ) ).
thf(fact_243_insert,axiom,
! [Y_103: arrow_1389031998le_alt,X_203: arrow_1389031998le_alt,Xs_196: list_A1402527603le_alt] :
( ( listMe991720154le_alt @ X_203 @ Xs_196 )
=> ( listMe991720154le_alt @ X_203 @ ( cons_A1142416030le_alt @ Y_103 @ Xs_196 ) ) ) ).
thf(fact_244_insert,axiom,
! [Y_103: nat,X_203: nat,Xs_196: list_nat] :
( ( listMem_nat @ X_203 @ Xs_196 )
=> ( listMem_nat @ X_203 @ ( cons_nat @ Y_103 @ Xs_196 ) ) ) ).
thf(fact_245_member__rec_I1_J,axiom,
! [X_202: $o,Xs_195: list_o,Y_102: $o] :
( ( member_o_1 @ ( cons_o @ X_202 @ Xs_195 ) @ Y_102 )
<=> ( ( X_202
<=> Y_102 )
| ( member_o_1 @ Xs_195 @ Y_102 ) ) ) ).
thf(fact_246_member__rec_I1_J,axiom,
! [X_202: arrow_1389031998le_alt,Xs_195: list_A1402527603le_alt,Y_102: arrow_1389031998le_alt] :
( ( member271157507le_alt @ ( cons_A1142416030le_alt @ X_202 @ Xs_195 ) @ Y_102 )
<=> ( ( X_202 = Y_102 )
| ( member271157507le_alt @ Xs_195 @ Y_102 ) ) ) ).
thf(fact_247_member__rec_I1_J,axiom,
! [X_202: nat,Xs_195: list_nat,Y_102: nat] :
( ( member_nat_1 @ ( cons_nat @ X_202 @ Xs_195 ) @ Y_102 )
<=> ( ( X_202 = Y_102 )
| ( member_nat_1 @ Xs_195 @ Y_102 ) ) ) ).
thf(fact_248_elem,axiom,
! [X_201: $o,Xs_194: list_o] : ( listMem_o @ X_201 @ ( cons_o @ X_201 @ Xs_194 ) ) ).
thf(fact_249_elem,axiom,
! [X_201: arrow_1389031998le_alt,Xs_194: list_A1402527603le_alt] : ( listMe991720154le_alt @ X_201 @ ( cons_A1142416030le_alt @ X_201 @ Xs_194 ) ) ).
thf(fact_250_elem,axiom,
! [X_201: nat,Xs_194: list_nat] : ( listMem_nat @ X_201 @ ( cons_nat @ X_201 @ Xs_194 ) ) ).
thf(fact_251_dropWhile_Osimps_I2_J,axiom,
! [Xs_193: list_o,P_55: $o > $o,X_200: $o] :
( ( ( P_55 @ X_200 )
=> ( ( dropWhile_o @ P_55 @ ( cons_o @ X_200 @ Xs_193 ) )
= ( dropWhile_o @ P_55 @ Xs_193 ) ) )
& ( ~ ( P_55 @ X_200 )
=> ( ( dropWhile_o @ P_55 @ ( cons_o @ X_200 @ Xs_193 ) )
= ( cons_o @ X_200 @ Xs_193 ) ) ) ) ).
thf(fact_252_dropWhile_Osimps_I2_J,axiom,
! [Xs_193: list_A1402527603le_alt,P_55: arrow_1389031998le_alt > $o,X_200: arrow_1389031998le_alt] :
( ( ( P_55 @ X_200 )
=> ( ( dropWh82971279le_alt @ P_55 @ ( cons_A1142416030le_alt @ X_200 @ Xs_193 ) )
= ( dropWh82971279le_alt @ P_55 @ Xs_193 ) ) )
& ( ~ ( P_55 @ X_200 )
=> ( ( dropWh82971279le_alt @ P_55 @ ( cons_A1142416030le_alt @ X_200 @ Xs_193 ) )
= ( cons_A1142416030le_alt @ X_200 @ Xs_193 ) ) ) ) ).
thf(fact_253_dropWhile_Osimps_I2_J,axiom,
! [Xs_193: list_nat,P_55: nat > $o,X_200: nat] :
( ( ( P_55 @ X_200 )
=> ( ( dropWhile_nat @ P_55 @ ( cons_nat @ X_200 @ Xs_193 ) )
= ( dropWhile_nat @ P_55 @ Xs_193 ) ) )
& ( ~ ( P_55 @ X_200 )
=> ( ( dropWhile_nat @ P_55 @ ( cons_nat @ X_200 @ Xs_193 ) )
= ( cons_nat @ X_200 @ Xs_193 ) ) ) ) ).
thf(fact_254_dropWhile_Osimps_I1_J,axiom,
! [P_54: arrow_1389031998le_alt > $o] :
( ( dropWh82971279le_alt @ P_54 @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_255_dropWhile_Osimps_I1_J,axiom,
! [P_54: nat > $o] :
( ( dropWhile_nat @ P_54 @ nil_nat )
= nil_nat ) ).
thf(fact_256_distinct__dropWhile,axiom,
! [P_53: arrow_1389031998le_alt > $o,Xs_192: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_192 )
=> ( distin1149997281le_alt @ ( dropWh82971279le_alt @ P_53 @ Xs_192 ) ) ) ).
thf(fact_257_distinct__dropWhile,axiom,
! [P_53: nat > $o,Xs_192: list_nat] :
( ( distinct_nat @ Xs_192 )
=> ( distinct_nat @ ( dropWhile_nat @ P_53 @ Xs_192 ) ) ) ).
thf(fact_258_sorted__dropWhile,axiom,
! [P_52: nat > $o,Xs_191: list_nat] :
( ( linorder_sorted_nat @ Xs_191 )
=> ( linorder_sorted_nat @ ( dropWhile_nat @ P_52 @ Xs_191 ) ) ) ).
thf(fact_259_dropWhile__eq__Cons__conv,axiom,
! [P_51: $o > $o,Xs_190: list_o,Y_101: $o,Ys_31: list_o] :
( ( ( dropWhile_o @ P_51 @ Xs_190 )
= ( cons_o @ Y_101 @ Ys_31 ) )
<=> ( ( Xs_190
= ( append_o @ ( takeWhile_o @ P_51 @ Xs_190 ) @ ( cons_o @ Y_101 @ Ys_31 ) ) )
& ~ ( P_51 @ Y_101 ) ) ) ).
thf(fact_260_dropWhile__eq__Cons__conv,axiom,
! [P_51: arrow_1389031998le_alt > $o,Xs_190: list_A1402527603le_alt,Y_101: arrow_1389031998le_alt,Ys_31: list_A1402527603le_alt] :
( ( ( dropWh82971279le_alt @ P_51 @ Xs_190 )
= ( cons_A1142416030le_alt @ Y_101 @ Ys_31 ) )
<=> ( ( Xs_190
= ( append1092755459le_alt @ ( takeWh462480871le_alt @ P_51 @ Xs_190 ) @ ( cons_A1142416030le_alt @ Y_101 @ Ys_31 ) ) )
& ~ ( P_51 @ Y_101 ) ) ) ).
thf(fact_261_dropWhile__eq__Cons__conv,axiom,
! [P_51: nat > $o,Xs_190: list_nat,Y_101: nat,Ys_31: list_nat] :
( ( ( dropWhile_nat @ P_51 @ Xs_190 )
= ( cons_nat @ Y_101 @ Ys_31 ) )
<=> ( ( Xs_190
= ( append_nat @ ( takeWhile_nat @ P_51 @ Xs_190 ) @ ( cons_nat @ Y_101 @ Ys_31 ) ) )
& ~ ( P_51 @ Y_101 ) ) ) ).
thf(fact_262_ListMem_Osimps,axiom,
! [A1_1: $o,A2_1: list_o] :
( ( listMem_o @ A1_1 @ A2_1 )
<=> ( ? [X_4: $o,Xs_23: list_o] :
( ( A1_1
<=> X_4 )
& ( A2_1
= ( cons_o @ X_4 @ Xs_23 ) ) )
| ? [X_4: $o,Xs_23: list_o,Y_40: $o] :
( ( A1_1
<=> X_4 )
& ( A2_1
= ( cons_o @ Y_40 @ Xs_23 ) )
& ( listMem_o @ X_4 @ Xs_23 ) ) ) ) ).
thf(fact_263_ListMem_Osimps,axiom,
! [A1_1: arrow_1389031998le_alt,A2_1: list_A1402527603le_alt] :
( ( listMe991720154le_alt @ A1_1 @ A2_1 )
<=> ( ? [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt] :
( ( A1_1 = X_4 )
& ( A2_1
= ( cons_A1142416030le_alt @ X_4 @ Xs_23 ) ) )
| ? [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt,Y_40: arrow_1389031998le_alt] :
( ( A1_1 = X_4 )
& ( A2_1
= ( cons_A1142416030le_alt @ Y_40 @ Xs_23 ) )
& ( listMe991720154le_alt @ X_4 @ Xs_23 ) ) ) ) ).
thf(fact_264_ListMem_Osimps,axiom,
! [A1_1: nat,A2_1: list_nat] :
( ( listMem_nat @ A1_1 @ A2_1 )
<=> ( ? [X_4: nat,Xs_23: list_nat] :
( ( A1_1 = X_4 )
& ( A2_1
= ( cons_nat @ X_4 @ Xs_23 ) ) )
| ? [X_4: nat,Xs_23: list_nat,Y_40: nat] :
( ( A1_1 = X_4 )
& ( A2_1
= ( cons_nat @ Y_40 @ Xs_23 ) )
& ( listMem_nat @ X_4 @ Xs_23 ) ) ) ) ).
thf(fact_265_takeWhile__dropWhile__id,axiom,
! [P_50: arrow_1389031998le_alt > $o,Xs_189: list_A1402527603le_alt] :
( ( append1092755459le_alt @ ( takeWh462480871le_alt @ P_50 @ Xs_189 ) @ ( dropWh82971279le_alt @ P_50 @ Xs_189 ) )
= Xs_189 ) ).
thf(fact_266_takeWhile__dropWhile__id,axiom,
! [P_50: nat > $o,Xs_189: list_nat] :
( ( append_nat @ ( takeWhile_nat @ P_50 @ Xs_189 ) @ ( dropWhile_nat @ P_50 @ Xs_189 ) )
= Xs_189 ) ).
thf(fact_267_hd__replicate,axiom,
! [X_199: arrow_1389031998le_alt,N_65: nat] :
( ( N_65 != zero_zero_nat )
=> ( ( hd_Arr731872705le_alt @ ( replic277728168le_alt @ N_65 @ X_199 ) )
= X_199 ) ) ).
thf(fact_268_hd__replicate,axiom,
! [X_199: nat,N_65: nat] :
( ( N_65 != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N_65 @ X_199 ) )
= X_199 ) ) ).
thf(fact_269_last__replicate,axiom,
! [X_198: nat,N_64: nat] :
( ( N_64 != zero_zero_nat )
=> ( ( last_nat @ ( replicate_nat @ N_64 @ X_198 ) )
= X_198 ) ) ).
thf(fact_270_last__replicate,axiom,
! [X_198: arrow_1389031998le_alt,N_64: nat] :
( ( N_64 != zero_zero_nat )
=> ( ( last_A2130988295le_alt @ ( replic277728168le_alt @ N_64 @ X_198 ) )
= X_198 ) ) ).
thf(fact_271_list__induct2_H,axiom,
! [Xs_188: list_A1402527603le_alt,Ys_30: list_o,P_49: list_A1402527603le_alt > list_o > $o] :
( ( P_49 @ nil_Ar52383470le_alt @ nil_o )
=> ( ! [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt] : ( P_49 @ ( cons_A1142416030le_alt @ X_4 @ Xs_23 ) @ nil_o )
=> ( ! [Y_40: $o,Ys_7: list_o] : ( P_49 @ nil_Ar52383470le_alt @ ( cons_o @ Y_40 @ Ys_7 ) )
=> ( ! [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt,Y_40: $o,Ys_7: list_o] :
( ( P_49 @ Xs_23 @ Ys_7 )
=> ( P_49 @ ( cons_A1142416030le_alt @ X_4 @ Xs_23 ) @ ( cons_o @ Y_40 @ Ys_7 ) ) )
=> ( P_49 @ Xs_188 @ Ys_30 ) ) ) ) ) ).
thf(fact_272_list__induct2_H,axiom,
! [Xs_188: list_nat,Ys_30: list_o,P_49: list_nat > list_o > $o] :
( ( P_49 @ nil_nat @ nil_o )
=> ( ! [X_4: nat,Xs_23: list_nat] : ( P_49 @ ( cons_nat @ X_4 @ Xs_23 ) @ nil_o )
=> ( ! [Y_40: $o,Ys_7: list_o] : ( P_49 @ nil_nat @ ( cons_o @ Y_40 @ Ys_7 ) )
=> ( ! [X_4: nat,Xs_23: list_nat,Y_40: $o,Ys_7: list_o] :
( ( P_49 @ Xs_23 @ Ys_7 )
=> ( P_49 @ ( cons_nat @ X_4 @ Xs_23 ) @ ( cons_o @ Y_40 @ Ys_7 ) ) )
=> ( P_49 @ Xs_188 @ Ys_30 ) ) ) ) ) ).
thf(fact_273_list__induct2_H,axiom,
! [Xs_188: list_o,Ys_30: list_A1402527603le_alt,P_49: list_o > list_A1402527603le_alt > $o] :
( ( P_49 @ nil_o @ nil_Ar52383470le_alt )
=> ( ! [X_4: $o,Xs_23: list_o] : ( P_49 @ ( cons_o @ X_4 @ Xs_23 ) @ nil_Ar52383470le_alt )
=> ( ! [Y_40: arrow_1389031998le_alt,Ys_7: list_A1402527603le_alt] : ( P_49 @ nil_o @ ( cons_A1142416030le_alt @ Y_40 @ Ys_7 ) )
=> ( ! [X_4: $o,Xs_23: list_o,Y_40: arrow_1389031998le_alt,Ys_7: list_A1402527603le_alt] :
( ( P_49 @ Xs_23 @ Ys_7 )
=> ( P_49 @ ( cons_o @ X_4 @ Xs_23 ) @ ( cons_A1142416030le_alt @ Y_40 @ Ys_7 ) ) )
=> ( P_49 @ Xs_188 @ Ys_30 ) ) ) ) ) ).
thf(fact_274_list__induct2_H,axiom,
! [Xs_188: list_o,Ys_30: list_nat,P_49: list_o > list_nat > $o] :
( ( P_49 @ nil_o @ nil_nat )
=> ( ! [X_4: $o,Xs_23: list_o] : ( P_49 @ ( cons_o @ X_4 @ Xs_23 ) @ nil_nat )
=> ( ! [Y_40: nat,Ys_7: list_nat] : ( P_49 @ nil_o @ ( cons_nat @ Y_40 @ Ys_7 ) )
=> ( ! [X_4: $o,Xs_23: list_o,Y_40: nat,Ys_7: list_nat] :
( ( P_49 @ Xs_23 @ Ys_7 )
=> ( P_49 @ ( cons_o @ X_4 @ Xs_23 ) @ ( cons_nat @ Y_40 @ Ys_7 ) ) )
=> ( P_49 @ Xs_188 @ Ys_30 ) ) ) ) ) ).
thf(fact_275_listrelp_ONil,axiom,
! [R_3: arrow_1389031998le_alt > arrow_1389031998le_alt > $o] : ( listre2126835016le_alt @ R_3 @ nil_Ar52383470le_alt @ nil_Ar52383470le_alt ) ).
thf(fact_276_listrelp_ONil,axiom,
! [R_3: nat > nat > $o] : ( listrelp_nat @ R_3 @ nil_nat @ nil_nat ) ).
thf(fact_277_listrelp_OCons,axiom,
! [Xs_187: list_o,Ys_29: list_o,R_2: $o > $o > $o,X_197: $o,Y_100: $o] :
( ( R_2 @ X_197 @ Y_100 )
=> ( ( listrelp_o @ R_2 @ Xs_187 @ Ys_29 )
=> ( listrelp_o @ R_2 @ ( cons_o @ X_197 @ Xs_187 ) @ ( cons_o @ Y_100 @ Ys_29 ) ) ) ) ).
thf(fact_278_listrelp_OCons,axiom,
! [Xs_187: list_A1402527603le_alt,Ys_29: list_A1402527603le_alt,R_2: arrow_1389031998le_alt > arrow_1389031998le_alt > $o,X_197: arrow_1389031998le_alt,Y_100: arrow_1389031998le_alt] :
( ( R_2 @ X_197 @ Y_100 )
=> ( ( listre2126835016le_alt @ R_2 @ Xs_187 @ Ys_29 )
=> ( listre2126835016le_alt @ R_2 @ ( cons_A1142416030le_alt @ X_197 @ Xs_187 ) @ ( cons_A1142416030le_alt @ Y_100 @ Ys_29 ) ) ) ) ).
thf(fact_279_listrelp_OCons,axiom,
! [Xs_187: list_nat,Ys_29: list_nat,R_2: nat > nat > $o,X_197: nat,Y_100: nat] :
( ( R_2 @ X_197 @ Y_100 )
=> ( ( listrelp_nat @ R_2 @ Xs_187 @ Ys_29 )
=> ( listrelp_nat @ R_2 @ ( cons_nat @ X_197 @ Xs_187 ) @ ( cons_nat @ Y_100 @ Ys_29 ) ) ) ) ).
thf(fact_280_takeWhile_Osimps_I1_J,axiom,
! [P_48: arrow_1389031998le_alt > $o] :
( ( takeWh462480871le_alt @ P_48 @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_281_takeWhile_Osimps_I1_J,axiom,
! [P_48: nat > $o] :
( ( takeWhile_nat @ P_48 @ nil_nat )
= nil_nat ) ).
thf(fact_282_replicate__eq__replicate,axiom,
! [M_12: nat,X_196: nat,N_63: nat,Y_99: nat] :
( ( ( replicate_nat @ M_12 @ X_196 )
= ( replicate_nat @ N_63 @ Y_99 ) )
<=> ( ( M_12 = N_63 )
& ( ( M_12 != zero_zero_nat )
=> ( X_196 = Y_99 ) ) ) ) ).
thf(fact_283_replicate__eq__replicate,axiom,
! [M_12: nat,X_196: arrow_1389031998le_alt,N_63: nat,Y_99: arrow_1389031998le_alt] :
( ( ( replic277728168le_alt @ M_12 @ X_196 )
= ( replic277728168le_alt @ N_63 @ Y_99 ) )
<=> ( ( M_12 = N_63 )
& ( ( M_12 != zero_zero_nat )
=> ( X_196 = Y_99 ) ) ) ) ).
thf(fact_284_distinct__takeWhile,axiom,
! [P_47: arrow_1389031998le_alt > $o,Xs_186: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_186 )
=> ( distin1149997281le_alt @ ( takeWh462480871le_alt @ P_47 @ Xs_186 ) ) ) ).
thf(fact_285_distinct__takeWhile,axiom,
! [P_47: nat > $o,Xs_186: list_nat] :
( ( distinct_nat @ Xs_186 )
=> ( distinct_nat @ ( takeWhile_nat @ P_47 @ Xs_186 ) ) ) ).
thf(fact_286_sorted__takeWhile,axiom,
! [P_46: nat > $o,Xs_185: list_nat] :
( ( linorder_sorted_nat @ Xs_185 )
=> ( linorder_sorted_nat @ ( takeWhile_nat @ P_46 @ Xs_185 ) ) ) ).
thf(fact_287_replicate__0,axiom,
! [X_195: arrow_1389031998le_alt] :
( ( replic277728168le_alt @ zero_zero_nat @ X_195 )
= nil_Ar52383470le_alt ) ).
thf(fact_288_replicate__0,axiom,
! [X_195: nat] :
( ( replicate_nat @ zero_zero_nat @ X_195 )
= nil_nat ) ).
thf(fact_289_empty__replicate,axiom,
! [N_62: nat,X_194: arrow_1389031998le_alt] :
( ( nil_Ar52383470le_alt
= ( replic277728168le_alt @ N_62 @ X_194 ) )
<=> ( N_62 = zero_zero_nat ) ) ).
thf(fact_290_empty__replicate,axiom,
! [N_62: nat,X_194: nat] :
( ( nil_nat
= ( replicate_nat @ N_62 @ X_194 ) )
<=> ( N_62 = zero_zero_nat ) ) ).
thf(fact_291_replicate__empty,axiom,
! [N_61: nat,X_193: arrow_1389031998le_alt] :
( ( ( replic277728168le_alt @ N_61 @ X_193 )
= nil_Ar52383470le_alt )
<=> ( N_61 = zero_zero_nat ) ) ).
thf(fact_292_replicate__empty,axiom,
! [N_61: nat,X_193: nat] :
( ( ( replicate_nat @ N_61 @ X_193 )
= nil_nat )
<=> ( N_61 = zero_zero_nat ) ) ).
thf(fact_293_takeWhile_Osimps_I2_J,axiom,
! [Xs_184: list_o,P_45: $o > $o,X_192: $o] :
( ( ( P_45 @ X_192 )
=> ( ( takeWhile_o @ P_45 @ ( cons_o @ X_192 @ Xs_184 ) )
= ( cons_o @ X_192 @ ( takeWhile_o @ P_45 @ Xs_184 ) ) ) )
& ( ~ ( P_45 @ X_192 )
=> ( ( takeWhile_o @ P_45 @ ( cons_o @ X_192 @ Xs_184 ) )
= nil_o ) ) ) ).
thf(fact_294_takeWhile_Osimps_I2_J,axiom,
! [Xs_184: list_A1402527603le_alt,P_45: arrow_1389031998le_alt > $o,X_192: arrow_1389031998le_alt] :
( ( ( P_45 @ X_192 )
=> ( ( takeWh462480871le_alt @ P_45 @ ( cons_A1142416030le_alt @ X_192 @ Xs_184 ) )
= ( cons_A1142416030le_alt @ X_192 @ ( takeWh462480871le_alt @ P_45 @ Xs_184 ) ) ) )
& ( ~ ( P_45 @ X_192 )
=> ( ( takeWh462480871le_alt @ P_45 @ ( cons_A1142416030le_alt @ X_192 @ Xs_184 ) )
= nil_Ar52383470le_alt ) ) ) ).
thf(fact_295_takeWhile_Osimps_I2_J,axiom,
! [Xs_184: list_nat,P_45: nat > $o,X_192: nat] :
( ( ( P_45 @ X_192 )
=> ( ( takeWhile_nat @ P_45 @ ( cons_nat @ X_192 @ Xs_184 ) )
= ( cons_nat @ X_192 @ ( takeWhile_nat @ P_45 @ Xs_184 ) ) ) )
& ( ~ ( P_45 @ X_192 )
=> ( ( takeWhile_nat @ P_45 @ ( cons_nat @ X_192 @ Xs_184 ) )
= nil_nat ) ) ) ).
thf(fact_296_takeWhile__tail,axiom,
! [Xs_183: list_o,L_7: list_o,P_44: $o > $o,X_191: $o] :
( ~ ( P_44 @ X_191 )
=> ( ( takeWhile_o @ P_44 @ ( append_o @ Xs_183 @ ( cons_o @ X_191 @ L_7 ) ) )
= ( takeWhile_o @ P_44 @ Xs_183 ) ) ) ).
thf(fact_297_takeWhile__tail,axiom,
! [Xs_183: list_A1402527603le_alt,L_7: list_A1402527603le_alt,P_44: arrow_1389031998le_alt > $o,X_191: arrow_1389031998le_alt] :
( ~ ( P_44 @ X_191 )
=> ( ( takeWh462480871le_alt @ P_44 @ ( append1092755459le_alt @ Xs_183 @ ( cons_A1142416030le_alt @ X_191 @ L_7 ) ) )
= ( takeWh462480871le_alt @ P_44 @ Xs_183 ) ) ) ).
thf(fact_298_takeWhile__tail,axiom,
! [Xs_183: list_nat,L_7: list_nat,P_44: nat > $o,X_191: nat] :
( ~ ( P_44 @ X_191 )
=> ( ( takeWhile_nat @ P_44 @ ( append_nat @ Xs_183 @ ( cons_nat @ X_191 @ L_7 ) ) )
= ( takeWhile_nat @ P_44 @ Xs_183 ) ) ) ).
thf(fact_299_sublist__singleton,axiom,
! [X_190: $o,A_124: nat > $o] :
( ( ( member_nat @ zero_zero_nat @ A_124 )
=> ( ( sublist_o @ ( cons_o @ X_190 @ nil_o ) @ A_124 )
= ( cons_o @ X_190 @ nil_o ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A_124 )
=> ( ( sublist_o @ ( cons_o @ X_190 @ nil_o ) @ A_124 )
= nil_o ) ) ) ).
thf(fact_300_sublist__singleton,axiom,
! [X_190: arrow_1389031998le_alt,A_124: nat > $o] :
( ( ( member_nat @ zero_zero_nat @ A_124 )
=> ( ( sublis814142963le_alt @ ( cons_A1142416030le_alt @ X_190 @ nil_Ar52383470le_alt ) @ A_124 )
= ( cons_A1142416030le_alt @ X_190 @ nil_Ar52383470le_alt ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A_124 )
=> ( ( sublis814142963le_alt @ ( cons_A1142416030le_alt @ X_190 @ nil_Ar52383470le_alt ) @ A_124 )
= nil_Ar52383470le_alt ) ) ) ).
thf(fact_301_sublist__singleton,axiom,
! [X_190: nat,A_124: nat > $o] :
( ( ( member_nat @ zero_zero_nat @ A_124 )
=> ( ( sublist_nat @ ( cons_nat @ X_190 @ nil_nat ) @ A_124 )
= ( cons_nat @ X_190 @ nil_nat ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A_124 )
=> ( ( sublist_nat @ ( cons_nat @ X_190 @ nil_nat ) @ A_124 )
= nil_nat ) ) ) ).
thf(fact_302_listrelp_Osimps,axiom,
! [R_1: $o > $o > $o,A1: list_o,A2: list_o] :
( ( listrelp_o @ R_1 @ A1 @ A2 )
<=> ( ( ( A1 = nil_o )
& ( A2 = nil_o ) )
| ? [X_4: $o,Y_40: $o,Xs_23: list_o,Ys_7: list_o] :
( ( A1
= ( cons_o @ X_4 @ Xs_23 ) )
& ( A2
= ( cons_o @ Y_40 @ Ys_7 ) )
& ( R_1 @ X_4 @ Y_40 )
& ( listrelp_o @ R_1 @ Xs_23 @ Ys_7 ) ) ) ) ).
thf(fact_303_listrelp_Osimps,axiom,
! [R_1: arrow_1389031998le_alt > arrow_1389031998le_alt > $o,A1: list_A1402527603le_alt,A2: list_A1402527603le_alt] :
( ( listre2126835016le_alt @ R_1 @ A1 @ A2 )
<=> ( ( ( A1 = nil_Ar52383470le_alt )
& ( A2 = nil_Ar52383470le_alt ) )
| ? [X_4: arrow_1389031998le_alt,Y_40: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt,Ys_7: list_A1402527603le_alt] :
( ( A1
= ( cons_A1142416030le_alt @ X_4 @ Xs_23 ) )
& ( A2
= ( cons_A1142416030le_alt @ Y_40 @ Ys_7 ) )
& ( R_1 @ X_4 @ Y_40 )
& ( listre2126835016le_alt @ R_1 @ Xs_23 @ Ys_7 ) ) ) ) ).
thf(fact_304_listrelp_Osimps,axiom,
! [R_1: nat > nat > $o,A1: list_nat,A2: list_nat] :
( ( listrelp_nat @ R_1 @ A1 @ A2 )
<=> ( ( ( A1 = nil_nat )
& ( A2 = nil_nat ) )
| ? [X_4: nat,Y_40: nat,Xs_23: list_nat,Ys_7: list_nat] :
( ( A1
= ( cons_nat @ X_4 @ Xs_23 ) )
& ( A2
= ( cons_nat @ Y_40 @ Ys_7 ) )
& ( R_1 @ X_4 @ Y_40 )
& ( listrelp_nat @ R_1 @ Xs_23 @ Ys_7 ) ) ) ) ).
thf(fact_305_list_Osize_I1_J,axiom,
! [Fa_1: arrow_1389031998le_alt > nat] :
( ( list_s1163968880le_alt @ Fa_1 @ nil_Ar52383470le_alt )
= zero_zero_nat ) ).
thf(fact_306_list_Osize_I1_J,axiom,
! [Fa_1: nat > nat] :
( ( list_size_nat @ Fa_1 @ nil_nat )
= zero_zero_nat ) ).
thf(fact_307_zero__reorient,axiom,
! [X_189: nat] :
( ( zero_zero_nat = X_189 )
<=> ( X_189 = zero_zero_nat ) ) ).
thf(fact_308_hd__conv__nth,axiom,
! [Xs_182: list_o] :
( ( Xs_182 != nil_o )
=> ( ( hd_o @ Xs_182 )
<=> ( nth_o @ Xs_182 @ zero_zero_nat ) ) ) ).
thf(fact_309_hd__conv__nth,axiom,
! [Xs_182: list_A1402527603le_alt] :
( ( Xs_182 != nil_Ar52383470le_alt )
=> ( ( hd_Arr731872705le_alt @ Xs_182 )
= ( nth_Ar1039551983le_alt @ Xs_182 @ zero_zero_nat ) ) ) ).
thf(fact_310_hd__conv__nth,axiom,
! [Xs_182: list_nat] :
( ( Xs_182 != nil_nat )
=> ( ( hd_nat @ Xs_182 )
= ( nth_nat @ Xs_182 @ zero_zero_nat ) ) ) ).
thf(fact_311_insort__key_Osimps_I1_J,axiom,
! [F_29: nat > nat,X_188: nat] :
( ( linord1850739451at_nat @ F_29 @ X_188 @ nil_nat )
= ( cons_nat @ X_188 @ nil_nat ) ) ).
thf(fact_312_takeWhile__not__last,axiom,
! [Xs_181: list_A1402527603le_alt] :
( ( Xs_181 != nil_Ar52383470le_alt )
=> ( ( distin1149997281le_alt @ Xs_181 )
=> ( ( takeWh462480871le_alt
@ ^ [Y_40: arrow_1389031998le_alt] :
( (~)
@ ( Y_40
= ( last_A2130988295le_alt @ Xs_181 ) ) )
@ Xs_181 )
= ( butlas1188620858le_alt @ Xs_181 ) ) ) ) ).
thf(fact_313_takeWhile__not__last,axiom,
! [Xs_181: list_nat] :
( ( Xs_181 != nil_nat )
=> ( ( distinct_nat @ Xs_181 )
=> ( ( takeWhile_nat
@ ^ [Y_40: nat] :
( (~)
@ ( Y_40
= ( last_nat @ Xs_181 ) ) )
@ Xs_181 )
= ( butlast_nat @ Xs_181 ) ) ) ) ).
thf(fact_314_List_Oinsert__def,axiom,
! [X_187: $o,Xs_180: list_o] :
( ( ( member_o @ X_187 @ ( set_o @ Xs_180 ) )
=> ( ( insert_o @ X_187 @ Xs_180 )
= Xs_180 ) )
& ( ~ ( member_o @ X_187 @ ( set_o @ Xs_180 ) )
=> ( ( insert_o @ X_187 @ Xs_180 )
= ( cons_o @ X_187 @ Xs_180 ) ) ) ) ).
thf(fact_315_List_Oinsert__def,axiom,
! [X_187: arrow_1389031998le_alt,Xs_180: list_A1402527603le_alt] :
( ( ( member998036369le_alt @ X_187 @ ( set_Ar1491127311le_alt @ Xs_180 ) )
=> ( ( insert886756100le_alt @ X_187 @ Xs_180 )
= Xs_180 ) )
& ( ~ ( member998036369le_alt @ X_187 @ ( set_Ar1491127311le_alt @ Xs_180 ) )
=> ( ( insert886756100le_alt @ X_187 @ Xs_180 )
= ( cons_A1142416030le_alt @ X_187 @ Xs_180 ) ) ) ) ).
thf(fact_316_List_Oinsert__def,axiom,
! [X_187: nat,Xs_180: list_nat] :
( ( ( member_nat @ X_187 @ ( set_nat @ Xs_180 ) )
=> ( ( insert_nat @ X_187 @ Xs_180 )
= Xs_180 ) )
& ( ~ ( member_nat @ X_187 @ ( set_nat @ Xs_180 ) )
=> ( ( insert_nat @ X_187 @ Xs_180 )
= ( cons_nat @ X_187 @ Xs_180 ) ) ) ) ).
thf(fact_317_distinct__insort,axiom,
! [F_28: nat > nat,X_186: nat,Xs_179: list_nat] :
( ( distinct_nat @ ( linord1850739451at_nat @ F_28 @ X_186 @ Xs_179 ) )
<=> ( ~ ( member_nat @ X_186 @ ( set_nat @ Xs_179 ) )
& ( distinct_nat @ Xs_179 ) ) ) ).
thf(fact_318_insort__left__comm,axiom,
! [X_185: nat,Y_98: nat,Xs_178: list_nat] :
( ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ X_185
@ ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ Y_98
@ Xs_178 ) )
= ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ Y_98
@ ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ X_185
@ Xs_178 ) ) ) ).
thf(fact_319_notin__set__sublistI,axiom,
! [I_15: nat > $o,X_184: arrow_1389031998le_alt,Xs_177: list_A1402527603le_alt] :
( ~ ( member998036369le_alt @ X_184 @ ( set_Ar1491127311le_alt @ Xs_177 ) )
=> ~ ( member998036369le_alt @ X_184 @ ( set_Ar1491127311le_alt @ ( sublis814142963le_alt @ Xs_177 @ I_15 ) ) ) ) ).
thf(fact_320_notin__set__sublistI,axiom,
! [I_15: nat > $o,X_184: nat,Xs_177: list_nat] :
( ~ ( member_nat @ X_184 @ ( set_nat @ Xs_177 ) )
=> ~ ( member_nat @ X_184 @ ( set_nat @ ( sublist_nat @ Xs_177 @ I_15 ) ) ) ) ).
thf(fact_321_insort__key__left__comm,axiom,
! [Xs_176: list_nat,F_27: nat > nat,X_183: nat,Y_97: nat] :
( ( ( F_27 @ X_183 )
!= ( F_27 @ Y_97 ) )
=> ( ( linord1850739451at_nat @ F_27 @ Y_97 @ ( linord1850739451at_nat @ F_27 @ X_183 @ Xs_176 ) )
= ( linord1850739451at_nat @ F_27 @ X_183 @ ( linord1850739451at_nat @ F_27 @ Y_97 @ Xs_176 ) ) ) ) ).
thf(fact_322_in__set__sublistD,axiom,
! [X_182: arrow_1389031998le_alt,Xs_175: list_A1402527603le_alt,I_14: nat > $o] :
( ( member998036369le_alt @ X_182 @ ( set_Ar1491127311le_alt @ ( sublis814142963le_alt @ Xs_175 @ I_14 ) ) )
=> ( member998036369le_alt @ X_182 @ ( set_Ar1491127311le_alt @ Xs_175 ) ) ) ).
thf(fact_323_in__set__sublistD,axiom,
! [X_182: nat,Xs_175: list_nat,I_14: nat > $o] :
( ( member_nat @ X_182 @ ( set_nat @ ( sublist_nat @ Xs_175 @ I_14 ) ) )
=> ( member_nat @ X_182 @ ( set_nat @ Xs_175 ) ) ) ).
thf(fact_324_sorted__insort,axiom,
! [X_181: nat,Xs_174: list_nat] :
( ( linorder_sorted_nat
@ ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ X_181
@ Xs_174 ) )
<=> ( linorder_sorted_nat @ Xs_174 ) ) ).
thf(fact_325_tl__append,axiom,
! [Xs_173: list_A1402527603le_alt,Ys_28: list_A1402527603le_alt] :
( ( tl_Arr1379124165le_alt @ ( append1092755459le_alt @ Xs_173 @ Ys_28 ) )
= ( list_c1300280951le_alt @ ( tl_Arr1379124165le_alt @ Ys_28 )
@ ^ [Z_12: arrow_1389031998le_alt,Zs_5: list_A1402527603le_alt] : ( append1092755459le_alt @ Zs_5 @ Ys_28 )
@ Xs_173 ) ) ).
thf(fact_326_tl__append,axiom,
! [Xs_173: list_nat,Ys_28: list_nat] :
( ( tl_nat @ ( append_nat @ Xs_173 @ Ys_28 ) )
= ( list_c554461815at_nat @ ( tl_nat @ Ys_28 )
@ ^ [Z_12: nat,Zs_5: list_nat] : ( append_nat @ Zs_5 @ Ys_28 )
@ Xs_173 ) ) ).
thf(fact_327_set__ConsD,axiom,
! [Y_96: $o,X_180: $o,Xs_172: list_o] :
( ( member_o @ Y_96 @ ( set_o @ ( cons_o @ X_180 @ Xs_172 ) ) )
=> ( ( Y_96
<=> X_180 )
| ( member_o @ Y_96 @ ( set_o @ Xs_172 ) ) ) ) ).
thf(fact_328_set__ConsD,axiom,
! [Y_96: arrow_1389031998le_alt,X_180: arrow_1389031998le_alt,Xs_172: list_A1402527603le_alt] :
( ( member998036369le_alt @ Y_96 @ ( set_Ar1491127311le_alt @ ( cons_A1142416030le_alt @ X_180 @ Xs_172 ) ) )
=> ( ( Y_96 = X_180 )
| ( member998036369le_alt @ Y_96 @ ( set_Ar1491127311le_alt @ Xs_172 ) ) ) ) ).
thf(fact_329_set__ConsD,axiom,
! [Y_96: nat,X_180: nat,Xs_172: list_nat] :
( ( member_nat @ Y_96 @ ( set_nat @ ( cons_nat @ X_180 @ Xs_172 ) ) )
=> ( ( Y_96 = X_180 )
| ( member_nat @ Y_96 @ ( set_nat @ Xs_172 ) ) ) ) ).
thf(fact_330_set__rev,axiom,
! [Xs_171: list_A1402527603le_alt] :
( ( set_Ar1491127311le_alt @ ( rev_Ar2020079950le_alt @ Xs_171 ) )
= ( set_Ar1491127311le_alt @ Xs_171 ) ) ).
thf(fact_331_set__rev,axiom,
! [Xs_171: list_nat] :
( ( set_nat @ ( rev_nat @ Xs_171 ) )
= ( set_nat @ Xs_171 ) ) ).
thf(fact_332_takeWhile__eq__all__conv,axiom,
! [P_43: arrow_1389031998le_alt > $o,Xs_170: list_A1402527603le_alt] :
( ( ( takeWh462480871le_alt @ P_43 @ Xs_170 )
= Xs_170 )
<=> ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_170 ) )
=> ( P_43 @ X_4 ) ) ) ).
thf(fact_333_takeWhile__eq__all__conv,axiom,
! [P_43: nat > $o,Xs_170: list_nat] :
( ( ( takeWhile_nat @ P_43 @ Xs_170 )
= Xs_170 )
<=> ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_170 ) )
=> ( P_43 @ X_4 ) ) ) ).
thf(fact_334_set__takeWhileD,axiom,
! [X_179: arrow_1389031998le_alt,P_42: arrow_1389031998le_alt > $o,Xs_169: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_179 @ ( set_Ar1491127311le_alt @ ( takeWh462480871le_alt @ P_42 @ Xs_169 ) ) )
=> ( ( member998036369le_alt @ X_179 @ ( set_Ar1491127311le_alt @ Xs_169 ) )
& ( P_42 @ X_179 ) ) ) ).
thf(fact_335_set__takeWhileD,axiom,
! [X_179: nat,P_42: nat > $o,Xs_169: list_nat] :
( ( member_nat @ X_179 @ ( set_nat @ ( takeWhile_nat @ P_42 @ Xs_169 ) ) )
=> ( ( member_nat @ X_179 @ ( set_nat @ Xs_169 ) )
& ( P_42 @ X_179 ) ) ) ).
thf(fact_336_in__set__butlastD,axiom,
! [X_178: arrow_1389031998le_alt,Xs_168: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_178 @ ( set_Ar1491127311le_alt @ ( butlas1188620858le_alt @ Xs_168 ) ) )
=> ( member998036369le_alt @ X_178 @ ( set_Ar1491127311le_alt @ Xs_168 ) ) ) ).
thf(fact_337_in__set__butlastD,axiom,
! [X_178: nat,Xs_168: list_nat] :
( ( member_nat @ X_178 @ ( set_nat @ ( butlast_nat @ Xs_168 ) ) )
=> ( member_nat @ X_178 @ ( set_nat @ Xs_168 ) ) ) ).
thf(fact_338_insort__not__Nil,axiom,
! [F_26: nat > nat,A_123: nat,Xs_167: list_nat] :
( ( linord1850739451at_nat @ F_26 @ A_123 @ Xs_167 )
!= nil_nat ) ).
thf(fact_339_set__rotate1,axiom,
! [Xs_166: list_A1402527603le_alt] :
( ( set_Ar1491127311le_alt @ ( rotate1249022267le_alt @ Xs_166 ) )
= ( set_Ar1491127311le_alt @ Xs_166 ) ) ).
thf(fact_340_set__rotate1,axiom,
! [Xs_166: list_nat] :
( ( set_nat @ ( rotate1_nat @ Xs_166 ) )
= ( set_nat @ Xs_166 ) ) ).
thf(fact_341_in__set__insert,axiom,
! [X_177: arrow_1389031998le_alt,Xs_165: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_177 @ ( set_Ar1491127311le_alt @ Xs_165 ) )
=> ( ( insert886756100le_alt @ X_177 @ Xs_165 )
= Xs_165 ) ) ).
thf(fact_342_in__set__insert,axiom,
! [X_177: nat,Xs_165: list_nat] :
( ( member_nat @ X_177 @ ( set_nat @ Xs_165 ) )
=> ( ( insert_nat @ X_177 @ Xs_165 )
= Xs_165 ) ) ).
thf(fact_343_member__set,axiom,
member271157507le_alt = set_Ar1491127311le_alt ).
thf(fact_344_member__set,axiom,
member_nat_1 = set_nat ).
thf(fact_345_list__ex1__iff,axiom,
! [P_41: arrow_1389031998le_alt > $o,Xs_164: list_A1402527603le_alt] :
( ( list_e1152917937le_alt @ P_41 @ Xs_164 )
<=> ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_164 ) )
& ( P_41 @ X_4 )
& ! [Y_40: arrow_1389031998le_alt] :
( ( ( member998036369le_alt @ Y_40 @ ( set_Ar1491127311le_alt @ Xs_164 ) )
& ( P_41 @ Y_40 ) )
=> ( Y_40 = X_4 ) ) ) ) ).
thf(fact_346_list__ex1__iff,axiom,
! [P_41: nat > $o,Xs_164: list_nat] :
( ( list_ex1_nat @ P_41 @ Xs_164 )
<=> ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_164 ) )
& ( P_41 @ X_4 )
& ! [Y_40: nat] :
( ( ( member_nat @ Y_40 @ ( set_nat @ Xs_164 ) )
& ( P_41 @ Y_40 ) )
=> ( Y_40 = X_4 ) ) ) ) ).
thf(fact_347_List_Omember__def,axiom,
! [Xs_163: list_A1402527603le_alt,X_176: arrow_1389031998le_alt] :
( ( member271157507le_alt @ Xs_163 @ X_176 )
<=> ( member998036369le_alt @ X_176 @ ( set_Ar1491127311le_alt @ Xs_163 ) ) ) ).
thf(fact_348_List_Omember__def,axiom,
! [Xs_163: list_nat,X_176: nat] :
( ( member_nat_1 @ Xs_163 @ X_176 )
<=> ( member_nat @ X_176 @ ( set_nat @ Xs_163 ) ) ) ).
thf(fact_349_ListMem__iff,axiom,
! [X_175: arrow_1389031998le_alt,Xs_162: list_A1402527603le_alt] :
( ( listMe991720154le_alt @ X_175 @ Xs_162 )
<=> ( member998036369le_alt @ X_175 @ ( set_Ar1491127311le_alt @ Xs_162 ) ) ) ).
thf(fact_350_ListMem__iff,axiom,
! [X_175: nat,Xs_162: list_nat] :
( ( listMem_nat @ X_175 @ Xs_162 )
<=> ( member_nat @ X_175 @ ( set_nat @ Xs_162 ) ) ) ).
thf(fact_351_in__set__member,axiom,
! [X_174: arrow_1389031998le_alt,Xs_161: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_174 @ ( set_Ar1491127311le_alt @ Xs_161 ) )
<=> ( member271157507le_alt @ Xs_161 @ X_174 ) ) ).
thf(fact_352_in__set__member,axiom,
! [X_174: nat,Xs_161: list_nat] :
( ( member_nat @ X_174 @ ( set_nat @ Xs_161 ) )
<=> ( member_nat_1 @ Xs_161 @ X_174 ) ) ).
thf(fact_353_dropWhile__neq__rev,axiom,
! [X_173: $o,Xs_160: list_o] :
( ( distinct_o @ Xs_160 )
=> ( ( member_o @ X_173 @ ( set_o @ Xs_160 ) )
=> ( ( dropWhile_o
@ ^ [Y_40: $o] : ( (~) @ ( Y_40 = X_173 ) )
@ ( rev_o @ Xs_160 ) )
= ( cons_o @ X_173
@ ( rev_o
@ ( takeWhile_o
@ ^ [Y_40: $o] : ( (~) @ ( Y_40 = X_173 ) )
@ Xs_160 ) ) ) ) ) ) ).
thf(fact_354_dropWhile__neq__rev,axiom,
! [X_173: arrow_1389031998le_alt,Xs_160: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_160 )
=> ( ( member998036369le_alt @ X_173 @ ( set_Ar1491127311le_alt @ Xs_160 ) )
=> ( ( dropWh82971279le_alt
@ ^ [Y_40: arrow_1389031998le_alt] : ( (~) @ ( Y_40 = X_173 ) )
@ ( rev_Ar2020079950le_alt @ Xs_160 ) )
= ( cons_A1142416030le_alt @ X_173
@ ( rev_Ar2020079950le_alt
@ ( takeWh462480871le_alt
@ ^ [Y_40: arrow_1389031998le_alt] : ( (~) @ ( Y_40 = X_173 ) )
@ Xs_160 ) ) ) ) ) ) ).
thf(fact_355_dropWhile__neq__rev,axiom,
! [X_173: nat,Xs_160: list_nat] :
( ( distinct_nat @ Xs_160 )
=> ( ( member_nat @ X_173 @ ( set_nat @ Xs_160 ) )
=> ( ( dropWhile_nat
@ ^ [Y_40: nat] : ( (~) @ ( Y_40 = X_173 ) )
@ ( rev_nat @ Xs_160 ) )
= ( cons_nat @ X_173
@ ( rev_nat
@ ( takeWhile_nat
@ ^ [Y_40: nat] : ( (~) @ ( Y_40 = X_173 ) )
@ Xs_160 ) ) ) ) ) ) ).
thf(fact_356_takeWhile__neq__rev,axiom,
! [X_172: arrow_1389031998le_alt,Xs_159: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_159 )
=> ( ( member998036369le_alt @ X_172 @ ( set_Ar1491127311le_alt @ Xs_159 ) )
=> ( ( takeWh462480871le_alt
@ ^ [Y_40: arrow_1389031998le_alt] : ( (~) @ ( Y_40 = X_172 ) )
@ ( rev_Ar2020079950le_alt @ Xs_159 ) )
= ( rev_Ar2020079950le_alt
@ ( tl_Arr1379124165le_alt
@ ( dropWh82971279le_alt
@ ^ [Y_40: arrow_1389031998le_alt] : ( (~) @ ( Y_40 = X_172 ) )
@ Xs_159 ) ) ) ) ) ) ).
thf(fact_357_takeWhile__neq__rev,axiom,
! [X_172: nat,Xs_159: list_nat] :
( ( distinct_nat @ Xs_159 )
=> ( ( member_nat @ X_172 @ ( set_nat @ Xs_159 ) )
=> ( ( takeWhile_nat
@ ^ [Y_40: nat] : ( (~) @ ( Y_40 = X_172 ) )
@ ( rev_nat @ Xs_159 ) )
= ( rev_nat
@ ( tl_nat
@ ( dropWhile_nat
@ ^ [Y_40: nat] : ( (~) @ ( Y_40 = X_172 ) )
@ Xs_159 ) ) ) ) ) ) ).
thf(fact_358_sublist__nil,axiom,
! [A_122: nat > $o] :
( ( sublis814142963le_alt @ nil_Ar52383470le_alt @ A_122 )
= nil_Ar52383470le_alt ) ).
thf(fact_359_sublist__nil,axiom,
! [A_122: nat > $o] :
( ( sublist_nat @ nil_nat @ A_122 )
= nil_nat ) ).
thf(fact_360_distinct__sublistI,axiom,
! [I_13: nat > $o,Xs_158: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_158 )
=> ( distin1149997281le_alt @ ( sublis814142963le_alt @ Xs_158 @ I_13 ) ) ) ).
thf(fact_361_distinct__sublistI,axiom,
! [I_13: nat > $o,Xs_158: list_nat] :
( ( distinct_nat @ Xs_158 )
=> ( distinct_nat @ ( sublist_nat @ Xs_158 @ I_13 ) ) ) ).
thf(fact_362_rotate1__def,axiom,
! [Xs_157: list_o] :
( ( rotate1_o @ Xs_157 )
= ( list_case_list_o_o @ nil_o
@ ^ [X_4: $o,Xs_23: list_o] : ( append_o @ Xs_23 @ ( cons_o @ X_4 @ nil_o ) )
@ Xs_157 ) ) ).
thf(fact_363_rotate1__def,axiom,
! [Xs_157: list_A1402527603le_alt] :
( ( rotate1249022267le_alt @ Xs_157 )
= ( list_c1300280951le_alt @ nil_Ar52383470le_alt
@ ^ [X_4: arrow_1389031998le_alt,Xs_23: list_A1402527603le_alt] : ( append1092755459le_alt @ Xs_23 @ ( cons_A1142416030le_alt @ X_4 @ nil_Ar52383470le_alt ) )
@ Xs_157 ) ) ).
thf(fact_364_rotate1__def,axiom,
! [Xs_157: list_nat] :
( ( rotate1_nat @ Xs_157 )
= ( list_c554461815at_nat @ nil_nat
@ ^ [X_4: nat,Xs_23: list_nat] : ( append_nat @ Xs_23 @ ( cons_nat @ X_4 @ nil_nat ) )
@ Xs_157 ) ) ).
thf(fact_365_nth__Cons__0,axiom,
! [X_171: $o,Xs_156: list_o] :
( ( nth_o @ ( cons_o @ X_171 @ Xs_156 ) @ zero_zero_nat )
<=> X_171 ) ).
thf(fact_366_nth__Cons__0,axiom,
! [X_171: arrow_1389031998le_alt,Xs_156: list_A1402527603le_alt] :
( ( nth_Ar1039551983le_alt @ ( cons_A1142416030le_alt @ X_171 @ Xs_156 ) @ zero_zero_nat )
= X_171 ) ).
thf(fact_367_nth__Cons__0,axiom,
! [X_171: nat,Xs_156: list_nat] :
( ( nth_nat @ ( cons_nat @ X_171 @ Xs_156 ) @ zero_zero_nat )
= X_171 ) ).
thf(fact_368_distinct_Osimps_I2_J,axiom,
! [X_170: $o,Xs_155: list_o] :
( ( distinct_o @ ( cons_o @ X_170 @ Xs_155 ) )
<=> ( ~ ( member_o @ X_170 @ ( set_o @ Xs_155 ) )
& ( distinct_o @ Xs_155 ) ) ) ).
thf(fact_369_distinct_Osimps_I2_J,axiom,
! [X_170: arrow_1389031998le_alt,Xs_155: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ ( cons_A1142416030le_alt @ X_170 @ Xs_155 ) )
<=> ( ~ ( member998036369le_alt @ X_170 @ ( set_Ar1491127311le_alt @ Xs_155 ) )
& ( distin1149997281le_alt @ Xs_155 ) ) ) ).
thf(fact_370_distinct_Osimps_I2_J,axiom,
! [X_170: nat,Xs_155: list_nat] :
( ( distinct_nat @ ( cons_nat @ X_170 @ Xs_155 ) )
<=> ( ~ ( member_nat @ X_170 @ ( set_nat @ Xs_155 ) )
& ( distinct_nat @ Xs_155 ) ) ) ).
thf(fact_371_in__set__replicate,axiom,
! [X_169: arrow_1389031998le_alt,N_60: nat,Y_95: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_169 @ ( set_Ar1491127311le_alt @ ( replic277728168le_alt @ N_60 @ Y_95 ) ) )
<=> ( ( X_169 = Y_95 )
& ( N_60 != zero_zero_nat ) ) ) ).
thf(fact_372_in__set__replicate,axiom,
! [X_169: nat,N_60: nat,Y_95: nat] :
( ( member_nat @ X_169 @ ( set_nat @ ( replicate_nat @ N_60 @ Y_95 ) ) )
<=> ( ( X_169 = Y_95 )
& ( N_60 != zero_zero_nat ) ) ) ).
thf(fact_373_Bex__set__replicate,axiom,
! [P_40: arrow_1389031998le_alt > $o,N_59: nat,A_121: arrow_1389031998le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ ( replic277728168le_alt @ N_59 @ A_121 ) ) )
& ( P_40 @ X_4 ) )
<=> ( ( P_40 @ A_121 )
& ( N_59 != zero_zero_nat ) ) ) ).
thf(fact_374_Bex__set__replicate,axiom,
! [P_40: nat > $o,N_59: nat,A_121: nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ ( replicate_nat @ N_59 @ A_121 ) ) )
& ( P_40 @ X_4 ) )
<=> ( ( P_40 @ A_121 )
& ( N_59 != zero_zero_nat ) ) ) ).
thf(fact_375_Ball__set__replicate,axiom,
! [P_39: arrow_1389031998le_alt > $o,N_58: nat,A_120: arrow_1389031998le_alt] :
( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ ( replic277728168le_alt @ N_58 @ A_120 ) ) )
=> ( P_39 @ X_4 ) )
<=> ( ( P_39 @ A_120 )
| ( N_58 = zero_zero_nat ) ) ) ).
thf(fact_376_Ball__set__replicate,axiom,
! [P_39: nat > $o,N_58: nat,A_120: nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ ( replicate_nat @ N_58 @ A_120 ) ) )
=> ( P_39 @ X_4 ) )
<=> ( ( P_39 @ A_120 )
| ( N_58 = zero_zero_nat ) ) ) ).
thf(fact_377_sorted__distinct__set__unique,axiom,
! [Ys_27: list_nat,Xs_154: list_nat] :
( ( linorder_sorted_nat @ Xs_154 )
=> ( ( distinct_nat @ Xs_154 )
=> ( ( linorder_sorted_nat @ Ys_27 )
=> ( ( distinct_nat @ Ys_27 )
=> ( ( ( set_nat @ Xs_154 )
= ( set_nat @ Ys_27 ) )
=> ( Xs_154 = Ys_27 ) ) ) ) ) ) ).
thf(fact_378_takeWhile__append1,axiom,
! [Ys_26: list_A1402527603le_alt,P_38: arrow_1389031998le_alt > $o,X_168: arrow_1389031998le_alt,Xs_153: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_168 @ ( set_Ar1491127311le_alt @ Xs_153 ) )
=> ( ~ ( P_38 @ X_168 )
=> ( ( takeWh462480871le_alt @ P_38 @ ( append1092755459le_alt @ Xs_153 @ Ys_26 ) )
= ( takeWh462480871le_alt @ P_38 @ Xs_153 ) ) ) ) ).
thf(fact_379_takeWhile__append1,axiom,
! [Ys_26: list_nat,P_38: nat > $o,X_168: nat,Xs_153: list_nat] :
( ( member_nat @ X_168 @ ( set_nat @ Xs_153 ) )
=> ( ~ ( P_38 @ X_168 )
=> ( ( takeWhile_nat @ P_38 @ ( append_nat @ Xs_153 @ Ys_26 ) )
= ( takeWhile_nat @ P_38 @ Xs_153 ) ) ) ) ).
thf(fact_380_last__in__set,axiom,
! [As: list_A1402527603le_alt] :
( ( As != nil_Ar52383470le_alt )
=> ( member998036369le_alt @ ( last_A2130988295le_alt @ As ) @ ( set_Ar1491127311le_alt @ As ) ) ) ).
thf(fact_381_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat @ As ) ) ) ).
thf(fact_382_dropWhile__eq__Nil__conv,axiom,
! [P_37: arrow_1389031998le_alt > $o,Xs_152: list_A1402527603le_alt] :
( ( ( dropWh82971279le_alt @ P_37 @ Xs_152 )
= nil_Ar52383470le_alt )
<=> ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_152 ) )
=> ( P_37 @ X_4 ) ) ) ).
thf(fact_383_dropWhile__eq__Nil__conv,axiom,
! [P_37: nat > $o,Xs_152: list_nat] :
( ( ( dropWhile_nat @ P_37 @ Xs_152 )
= nil_nat )
<=> ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_152 ) )
=> ( P_37 @ X_4 ) ) ) ).
thf(fact_384_in__set__butlast__appendI,axiom,
! [Ys_25: list_A1402527603le_alt,X_167: arrow_1389031998le_alt,Xs_151: list_A1402527603le_alt] :
( ( ( member998036369le_alt @ X_167 @ ( set_Ar1491127311le_alt @ ( butlas1188620858le_alt @ Xs_151 ) ) )
| ( member998036369le_alt @ X_167 @ ( set_Ar1491127311le_alt @ ( butlas1188620858le_alt @ Ys_25 ) ) ) )
=> ( member998036369le_alt @ X_167 @ ( set_Ar1491127311le_alt @ ( butlas1188620858le_alt @ ( append1092755459le_alt @ Xs_151 @ Ys_25 ) ) ) ) ) ).
thf(fact_385_in__set__butlast__appendI,axiom,
! [Ys_25: list_nat,X_167: nat,Xs_151: list_nat] :
( ( ( member_nat @ X_167 @ ( set_nat @ ( butlast_nat @ Xs_151 ) ) )
| ( member_nat @ X_167 @ ( set_nat @ ( butlast_nat @ Ys_25 ) ) ) )
=> ( member_nat @ X_167 @ ( set_nat @ ( butlast_nat @ ( append_nat @ Xs_151 @ Ys_25 ) ) ) ) ) ).
thf(fact_386_hd__in__set,axiom,
! [Xs_150: list_A1402527603le_alt] :
( ( Xs_150 != nil_Ar52383470le_alt )
=> ( member998036369le_alt @ ( hd_Arr731872705le_alt @ Xs_150 ) @ ( set_Ar1491127311le_alt @ Xs_150 ) ) ) ).
thf(fact_387_hd__in__set,axiom,
! [Xs_150: list_nat] :
( ( Xs_150 != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs_150 ) @ ( set_nat @ Xs_150 ) ) ) ).
thf(fact_388_dropWhile__append1,axiom,
! [Ys_24: list_A1402527603le_alt,P_36: arrow_1389031998le_alt > $o,X_166: arrow_1389031998le_alt,Xs_149: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_166 @ ( set_Ar1491127311le_alt @ Xs_149 ) )
=> ( ~ ( P_36 @ X_166 )
=> ( ( dropWh82971279le_alt @ P_36 @ ( append1092755459le_alt @ Xs_149 @ Ys_24 ) )
= ( append1092755459le_alt @ ( dropWh82971279le_alt @ P_36 @ Xs_149 ) @ Ys_24 ) ) ) ) ).
thf(fact_389_dropWhile__append1,axiom,
! [Ys_24: list_nat,P_36: nat > $o,X_166: nat,Xs_149: list_nat] :
( ( member_nat @ X_166 @ ( set_nat @ Xs_149 ) )
=> ( ~ ( P_36 @ X_166 )
=> ( ( dropWhile_nat @ P_36 @ ( append_nat @ Xs_149 @ Ys_24 ) )
= ( append_nat @ ( dropWhile_nat @ P_36 @ Xs_149 ) @ Ys_24 ) ) ) ) ).
thf(fact_390_not__in__set__insert,axiom,
! [X_165: $o,Xs_148: list_o] :
( ~ ( member_o @ X_165 @ ( set_o @ Xs_148 ) )
=> ( ( insert_o @ X_165 @ Xs_148 )
= ( cons_o @ X_165 @ Xs_148 ) ) ) ).
thf(fact_391_not__in__set__insert,axiom,
! [X_165: arrow_1389031998le_alt,Xs_148: list_A1402527603le_alt] :
( ~ ( member998036369le_alt @ X_165 @ ( set_Ar1491127311le_alt @ Xs_148 ) )
=> ( ( insert886756100le_alt @ X_165 @ Xs_148 )
= ( cons_A1142416030le_alt @ X_165 @ Xs_148 ) ) ) ).
thf(fact_392_not__in__set__insert,axiom,
! [X_165: nat,Xs_148: list_nat] :
( ~ ( member_nat @ X_165 @ ( set_nat @ Xs_148 ) )
=> ( ( insert_nat @ X_165 @ Xs_148 )
= ( cons_nat @ X_165 @ Xs_148 ) ) ) ).
thf(fact_393_dropWhile__append2,axiom,
! [Ys_23: list_A1402527603le_alt,P_35: arrow_1389031998le_alt > $o,Xs_147: list_A1402527603le_alt] :
( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_147 ) )
=> ( P_35 @ X_4 ) )
=> ( ( dropWh82971279le_alt @ P_35 @ ( append1092755459le_alt @ Xs_147 @ Ys_23 ) )
= ( dropWh82971279le_alt @ P_35 @ Ys_23 ) ) ) ).
thf(fact_394_dropWhile__append2,axiom,
! [Ys_23: list_nat,P_35: nat > $o,Xs_147: list_nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_147 ) )
=> ( P_35 @ X_4 ) )
=> ( ( dropWhile_nat @ P_35 @ ( append_nat @ Xs_147 @ Ys_23 ) )
= ( dropWhile_nat @ P_35 @ Ys_23 ) ) ) ).
thf(fact_395_takeWhile__append2,axiom,
! [Ys_22: list_A1402527603le_alt,P_34: arrow_1389031998le_alt > $o,Xs_146: list_A1402527603le_alt] :
( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_146 ) )
=> ( P_34 @ X_4 ) )
=> ( ( takeWh462480871le_alt @ P_34 @ ( append1092755459le_alt @ Xs_146 @ Ys_22 ) )
= ( append1092755459le_alt @ Xs_146 @ ( takeWh462480871le_alt @ P_34 @ Ys_22 ) ) ) ) ).
thf(fact_396_takeWhile__append2,axiom,
! [Ys_22: list_nat,P_34: nat > $o,Xs_146: list_nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_146 ) )
=> ( P_34 @ X_4 ) )
=> ( ( takeWhile_nat @ P_34 @ ( append_nat @ Xs_146 @ Ys_22 ) )
= ( append_nat @ Xs_146 @ ( takeWhile_nat @ P_34 @ Ys_22 ) ) ) ) ).
thf(fact_397_in__set__conv__decomp,axiom,
! [X_164: $o,Xs_145: list_o] :
( ( member_o @ X_164 @ ( set_o @ Xs_145 ) )
<=> ? [Ys_7: list_o,Zs_5: list_o] :
( Xs_145
= ( append_o @ Ys_7 @ ( cons_o @ X_164 @ Zs_5 ) ) ) ) ).
thf(fact_398_in__set__conv__decomp,axiom,
! [X_164: arrow_1389031998le_alt,Xs_145: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_164 @ ( set_Ar1491127311le_alt @ Xs_145 ) )
<=> ? [Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt] :
( Xs_145
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_164 @ Zs_5 ) ) ) ) ).
thf(fact_399_in__set__conv__decomp,axiom,
! [X_164: nat,Xs_145: list_nat] :
( ( member_nat @ X_164 @ ( set_nat @ Xs_145 ) )
<=> ? [Ys_7: list_nat,Zs_5: list_nat] :
( Xs_145
= ( append_nat @ Ys_7 @ ( cons_nat @ X_164 @ Zs_5 ) ) ) ) ).
thf(fact_400_in__set__conv__decomp__last,axiom,
! [X_163: $o,Xs_144: list_o] :
( ( member_o @ X_163 @ ( set_o @ Xs_144 ) )
<=> ? [Ys_7: list_o,Zs_5: list_o] :
( ( Xs_144
= ( append_o @ Ys_7 @ ( cons_o @ X_163 @ Zs_5 ) ) )
& ~ ( member_o @ X_163 @ ( set_o @ Zs_5 ) ) ) ) ).
thf(fact_401_in__set__conv__decomp__last,axiom,
! [X_163: arrow_1389031998le_alt,Xs_144: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_163 @ ( set_Ar1491127311le_alt @ Xs_144 ) )
<=> ? [Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_144
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_163 @ Zs_5 ) ) )
& ~ ( member998036369le_alt @ X_163 @ ( set_Ar1491127311le_alt @ Zs_5 ) ) ) ) ).
thf(fact_402_in__set__conv__decomp__last,axiom,
! [X_163: nat,Xs_144: list_nat] :
( ( member_nat @ X_163 @ ( set_nat @ Xs_144 ) )
<=> ? [Ys_7: list_nat,Zs_5: list_nat] :
( ( Xs_144
= ( append_nat @ Ys_7 @ ( cons_nat @ X_163 @ Zs_5 ) ) )
& ~ ( member_nat @ X_163 @ ( set_nat @ Zs_5 ) ) ) ) ).
thf(fact_403_in__set__conv__decomp__first,axiom,
! [X_162: $o,Xs_143: list_o] :
( ( member_o @ X_162 @ ( set_o @ Xs_143 ) )
<=> ? [Ys_7: list_o,Zs_5: list_o] :
( ( Xs_143
= ( append_o @ Ys_7 @ ( cons_o @ X_162 @ Zs_5 ) ) )
& ~ ( member_o @ X_162 @ ( set_o @ Ys_7 ) ) ) ) ).
thf(fact_404_in__set__conv__decomp__first,axiom,
! [X_162: arrow_1389031998le_alt,Xs_143: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_162 @ ( set_Ar1491127311le_alt @ Xs_143 ) )
<=> ? [Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_143
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_162 @ Zs_5 ) ) )
& ~ ( member998036369le_alt @ X_162 @ ( set_Ar1491127311le_alt @ Ys_7 ) ) ) ) ).
thf(fact_405_in__set__conv__decomp__first,axiom,
! [X_162: nat,Xs_143: list_nat] :
( ( member_nat @ X_162 @ ( set_nat @ Xs_143 ) )
<=> ? [Ys_7: list_nat,Zs_5: list_nat] :
( ( Xs_143
= ( append_nat @ Ys_7 @ ( cons_nat @ X_162 @ Zs_5 ) ) )
& ~ ( member_nat @ X_162 @ ( set_nat @ Ys_7 ) ) ) ) ).
thf(fact_406_split__list__propE,axiom,
! [P_33: $o > $o,Xs_142: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_142 ) )
& ( P_33 @ X_4 ) )
=> ~ ! [Ys_7: list_o,X_4: $o] :
( ? [Zs_5: list_o] :
( Xs_142
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
=> ~ ( P_33 @ X_4 ) ) ) ).
thf(fact_407_split__list__propE,axiom,
! [P_33: arrow_1389031998le_alt > $o,Xs_142: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_142 ) )
& ( P_33 @ X_4 ) )
=> ~ ! [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt] :
( ? [Zs_5: list_A1402527603le_alt] :
( Xs_142
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
=> ~ ( P_33 @ X_4 ) ) ) ).
thf(fact_408_split__list__propE,axiom,
! [P_33: nat > $o,Xs_142: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_142 ) )
& ( P_33 @ X_4 ) )
=> ~ ! [Ys_7: list_nat,X_4: nat] :
( ? [Zs_5: list_nat] :
( Xs_142
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
=> ~ ( P_33 @ X_4 ) ) ) ).
thf(fact_409_dropWhile__cong,axiom,
! [Q_3: arrow_1389031998le_alt > $o,P_32: arrow_1389031998le_alt > $o,L_6: list_A1402527603le_alt,K_8: list_A1402527603le_alt] :
( ( L_6 = K_8 )
=> ( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ L_6 ) )
=> ( ( P_32 @ X_4 )
<=> ( Q_3 @ X_4 ) ) )
=> ( ( dropWh82971279le_alt @ P_32 @ L_6 )
= ( dropWh82971279le_alt @ Q_3 @ K_8 ) ) ) ) ).
thf(fact_410_dropWhile__cong,axiom,
! [Q_3: nat > $o,P_32: nat > $o,L_6: list_nat,K_8: list_nat] :
( ( L_6 = K_8 )
=> ( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ L_6 ) )
=> ( ( P_32 @ X_4 )
<=> ( Q_3 @ X_4 ) ) )
=> ( ( dropWhile_nat @ P_32 @ L_6 )
= ( dropWhile_nat @ Q_3 @ K_8 ) ) ) ) ).
thf(fact_411_takeWhile__cong,axiom,
! [Q_2: arrow_1389031998le_alt > $o,P_31: arrow_1389031998le_alt > $o,L_5: list_A1402527603le_alt,K_7: list_A1402527603le_alt] :
( ( L_5 = K_7 )
=> ( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ L_5 ) )
=> ( ( P_31 @ X_4 )
<=> ( Q_2 @ X_4 ) ) )
=> ( ( takeWh462480871le_alt @ P_31 @ L_5 )
= ( takeWh462480871le_alt @ Q_2 @ K_7 ) ) ) ) ).
thf(fact_412_takeWhile__cong,axiom,
! [Q_2: nat > $o,P_31: nat > $o,L_5: list_nat,K_7: list_nat] :
( ( L_5 = K_7 )
=> ( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ L_5 ) )
=> ( ( P_31 @ X_4 )
<=> ( Q_2 @ X_4 ) ) )
=> ( ( takeWhile_nat @ P_31 @ L_5 )
= ( takeWhile_nat @ Q_2 @ K_7 ) ) ) ) ).
thf(fact_413_map__filter__simps_I1_J,axiom,
! [F_25: arrow_1389031998le_alt > option_o,X_161: arrow_1389031998le_alt,Xs_141: list_A1402527603le_alt] :
( ( map_fi312952763_alt_o @ F_25 @ ( cons_A1142416030le_alt @ X_161 @ Xs_141 ) )
= ( option_case_list_o_o @ ( map_fi312952763_alt_o @ F_25 @ Xs_141 )
@ ^ [Y_40: $o] : ( cons_o @ Y_40 @ ( map_fi312952763_alt_o @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_414_map__filter__simps_I1_J,axiom,
! [F_25: arrow_1389031998le_alt > option_nat,X_161: arrow_1389031998le_alt,Xs_141: list_A1402527603le_alt] :
( ( map_fi1378126230lt_nat @ F_25 @ ( cons_A1142416030le_alt @ X_161 @ Xs_141 ) )
= ( option306705938at_nat @ ( map_fi1378126230lt_nat @ F_25 @ Xs_141 )
@ ^ [Y_40: nat] : ( cons_nat @ Y_40 @ ( map_fi1378126230lt_nat @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_415_map__filter__simps_I1_J,axiom,
! [F_25: arrow_1389031998le_alt > option1949451059le_alt,X_161: arrow_1389031998le_alt,Xs_141: list_A1402527603le_alt] :
( ( map_fi193489098le_alt @ F_25 @ ( cons_A1142416030le_alt @ X_161 @ Xs_141 ) )
= ( option1388650770le_alt @ ( map_fi193489098le_alt @ F_25 @ Xs_141 )
@ ^ [Y_40: arrow_1389031998le_alt] : ( cons_A1142416030le_alt @ Y_40 @ ( map_fi193489098le_alt @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_416_map__filter__simps_I1_J,axiom,
! [F_25: nat > option_o,X_161: nat,Xs_141: list_nat] :
( ( map_filter_nat_o @ F_25 @ ( cons_nat @ X_161 @ Xs_141 ) )
= ( option_case_list_o_o @ ( map_filter_nat_o @ F_25 @ Xs_141 )
@ ^ [Y_40: $o] : ( cons_o @ Y_40 @ ( map_filter_nat_o @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_417_map__filter__simps_I1_J,axiom,
! [F_25: nat > option_nat,X_161: nat,Xs_141: list_nat] :
( ( map_filter_nat_nat @ F_25 @ ( cons_nat @ X_161 @ Xs_141 ) )
= ( option306705938at_nat @ ( map_filter_nat_nat @ F_25 @ Xs_141 )
@ ^ [Y_40: nat] : ( cons_nat @ Y_40 @ ( map_filter_nat_nat @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_418_map__filter__simps_I1_J,axiom,
! [F_25: nat > option1949451059le_alt,X_161: nat,Xs_141: list_nat] :
( ( map_fi228886654le_alt @ F_25 @ ( cons_nat @ X_161 @ Xs_141 ) )
= ( option1388650770le_alt @ ( map_fi228886654le_alt @ F_25 @ Xs_141 )
@ ^ [Y_40: arrow_1389031998le_alt] : ( cons_A1142416030le_alt @ Y_40 @ ( map_fi228886654le_alt @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_419_map__filter__simps_I1_J,axiom,
! [F_25: $o > option1949451059le_alt,X_161: $o,Xs_141: list_o] :
( ( map_fi875972505le_alt @ F_25 @ ( cons_o @ X_161 @ Xs_141 ) )
= ( option1388650770le_alt @ ( map_fi875972505le_alt @ F_25 @ Xs_141 )
@ ^ [Y_40: arrow_1389031998le_alt] : ( cons_A1142416030le_alt @ Y_40 @ ( map_fi875972505le_alt @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_420_map__filter__simps_I1_J,axiom,
! [F_25: $o > option_nat,X_161: $o,Xs_141: list_o] :
( ( map_filter_o_nat @ F_25 @ ( cons_o @ X_161 @ Xs_141 ) )
= ( option306705938at_nat @ ( map_filter_o_nat @ F_25 @ Xs_141 )
@ ^ [Y_40: nat] : ( cons_nat @ Y_40 @ ( map_filter_o_nat @ F_25 @ Xs_141 ) )
@ ( F_25 @ X_161 ) ) ) ).
thf(fact_421_split__list__first__propE,axiom,
! [P_30: $o > $o,Xs_140: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_140 ) )
& ( P_30 @ X_4 ) )
=> ~ ! [Ys_7: list_o,X_4: $o] :
( ? [Zs_5: list_o] :
( Xs_140
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
=> ( ( P_30 @ X_4 )
=> ~ ! [Xa: $o] :
( ( member_o @ Xa @ ( set_o @ Ys_7 ) )
=> ~ ( P_30 @ Xa ) ) ) ) ) ).
thf(fact_422_split__list__first__propE,axiom,
! [P_30: arrow_1389031998le_alt > $o,Xs_140: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_140 ) )
& ( P_30 @ X_4 ) )
=> ~ ! [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt] :
( ? [Zs_5: list_A1402527603le_alt] :
( Xs_140
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
=> ( ( P_30 @ X_4 )
=> ~ ! [Xa: arrow_1389031998le_alt] :
( ( member998036369le_alt @ Xa @ ( set_Ar1491127311le_alt @ Ys_7 ) )
=> ~ ( P_30 @ Xa ) ) ) ) ) ).
thf(fact_423_split__list__first__propE,axiom,
! [P_30: nat > $o,Xs_140: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_140 ) )
& ( P_30 @ X_4 ) )
=> ~ ! [Ys_7: list_nat,X_4: nat] :
( ? [Zs_5: list_nat] :
( Xs_140
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
=> ( ( P_30 @ X_4 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Ys_7 ) )
=> ~ ( P_30 @ Xa ) ) ) ) ) ).
thf(fact_424_split__list__last__propE,axiom,
! [P_29: $o > $o,Xs_139: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_139 ) )
& ( P_29 @ X_4 ) )
=> ~ ! [Ys_7: list_o,X_4: $o,Zs_5: list_o] :
( ( Xs_139
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
=> ( ( P_29 @ X_4 )
=> ~ ! [Xa: $o] :
( ( member_o @ Xa @ ( set_o @ Zs_5 ) )
=> ~ ( P_29 @ Xa ) ) ) ) ) ).
thf(fact_425_split__list__last__propE,axiom,
! [P_29: arrow_1389031998le_alt > $o,Xs_139: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_139 ) )
& ( P_29 @ X_4 ) )
=> ~ ! [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_139
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
=> ( ( P_29 @ X_4 )
=> ~ ! [Xa: arrow_1389031998le_alt] :
( ( member998036369le_alt @ Xa @ ( set_Ar1491127311le_alt @ Zs_5 ) )
=> ~ ( P_29 @ Xa ) ) ) ) ) ).
thf(fact_426_split__list__last__propE,axiom,
! [P_29: nat > $o,Xs_139: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_139 ) )
& ( P_29 @ X_4 ) )
=> ~ ! [Ys_7: list_nat,X_4: nat,Zs_5: list_nat] :
( ( Xs_139
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
=> ( ( P_29 @ X_4 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Zs_5 ) )
=> ~ ( P_29 @ Xa ) ) ) ) ) ).
thf(fact_427_split__list__first,axiom,
! [X_160: $o,Xs_138: list_o] :
( ( member_o @ X_160 @ ( set_o @ Xs_138 ) )
=> ? [Ys_7: list_o,Zs_5: list_o] :
( ( Xs_138
= ( append_o @ Ys_7 @ ( cons_o @ X_160 @ Zs_5 ) ) )
& ~ ( member_o @ X_160 @ ( set_o @ Ys_7 ) ) ) ) ).
thf(fact_428_split__list__first,axiom,
! [X_160: arrow_1389031998le_alt,Xs_138: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_160 @ ( set_Ar1491127311le_alt @ Xs_138 ) )
=> ? [Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_138
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_160 @ Zs_5 ) ) )
& ~ ( member998036369le_alt @ X_160 @ ( set_Ar1491127311le_alt @ Ys_7 ) ) ) ) ).
thf(fact_429_split__list__first,axiom,
! [X_160: nat,Xs_138: list_nat] :
( ( member_nat @ X_160 @ ( set_nat @ Xs_138 ) )
=> ? [Ys_7: list_nat,Zs_5: list_nat] :
( ( Xs_138
= ( append_nat @ Ys_7 @ ( cons_nat @ X_160 @ Zs_5 ) ) )
& ~ ( member_nat @ X_160 @ ( set_nat @ Ys_7 ) ) ) ) ).
thf(fact_430_split__list__last,axiom,
! [X_159: $o,Xs_137: list_o] :
( ( member_o @ X_159 @ ( set_o @ Xs_137 ) )
=> ? [Ys_7: list_o,Zs_5: list_o] :
( ( Xs_137
= ( append_o @ Ys_7 @ ( cons_o @ X_159 @ Zs_5 ) ) )
& ~ ( member_o @ X_159 @ ( set_o @ Zs_5 ) ) ) ) ).
thf(fact_431_split__list__last,axiom,
! [X_159: arrow_1389031998le_alt,Xs_137: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_159 @ ( set_Ar1491127311le_alt @ Xs_137 ) )
=> ? [Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_137
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_159 @ Zs_5 ) ) )
& ~ ( member998036369le_alt @ X_159 @ ( set_Ar1491127311le_alt @ Zs_5 ) ) ) ) ).
thf(fact_432_split__list__last,axiom,
! [X_159: nat,Xs_137: list_nat] :
( ( member_nat @ X_159 @ ( set_nat @ Xs_137 ) )
=> ? [Ys_7: list_nat,Zs_5: list_nat] :
( ( Xs_137
= ( append_nat @ Ys_7 @ ( cons_nat @ X_159 @ Zs_5 ) ) )
& ~ ( member_nat @ X_159 @ ( set_nat @ Zs_5 ) ) ) ) ).
thf(fact_433_split__list,axiom,
! [X_158: $o,Xs_136: list_o] :
( ( member_o @ X_158 @ ( set_o @ Xs_136 ) )
=> ? [Ys_7: list_o,Zs_5: list_o] :
( Xs_136
= ( append_o @ Ys_7 @ ( cons_o @ X_158 @ Zs_5 ) ) ) ) ).
thf(fact_434_split__list,axiom,
! [X_158: arrow_1389031998le_alt,Xs_136: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_158 @ ( set_Ar1491127311le_alt @ Xs_136 ) )
=> ? [Ys_7: list_A1402527603le_alt,Zs_5: list_A1402527603le_alt] :
( Xs_136
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_158 @ Zs_5 ) ) ) ) ).
thf(fact_435_split__list,axiom,
! [X_158: nat,Xs_136: list_nat] :
( ( member_nat @ X_158 @ ( set_nat @ Xs_136 ) )
=> ? [Ys_7: list_nat,Zs_5: list_nat] :
( Xs_136
= ( append_nat @ Ys_7 @ ( cons_nat @ X_158 @ Zs_5 ) ) ) ) ).
thf(fact_436_split__list__first__prop__iff,axiom,
! [P_28: $o > $o,Xs_135: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_135 ) )
& ( P_28 @ X_4 ) )
<=> ? [Ys_7: list_o,X_4: $o] :
( ? [Zs_5: list_o] :
( Xs_135
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
& ( P_28 @ X_4 )
& ! [Xa: $o] :
( ( member_o @ Xa @ ( set_o @ Ys_7 ) )
=> ~ ( P_28 @ Xa ) ) ) ) ).
thf(fact_437_split__list__first__prop__iff,axiom,
! [P_28: arrow_1389031998le_alt > $o,Xs_135: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_135 ) )
& ( P_28 @ X_4 ) )
<=> ? [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt] :
( ? [Zs_5: list_A1402527603le_alt] :
( Xs_135
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
& ( P_28 @ X_4 )
& ! [Xa: arrow_1389031998le_alt] :
( ( member998036369le_alt @ Xa @ ( set_Ar1491127311le_alt @ Ys_7 ) )
=> ~ ( P_28 @ Xa ) ) ) ) ).
thf(fact_438_split__list__first__prop__iff,axiom,
! [P_28: nat > $o,Xs_135: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_135 ) )
& ( P_28 @ X_4 ) )
<=> ? [Ys_7: list_nat,X_4: nat] :
( ? [Zs_5: list_nat] :
( Xs_135
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
& ( P_28 @ X_4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Ys_7 ) )
=> ~ ( P_28 @ Xa ) ) ) ) ).
thf(fact_439_split__list__last__prop__iff,axiom,
! [P_27: $o > $o,Xs_134: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_134 ) )
& ( P_27 @ X_4 ) )
<=> ? [Ys_7: list_o,X_4: $o,Zs_5: list_o] :
( ( Xs_134
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
& ( P_27 @ X_4 )
& ! [Xa: $o] :
( ( member_o @ Xa @ ( set_o @ Zs_5 ) )
=> ~ ( P_27 @ Xa ) ) ) ) ).
thf(fact_440_split__list__last__prop__iff,axiom,
! [P_27: arrow_1389031998le_alt > $o,Xs_134: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_134 ) )
& ( P_27 @ X_4 ) )
<=> ? [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_134
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
& ( P_27 @ X_4 )
& ! [Xa: arrow_1389031998le_alt] :
( ( member998036369le_alt @ Xa @ ( set_Ar1491127311le_alt @ Zs_5 ) )
=> ~ ( P_27 @ Xa ) ) ) ) ).
thf(fact_441_split__list__last__prop__iff,axiom,
! [P_27: nat > $o,Xs_134: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_134 ) )
& ( P_27 @ X_4 ) )
<=> ? [Ys_7: list_nat,X_4: nat,Zs_5: list_nat] :
( ( Xs_134
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
& ( P_27 @ X_4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Zs_5 ) )
=> ~ ( P_27 @ Xa ) ) ) ) ).
thf(fact_442_sublist__Cons,axiom,
! [X_157: $o,L_4: list_o,A_119: nat > $o] :
( ( sublist_o @ ( cons_o @ X_157 @ L_4 ) @ A_119 )
= ( append_o @ ( if_list_o @ ( member_nat @ zero_zero_nat @ A_119 ) @ ( cons_o @ X_157 @ nil_o ) @ nil_o )
@ ( sublist_o @ L_4
@ ( collect_nat
@ ^ [J_1: nat] : ( member_nat @ ( suc @ J_1 ) @ A_119 ) ) ) ) ) ).
thf(fact_443_sublist__Cons,axiom,
! [X_157: arrow_1389031998le_alt,L_4: list_A1402527603le_alt,A_119: nat > $o] :
( ( sublis814142963le_alt @ ( cons_A1142416030le_alt @ X_157 @ L_4 ) @ A_119 )
= ( append1092755459le_alt @ ( if_lis613229059le_alt @ ( member_nat @ zero_zero_nat @ A_119 ) @ ( cons_A1142416030le_alt @ X_157 @ nil_Ar52383470le_alt ) @ nil_Ar52383470le_alt )
@ ( sublis814142963le_alt @ L_4
@ ( collect_nat
@ ^ [J_1: nat] : ( member_nat @ ( suc @ J_1 ) @ A_119 ) ) ) ) ) ).
thf(fact_444_sublist__Cons,axiom,
! [X_157: nat,L_4: list_nat,A_119: nat > $o] :
( ( sublist_nat @ ( cons_nat @ X_157 @ L_4 ) @ A_119 )
= ( append_nat @ ( if_list_nat @ ( member_nat @ zero_zero_nat @ A_119 ) @ ( cons_nat @ X_157 @ nil_nat ) @ nil_nat )
@ ( sublist_nat @ L_4
@ ( collect_nat
@ ^ [J_1: nat] : ( member_nat @ ( suc @ J_1 ) @ A_119 ) ) ) ) ) ).
thf(fact_445_nth__Cons__Suc,axiom,
! [X_156: $o,Xs_133: list_o,N_57: nat] :
( ( nth_o @ ( cons_o @ X_156 @ Xs_133 ) @ ( suc @ N_57 ) )
<=> ( nth_o @ Xs_133 @ N_57 ) ) ).
thf(fact_446_nth__Cons__Suc,axiom,
! [X_156: arrow_1389031998le_alt,Xs_133: list_A1402527603le_alt,N_57: nat] :
( ( nth_Ar1039551983le_alt @ ( cons_A1142416030le_alt @ X_156 @ Xs_133 ) @ ( suc @ N_57 ) )
= ( nth_Ar1039551983le_alt @ Xs_133 @ N_57 ) ) ).
thf(fact_447_nth__Cons__Suc,axiom,
! [X_156: nat,Xs_133: list_nat,N_57: nat] :
( ( nth_nat @ ( cons_nat @ X_156 @ Xs_133 ) @ ( suc @ N_57 ) )
= ( nth_nat @ Xs_133 @ N_57 ) ) ).
thf(fact_448_replicate__Suc,axiom,
! [N_56: nat,X_155: $o] :
( ( replicate_o @ ( suc @ N_56 ) @ X_155 )
= ( cons_o @ X_155 @ ( replicate_o @ N_56 @ X_155 ) ) ) ).
thf(fact_449_replicate__Suc,axiom,
! [N_56: nat,X_155: arrow_1389031998le_alt] :
( ( replic277728168le_alt @ ( suc @ N_56 ) @ X_155 )
= ( cons_A1142416030le_alt @ X_155 @ ( replic277728168le_alt @ N_56 @ X_155 ) ) ) ).
thf(fact_450_replicate__Suc,axiom,
! [N_56: nat,X_155: nat] :
( ( replicate_nat @ ( suc @ N_56 ) @ X_155 )
= ( cons_nat @ X_155 @ ( replicate_nat @ N_56 @ X_155 ) ) ) ).
thf(fact_451_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_452_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_453_nat_Osimps_I3_J,axiom,
! [Nat_4: nat] :
( ( suc @ Nat_4 )
!= zero_zero_nat ) ).
thf(fact_454_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_455_nat_Osimps_I2_J,axiom,
! [Nat_2: nat] :
( zero_zero_nat
!= ( suc @ Nat_2 ) ) ).
thf(fact_456_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_457_nth_Osimps,axiom,
! [X_154: list_nat,Xs_132: list_list_nat,N_55: nat] :
( ( nth_list_nat @ ( cons_list_nat @ X_154 @ Xs_132 ) @ N_55 )
= ( nat_case_list_nat @ X_154 @ ( nth_list_nat @ Xs_132 ) @ N_55 ) ) ).
thf(fact_458_nth_Osimps,axiom,
! [X_154: list_A1402527603le_alt,Xs_132: list_l1230899348le_alt,N_55: nat] :
( ( nth_li1391746178le_alt @ ( cons_l2069870355le_alt @ X_154 @ Xs_132 ) @ N_55 )
= ( nat_ca1434653759le_alt @ X_154 @ ( nth_li1391746178le_alt @ Xs_132 ) @ N_55 ) ) ).
thf(fact_459_nth_Osimps,axiom,
! [X_154: $o,Xs_132: list_o,N_55: nat] :
( ( nth_o @ ( cons_o @ X_154 @ Xs_132 ) @ N_55 )
<=> ( nat_case_o @ X_154 @ ( nth_o @ Xs_132 ) @ N_55 ) ) ).
thf(fact_460_nth_Osimps,axiom,
! [X_154: arrow_1389031998le_alt,Xs_132: list_A1402527603le_alt,N_55: nat] :
( ( nth_Ar1039551983le_alt @ ( cons_A1142416030le_alt @ X_154 @ Xs_132 ) @ N_55 )
= ( nat_ca317420658le_alt @ X_154 @ ( nth_Ar1039551983le_alt @ Xs_132 ) @ N_55 ) ) ).
thf(fact_461_nth_Osimps,axiom,
! [X_154: nat,Xs_132: list_nat,N_55: nat] :
( ( nth_nat @ ( cons_nat @ X_154 @ Xs_132 ) @ N_55 )
= ( nat_case_nat @ X_154 @ ( nth_nat @ Xs_132 ) @ N_55 ) ) ).
thf(fact_462_nat__case__0,axiom,
! [F1_1: list_nat,F2_1: nat > list_nat] :
( ( nat_case_list_nat @ F1_1 @ F2_1 @ zero_zero_nat )
= F1_1 ) ).
thf(fact_463_nat__case__0,axiom,
! [F1_1: list_A1402527603le_alt,F2_1: nat > list_A1402527603le_alt] :
( ( nat_ca1434653759le_alt @ F1_1 @ F2_1 @ zero_zero_nat )
= F1_1 ) ).
thf(fact_464_nat__case__0,axiom,
! [F1_1: arrow_1389031998le_alt,F2_1: nat > arrow_1389031998le_alt] :
( ( nat_ca317420658le_alt @ F1_1 @ F2_1 @ zero_zero_nat )
= F1_1 ) ).
thf(fact_465_nat__case__0,axiom,
! [F1_1: nat,F2_1: nat > nat] :
( ( nat_case_nat @ F1_1 @ F2_1 @ zero_zero_nat )
= F1_1 ) ).
thf(fact_466_nat__case__0,axiom,
! [F1_1: $o,F2_1: nat > $o] :
( ( nat_case_o @ F1_1 @ F2_1 @ zero_zero_nat )
<=> F1_1 ) ).
thf(fact_467_nat__case__Suc,axiom,
! [F1: list_nat,F2: nat > list_nat,Nat_3: nat] :
( ( nat_case_list_nat @ F1 @ F2 @ ( suc @ Nat_3 ) )
= ( F2 @ Nat_3 ) ) ).
thf(fact_468_nat__case__Suc,axiom,
! [F1: list_A1402527603le_alt,F2: nat > list_A1402527603le_alt,Nat_3: nat] :
( ( nat_ca1434653759le_alt @ F1 @ F2 @ ( suc @ Nat_3 ) )
= ( F2 @ Nat_3 ) ) ).
thf(fact_469_nat__case__Suc,axiom,
! [F1: arrow_1389031998le_alt,F2: nat > arrow_1389031998le_alt,Nat_3: nat] :
( ( nat_ca317420658le_alt @ F1 @ F2 @ ( suc @ Nat_3 ) )
= ( F2 @ Nat_3 ) ) ).
thf(fact_470_nat__case__Suc,axiom,
! [F1: nat,F2: nat > nat,Nat_3: nat] :
( ( nat_case_nat @ F1 @ F2 @ ( suc @ Nat_3 ) )
= ( F2 @ Nat_3 ) ) ).
thf(fact_471_nat__case__Suc,axiom,
! [F1: $o,F2: nat > $o,Nat_3: nat] :
( ( nat_case_o @ F1 @ F2 @ ( suc @ Nat_3 ) )
<=> ( F2 @ Nat_3 ) ) ).
thf(fact_472_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_473_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_474_nat_Oinject,axiom,
! [Nat: nat,Nat_2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat_2 ) )
<=> ( Nat = Nat_2 ) ) ).
thf(fact_475_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_476_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_6: nat] :
( N
= ( suc @ M_6 ) ) ) ).
thf(fact_477_nat__induct,axiom,
! [N: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N_23: nat] :
( ( P @ N_23 )
=> ( P @ ( suc @ N_23 ) ) )
=> ( P @ N ) ) ) ).
thf(fact_478_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_23: nat] :
( ( P @ ( suc @ N_23 ) )
=> ( P @ N_23 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_479_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat_1: nat] :
( Y
!= ( suc @ Nat_1 ) ) ) ).
thf(fact_480_take__Suc,axiom,
! [N_54: nat,Xs_131: list_o] :
( ( Xs_131 != nil_o )
=> ( ( take_o @ ( suc @ N_54 ) @ Xs_131 )
= ( cons_o @ ( hd_o @ Xs_131 ) @ ( take_o @ N_54 @ ( tl_o @ Xs_131 ) ) ) ) ) ).
thf(fact_481_take__Suc,axiom,
! [N_54: nat,Xs_131: list_A1402527603le_alt] :
( ( Xs_131 != nil_Ar52383470le_alt )
=> ( ( take_A1527720662le_alt @ ( suc @ N_54 ) @ Xs_131 )
= ( cons_A1142416030le_alt @ ( hd_Arr731872705le_alt @ Xs_131 ) @ ( take_A1527720662le_alt @ N_54 @ ( tl_Arr1379124165le_alt @ Xs_131 ) ) ) ) ) ).
thf(fact_482_take__Suc,axiom,
! [N_54: nat,Xs_131: list_nat] :
( ( Xs_131 != nil_nat )
=> ( ( take_nat @ ( suc @ N_54 ) @ Xs_131 )
= ( cons_nat @ ( hd_nat @ Xs_131 ) @ ( take_nat @ N_54 @ ( tl_nat @ Xs_131 ) ) ) ) ) ).
thf(fact_483_insort__insert__insort,axiom,
! [X_153: nat,Xs_130: list_nat] :
( ~ ( member_nat @ X_153 @ ( set_nat @ Xs_130 ) )
=> ( ( linord297828738at_nat
@ ^ [X_4: nat] : X_4
@ X_153
@ Xs_130 )
= ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ X_153
@ Xs_130 ) ) ) ).
thf(fact_484_list__ex1__simps_I2_J,axiom,
! [P_26: $o > $o,X_152: $o,Xs_129: list_o] :
( ( list_ex1_o @ P_26 @ ( cons_o @ X_152 @ Xs_129 ) )
<=> ( ( ( P_26 @ X_152 )
=> ( list_all_o
@ ^ [Y_40: $o] : ( (|) @ ( (~) @ ( P_26 @ Y_40 ) ) @ ( X_152 = Y_40 ) )
@ Xs_129 ) )
& ( ~ ( P_26 @ X_152 )
=> ( list_ex1_o @ P_26 @ Xs_129 ) ) ) ) ).
thf(fact_485_list__ex1__simps_I2_J,axiom,
! [P_26: arrow_1389031998le_alt > $o,X_152: arrow_1389031998le_alt,Xs_129: list_A1402527603le_alt] :
( ( list_e1152917937le_alt @ P_26 @ ( cons_A1142416030le_alt @ X_152 @ Xs_129 ) )
<=> ( ( ( P_26 @ X_152 )
=> ( list_a1483787182le_alt
@ ^ [Y_40: arrow_1389031998le_alt] : ( (|) @ ( (~) @ ( P_26 @ Y_40 ) ) @ ( X_152 = Y_40 ) )
@ Xs_129 ) )
& ( ~ ( P_26 @ X_152 )
=> ( list_e1152917937le_alt @ P_26 @ Xs_129 ) ) ) ) ).
thf(fact_486_list__ex1__simps_I2_J,axiom,
! [P_26: nat > $o,X_152: nat,Xs_129: list_nat] :
( ( list_ex1_nat @ P_26 @ ( cons_nat @ X_152 @ Xs_129 ) )
<=> ( ( ( P_26 @ X_152 )
=> ( list_all_nat
@ ^ [Y_40: nat] : ( (|) @ ( (~) @ ( P_26 @ Y_40 ) ) @ ( X_152 = Y_40 ) )
@ Xs_129 ) )
& ( ~ ( P_26 @ X_152 )
=> ( list_ex1_nat @ P_26 @ Xs_129 ) ) ) ) ).
thf(fact_487_insort__remove1,axiom,
! [A_118: nat,Xs_128: list_nat] :
( ( member_nat @ A_118 @ ( set_nat @ Xs_128 ) )
=> ( ( linorder_sorted_nat @ Xs_128 )
=> ( ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ A_118
@ ( remove1_nat @ A_118 @ Xs_128 ) )
= Xs_128 ) ) ) ).
thf(fact_488_remove1__commute,axiom,
! [X_151: nat,Y_94: nat,Zs_6: list_nat] :
( ( remove1_nat @ X_151 @ ( remove1_nat @ Y_94 @ Zs_6 ) )
= ( remove1_nat @ Y_94 @ ( remove1_nat @ X_151 @ Zs_6 ) ) ) ).
thf(fact_489_remove1__commute,axiom,
! [X_151: arrow_1389031998le_alt,Y_94: arrow_1389031998le_alt,Zs_6: list_A1402527603le_alt] :
( ( remove356532612le_alt @ X_151 @ ( remove356532612le_alt @ Y_94 @ Zs_6 ) )
= ( remove356532612le_alt @ Y_94 @ ( remove356532612le_alt @ X_151 @ Zs_6 ) ) ) ).
thf(fact_490_take__Nil,axiom,
! [N_53: nat] :
( ( take_A1527720662le_alt @ N_53 @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_491_take__Nil,axiom,
! [N_53: nat] :
( ( take_nat @ N_53 @ nil_nat )
= nil_nat ) ).
thf(fact_492_in__set__takeD,axiom,
! [X_150: arrow_1389031998le_alt,N_52: nat,Xs_127: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_150 @ ( set_Ar1491127311le_alt @ ( take_A1527720662le_alt @ N_52 @ Xs_127 ) ) )
=> ( member998036369le_alt @ X_150 @ ( set_Ar1491127311le_alt @ Xs_127 ) ) ) ).
thf(fact_493_in__set__takeD,axiom,
! [X_150: nat,N_52: nat,Xs_127: list_nat] :
( ( member_nat @ X_150 @ ( set_nat @ ( take_nat @ N_52 @ Xs_127 ) ) )
=> ( member_nat @ X_150 @ ( set_nat @ Xs_127 ) ) ) ).
thf(fact_494_distinct__take,axiom,
! [I_12: nat,Xs_126: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_126 )
=> ( distin1149997281le_alt @ ( take_A1527720662le_alt @ I_12 @ Xs_126 ) ) ) ).
thf(fact_495_distinct__take,axiom,
! [I_12: nat,Xs_126: list_nat] :
( ( distinct_nat @ Xs_126 )
=> ( distinct_nat @ ( take_nat @ I_12 @ Xs_126 ) ) ) ).
thf(fact_496_sorted__take,axiom,
! [N_51: nat,Xs_125: list_nat] :
( ( linorder_sorted_nat @ Xs_125 )
=> ( linorder_sorted_nat @ ( take_nat @ N_51 @ Xs_125 ) ) ) ).
thf(fact_497_remove1_Osimps_I2_J,axiom,
! [Xs_124: list_o,Y_93: $o,X_149: $o] :
( ( ( X_149
<=> Y_93 )
=> ( ( remove1_o @ X_149 @ ( cons_o @ Y_93 @ Xs_124 ) )
= Xs_124 ) )
& ( ( X_149
<=> ~ Y_93 )
=> ( ( remove1_o @ X_149 @ ( cons_o @ Y_93 @ Xs_124 ) )
= ( cons_o @ Y_93 @ ( remove1_o @ X_149 @ Xs_124 ) ) ) ) ) ).
thf(fact_498_remove1_Osimps_I2_J,axiom,
! [Xs_124: list_A1402527603le_alt,X_149: arrow_1389031998le_alt,Y_93: arrow_1389031998le_alt] :
( ( ( X_149 = Y_93 )
=> ( ( remove356532612le_alt @ X_149 @ ( cons_A1142416030le_alt @ Y_93 @ Xs_124 ) )
= Xs_124 ) )
& ( ( X_149 != Y_93 )
=> ( ( remove356532612le_alt @ X_149 @ ( cons_A1142416030le_alt @ Y_93 @ Xs_124 ) )
= ( cons_A1142416030le_alt @ Y_93 @ ( remove356532612le_alt @ X_149 @ Xs_124 ) ) ) ) ) ).
thf(fact_499_remove1_Osimps_I2_J,axiom,
! [Xs_124: list_nat,X_149: nat,Y_93: nat] :
( ( ( X_149 = Y_93 )
=> ( ( remove1_nat @ X_149 @ ( cons_nat @ Y_93 @ Xs_124 ) )
= Xs_124 ) )
& ( ( X_149 != Y_93 )
=> ( ( remove1_nat @ X_149 @ ( cons_nat @ Y_93 @ Xs_124 ) )
= ( cons_nat @ Y_93 @ ( remove1_nat @ X_149 @ Xs_124 ) ) ) ) ) ).
thf(fact_500_remove1_Osimps_I1_J,axiom,
! [X_148: arrow_1389031998le_alt] :
( ( remove356532612le_alt @ X_148 @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_501_remove1_Osimps_I1_J,axiom,
! [X_148: nat] :
( ( remove1_nat @ X_148 @ nil_nat )
= nil_nat ) ).
thf(fact_502_in__set__remove1,axiom,
! [Xs_123: list_A1402527603le_alt,A_117: arrow_1389031998le_alt,B_93: arrow_1389031998le_alt] :
( ( A_117 != B_93 )
=> ( ( member998036369le_alt @ A_117 @ ( set_Ar1491127311le_alt @ ( remove356532612le_alt @ B_93 @ Xs_123 ) ) )
<=> ( member998036369le_alt @ A_117 @ ( set_Ar1491127311le_alt @ Xs_123 ) ) ) ) ).
thf(fact_503_in__set__remove1,axiom,
! [Xs_123: list_nat,A_117: nat,B_93: nat] :
( ( A_117 != B_93 )
=> ( ( member_nat @ A_117 @ ( set_nat @ ( remove1_nat @ B_93 @ Xs_123 ) ) )
<=> ( member_nat @ A_117 @ ( set_nat @ Xs_123 ) ) ) ) ).
thf(fact_504_notin__set__remove1,axiom,
! [Y_92: arrow_1389031998le_alt,X_147: arrow_1389031998le_alt,Xs_122: list_A1402527603le_alt] :
( ~ ( member998036369le_alt @ X_147 @ ( set_Ar1491127311le_alt @ Xs_122 ) )
=> ~ ( member998036369le_alt @ X_147 @ ( set_Ar1491127311le_alt @ ( remove356532612le_alt @ Y_92 @ Xs_122 ) ) ) ) ).
thf(fact_505_notin__set__remove1,axiom,
! [Y_92: nat,X_147: nat,Xs_122: list_nat] :
( ~ ( member_nat @ X_147 @ ( set_nat @ Xs_122 ) )
=> ~ ( member_nat @ X_147 @ ( set_nat @ ( remove1_nat @ Y_92 @ Xs_122 ) ) ) ) ).
thf(fact_506_remove1__idem,axiom,
! [X_146: arrow_1389031998le_alt,Xs_121: list_A1402527603le_alt] :
( ~ ( member998036369le_alt @ X_146 @ ( set_Ar1491127311le_alt @ Xs_121 ) )
=> ( ( remove356532612le_alt @ X_146 @ Xs_121 )
= Xs_121 ) ) ).
thf(fact_507_remove1__idem,axiom,
! [X_146: nat,Xs_121: list_nat] :
( ~ ( member_nat @ X_146 @ ( set_nat @ Xs_121 ) )
=> ( ( remove1_nat @ X_146 @ Xs_121 )
= Xs_121 ) ) ).
thf(fact_508_distinct__remove1,axiom,
! [X_145: arrow_1389031998le_alt,Xs_120: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_120 )
=> ( distin1149997281le_alt @ ( remove356532612le_alt @ X_145 @ Xs_120 ) ) ) ).
thf(fact_509_distinct__remove1,axiom,
! [X_145: nat,Xs_120: list_nat] :
( ( distinct_nat @ Xs_120 )
=> ( distinct_nat @ ( remove1_nat @ X_145 @ Xs_120 ) ) ) ).
thf(fact_510_sorted__remove1,axiom,
! [A_116: nat,Xs_119: list_nat] :
( ( linorder_sorted_nat @ Xs_119 )
=> ( linorder_sorted_nat @ ( remove1_nat @ A_116 @ Xs_119 ) ) ) ).
thf(fact_511_list__all__simps_I1_J,axiom,
! [P_25: $o > $o,X_144: $o,Xs_118: list_o] :
( ( list_all_o @ P_25 @ ( cons_o @ X_144 @ Xs_118 ) )
<=> ( ( P_25 @ X_144 )
& ( list_all_o @ P_25 @ Xs_118 ) ) ) ).
thf(fact_512_list__all__simps_I1_J,axiom,
! [P_25: arrow_1389031998le_alt > $o,X_144: arrow_1389031998le_alt,Xs_118: list_A1402527603le_alt] :
( ( list_a1483787182le_alt @ P_25 @ ( cons_A1142416030le_alt @ X_144 @ Xs_118 ) )
<=> ( ( P_25 @ X_144 )
& ( list_a1483787182le_alt @ P_25 @ Xs_118 ) ) ) ).
thf(fact_513_list__all__simps_I1_J,axiom,
! [P_25: nat > $o,X_144: nat,Xs_118: list_nat] :
( ( list_all_nat @ P_25 @ ( cons_nat @ X_144 @ Xs_118 ) )
<=> ( ( P_25 @ X_144 )
& ( list_all_nat @ P_25 @ Xs_118 ) ) ) ).
thf(fact_514_list__all__simps_I2_J,axiom,
! [P_24: arrow_1389031998le_alt > $o] : ( list_a1483787182le_alt @ P_24 @ nil_Ar52383470le_alt ) ).
thf(fact_515_list__all__simps_I2_J,axiom,
! [P_24: nat > $o] : ( list_all_nat @ P_24 @ nil_nat ) ).
thf(fact_516_list__all__iff,axiom,
! [P_23: arrow_1389031998le_alt > $o,Xs_117: list_A1402527603le_alt] :
( ( list_a1483787182le_alt @ P_23 @ Xs_117 )
<=> ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_117 ) )
=> ( P_23 @ X_4 ) ) ) ).
thf(fact_517_list__all__iff,axiom,
! [P_23: nat > $o,Xs_117: list_nat] :
( ( list_all_nat @ P_23 @ Xs_117 )
<=> ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_117 ) )
=> ( P_23 @ X_4 ) ) ) ).
thf(fact_518_list__all__append,axiom,
! [P_22: arrow_1389031998le_alt > $o,Xs_116: list_A1402527603le_alt,Ys_21: list_A1402527603le_alt] :
( ( list_a1483787182le_alt @ P_22 @ ( append1092755459le_alt @ Xs_116 @ Ys_21 ) )
<=> ( ( list_a1483787182le_alt @ P_22 @ Xs_116 )
& ( list_a1483787182le_alt @ P_22 @ Ys_21 ) ) ) ).
thf(fact_519_list__all__append,axiom,
! [P_22: nat > $o,Xs_116: list_nat,Ys_21: list_nat] :
( ( list_all_nat @ P_22 @ ( append_nat @ Xs_116 @ Ys_21 ) )
<=> ( ( list_all_nat @ P_22 @ Xs_116 )
& ( list_all_nat @ P_22 @ Ys_21 ) ) ) ).
thf(fact_520_list__all__rev,axiom,
! [P_21: nat > $o,Xs_115: list_nat] :
( ( list_all_nat @ P_21 @ ( rev_nat @ Xs_115 ) )
<=> ( list_all_nat @ P_21 @ Xs_115 ) ) ).
thf(fact_521_list__all__rev,axiom,
! [P_21: arrow_1389031998le_alt > $o,Xs_115: list_A1402527603le_alt] :
( ( list_a1483787182le_alt @ P_21 @ ( rev_Ar2020079950le_alt @ Xs_115 ) )
<=> ( list_a1483787182le_alt @ P_21 @ Xs_115 ) ) ).
thf(fact_522_take__0,axiom,
! [Xs_114: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ zero_zero_nat @ Xs_114 )
= nil_Ar52383470le_alt ) ).
thf(fact_523_take__0,axiom,
! [Xs_114: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs_114 )
= nil_nat ) ).
thf(fact_524_take__eq__Nil,axiom,
! [N_50: nat,Xs_113: list_A1402527603le_alt] :
( ( ( take_A1527720662le_alt @ N_50 @ Xs_113 )
= nil_Ar52383470le_alt )
<=> ( ( N_50 = zero_zero_nat )
| ( Xs_113 = nil_Ar52383470le_alt ) ) ) ).
thf(fact_525_take__eq__Nil,axiom,
! [N_50: nat,Xs_113: list_nat] :
( ( ( take_nat @ N_50 @ Xs_113 )
= nil_nat )
<=> ( ( N_50 = zero_zero_nat )
| ( Xs_113 = nil_nat ) ) ) ).
thf(fact_526_take__Suc__Cons,axiom,
! [N_49: nat,X_143: $o,Xs_112: list_o] :
( ( take_o @ ( suc @ N_49 ) @ ( cons_o @ X_143 @ Xs_112 ) )
= ( cons_o @ X_143 @ ( take_o @ N_49 @ Xs_112 ) ) ) ).
thf(fact_527_take__Suc__Cons,axiom,
! [N_49: nat,X_143: arrow_1389031998le_alt,Xs_112: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ ( suc @ N_49 ) @ ( cons_A1142416030le_alt @ X_143 @ Xs_112 ) )
= ( cons_A1142416030le_alt @ X_143 @ ( take_A1527720662le_alt @ N_49 @ Xs_112 ) ) ) ).
thf(fact_528_take__Suc__Cons,axiom,
! [N_49: nat,X_143: nat,Xs_112: list_nat] :
( ( take_nat @ ( suc @ N_49 ) @ ( cons_nat @ X_143 @ Xs_112 ) )
= ( cons_nat @ X_143 @ ( take_nat @ N_49 @ Xs_112 ) ) ) ).
thf(fact_529_remove1__insort,axiom,
! [X_142: nat,Xs_111: list_nat] :
( ( remove1_nat @ X_142
@ ( linord1850739451at_nat
@ ^ [X_4: nat] : X_4
@ X_142
@ Xs_111 ) )
= Xs_111 ) ).
thf(fact_530_take__tl,axiom,
! [N_48: nat,Xs_110: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ N_48 @ ( tl_Arr1379124165le_alt @ Xs_110 ) )
= ( tl_Arr1379124165le_alt @ ( take_A1527720662le_alt @ ( suc @ N_48 ) @ Xs_110 ) ) ) ).
thf(fact_531_take__tl,axiom,
! [N_48: nat,Xs_110: list_nat] :
( ( take_nat @ N_48 @ ( tl_nat @ Xs_110 ) )
= ( tl_nat @ ( take_nat @ ( suc @ N_48 ) @ Xs_110 ) ) ) ).
thf(fact_532_take__Cons,axiom,
! [N_47: nat,X_141: $o,Xs_109: list_o] :
( ( take_o @ N_47 @ ( cons_o @ X_141 @ Xs_109 ) )
= ( nat_case_list_o @ nil_o
@ ^ [M_6: nat] : ( cons_o @ X_141 @ ( take_o @ M_6 @ Xs_109 ) )
@ N_47 ) ) ).
thf(fact_533_take__Cons,axiom,
! [N_47: nat,X_141: arrow_1389031998le_alt,Xs_109: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ N_47 @ ( cons_A1142416030le_alt @ X_141 @ Xs_109 ) )
= ( nat_ca1434653759le_alt @ nil_Ar52383470le_alt
@ ^ [M_6: nat] : ( cons_A1142416030le_alt @ X_141 @ ( take_A1527720662le_alt @ M_6 @ Xs_109 ) )
@ N_47 ) ) ).
thf(fact_534_take__Cons,axiom,
! [N_47: nat,X_141: nat,Xs_109: list_nat] :
( ( take_nat @ N_47 @ ( cons_nat @ X_141 @ Xs_109 ) )
= ( nat_case_list_nat @ nil_nat
@ ^ [M_6: nat] : ( cons_nat @ X_141 @ ( take_nat @ M_6 @ Xs_109 ) )
@ N_47 ) ) ).
thf(fact_535_distinct__insort__insert,axiom,
! [F_24: nat > nat,X_140: nat,Xs_108: list_nat] :
( ( distinct_nat @ Xs_108 )
=> ( distinct_nat @ ( linord297828738at_nat @ F_24 @ X_140 @ Xs_108 ) ) ) ).
thf(fact_536_remove1__append,axiom,
! [Ys_20: list_A1402527603le_alt,X_139: arrow_1389031998le_alt,Xs_107: list_A1402527603le_alt] :
( ( ( member998036369le_alt @ X_139 @ ( set_Ar1491127311le_alt @ Xs_107 ) )
=> ( ( remove356532612le_alt @ X_139 @ ( append1092755459le_alt @ Xs_107 @ Ys_20 ) )
= ( append1092755459le_alt @ ( remove356532612le_alt @ X_139 @ Xs_107 ) @ Ys_20 ) ) )
& ( ~ ( member998036369le_alt @ X_139 @ ( set_Ar1491127311le_alt @ Xs_107 ) )
=> ( ( remove356532612le_alt @ X_139 @ ( append1092755459le_alt @ Xs_107 @ Ys_20 ) )
= ( append1092755459le_alt @ Xs_107 @ ( remove356532612le_alt @ X_139 @ Ys_20 ) ) ) ) ) ).
thf(fact_537_remove1__append,axiom,
! [Ys_20: list_nat,X_139: nat,Xs_107: list_nat] :
( ( ( member_nat @ X_139 @ ( set_nat @ Xs_107 ) )
=> ( ( remove1_nat @ X_139 @ ( append_nat @ Xs_107 @ Ys_20 ) )
= ( append_nat @ ( remove1_nat @ X_139 @ Xs_107 ) @ Ys_20 ) ) )
& ( ~ ( member_nat @ X_139 @ ( set_nat @ Xs_107 ) )
=> ( ( remove1_nat @ X_139 @ ( append_nat @ Xs_107 @ Ys_20 ) )
= ( append_nat @ Xs_107 @ ( remove1_nat @ X_139 @ Ys_20 ) ) ) ) ) ).
thf(fact_538_insort__insert__triv,axiom,
! [X_138: nat,Xs_106: list_nat] :
( ( member_nat @ X_138 @ ( set_nat @ Xs_106 ) )
=> ( ( linord297828738at_nat
@ ^ [X_4: nat] : X_4
@ X_138
@ Xs_106 )
= Xs_106 ) ) ).
thf(fact_539_sorted__insort__insert,axiom,
! [X_137: nat,Xs_105: list_nat] :
( ( linorder_sorted_nat @ Xs_105 )
=> ( linorder_sorted_nat
@ ( linord297828738at_nat
@ ^ [X_4: nat] : X_4
@ X_137
@ Xs_105 ) ) ) ).
thf(fact_540_list__all__cong,axiom,
! [G_2: arrow_1389031998le_alt > $o,F_23: arrow_1389031998le_alt > $o,Xs_104: list_A1402527603le_alt,Ys_19: list_A1402527603le_alt] :
( ( Xs_104 = Ys_19 )
=> ( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Ys_19 ) )
=> ( ( F_23 @ X_4 )
<=> ( G_2 @ X_4 ) ) )
=> ( ( list_a1483787182le_alt @ F_23 @ Xs_104 )
<=> ( list_a1483787182le_alt @ G_2 @ Ys_19 ) ) ) ) ).
thf(fact_541_list__all__cong,axiom,
! [G_2: nat > $o,F_23: nat > $o,Xs_104: list_nat,Ys_19: list_nat] :
( ( Xs_104 = Ys_19 )
=> ( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Ys_19 ) )
=> ( ( F_23 @ X_4 )
<=> ( G_2 @ X_4 ) ) )
=> ( ( list_all_nat @ F_23 @ Xs_104 )
<=> ( list_all_nat @ G_2 @ Ys_19 ) ) ) ) ).
thf(fact_542_take__equalityI,axiom,
! [Xs_103: list_A1402527603le_alt,Ys_18: list_A1402527603le_alt] :
( ! [I_3: nat] :
( ( take_A1527720662le_alt @ I_3 @ Xs_103 )
= ( take_A1527720662le_alt @ I_3 @ Ys_18 ) )
=> ( Xs_103 = Ys_18 ) ) ).
thf(fact_543_take__equalityI,axiom,
! [Xs_103: list_nat,Ys_18: list_nat] :
( ! [I_3: nat] :
( ( take_nat @ I_3 @ Xs_103 )
= ( take_nat @ I_3 @ Ys_18 ) )
=> ( Xs_103 = Ys_18 ) ) ).
thf(fact_544_list_Osize_I2_J,axiom,
! [Fa: $o > nat,A_115: $o,List_1: list_o] :
( ( list_size_o @ Fa @ ( cons_o @ A_115 @ List_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( Fa @ A_115 ) @ ( list_size_o @ Fa @ List_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_545_list_Osize_I2_J,axiom,
! [Fa: arrow_1389031998le_alt > nat,A_115: arrow_1389031998le_alt,List_1: list_A1402527603le_alt] :
( ( list_s1163968880le_alt @ Fa @ ( cons_A1142416030le_alt @ A_115 @ List_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( Fa @ A_115 ) @ ( list_s1163968880le_alt @ Fa @ List_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_546_list_Osize_I2_J,axiom,
! [Fa: nat > nat,A_115: nat,List_1: list_nat] :
( ( list_size_nat @ Fa @ ( cons_nat @ A_115 @ List_1 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( Fa @ A_115 ) @ ( list_size_nat @ Fa @ List_1 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_547_split__list__prop,axiom,
! [P_20: arrow_1389031998le_alt > $o,Xs_102: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_102 ) )
& ( P_20 @ X_4 ) )
=> ? [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt] :
( ? [Zs_5: list_A1402527603le_alt] :
( Xs_102
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
& ( P_20 @ X_4 ) ) ) ).
thf(fact_548_split__list__prop,axiom,
! [P_20: nat > $o,Xs_102: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_102 ) )
& ( P_20 @ X_4 ) )
=> ? [Ys_7: list_nat,X_4: nat] :
( ? [Zs_5: list_nat] :
( Xs_102
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
& ( P_20 @ X_4 ) ) ) ).
thf(fact_549_split__list__prop,axiom,
! [P_20: $o > $o,Xs_102: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_102 ) )
& ( P_20 @ X_4 ) )
=> ? [Ys_7: list_o,X_4: $o] :
( ? [Zs_5: list_o] :
( Xs_102
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
& ( P_20 @ X_4 ) ) ) ).
thf(fact_550_split__list__last__prop,axiom,
! [P_19: arrow_1389031998le_alt > $o,Xs_101: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_101 ) )
& ( P_19 @ X_4 ) )
=> ? [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt,Zs_5: list_A1402527603le_alt] :
( ( Xs_101
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
& ( P_19 @ X_4 )
& ! [Xa: arrow_1389031998le_alt] :
( ( member998036369le_alt @ Xa @ ( set_Ar1491127311le_alt @ Zs_5 ) )
=> ~ ( P_19 @ Xa ) ) ) ) ).
thf(fact_551_split__list__last__prop,axiom,
! [P_19: nat > $o,Xs_101: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_101 ) )
& ( P_19 @ X_4 ) )
=> ? [Ys_7: list_nat,X_4: nat,Zs_5: list_nat] :
( ( Xs_101
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
& ( P_19 @ X_4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Zs_5 ) )
=> ~ ( P_19 @ Xa ) ) ) ) ).
thf(fact_552_split__list__last__prop,axiom,
! [P_19: $o > $o,Xs_101: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_101 ) )
& ( P_19 @ X_4 ) )
=> ? [Ys_7: list_o,X_4: $o,Zs_5: list_o] :
( ( Xs_101
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
& ( P_19 @ X_4 )
& ! [Xa: $o] :
( ( member_o @ Xa @ ( set_o @ Zs_5 ) )
=> ~ ( P_19 @ Xa ) ) ) ) ).
thf(fact_553_split__list__first__prop,axiom,
! [P_18: arrow_1389031998le_alt > $o,Xs_100: list_A1402527603le_alt] :
( ? [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_100 ) )
& ( P_18 @ X_4 ) )
=> ? [Ys_7: list_A1402527603le_alt,X_4: arrow_1389031998le_alt] :
( ? [Zs_5: list_A1402527603le_alt] :
( Xs_100
= ( append1092755459le_alt @ Ys_7 @ ( cons_A1142416030le_alt @ X_4 @ Zs_5 ) ) )
& ( P_18 @ X_4 )
& ! [Xa: arrow_1389031998le_alt] :
( ( member998036369le_alt @ Xa @ ( set_Ar1491127311le_alt @ Ys_7 ) )
=> ~ ( P_18 @ Xa ) ) ) ) ).
thf(fact_554_split__list__first__prop,axiom,
! [P_18: nat > $o,Xs_100: list_nat] :
( ? [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_100 ) )
& ( P_18 @ X_4 ) )
=> ? [Ys_7: list_nat,X_4: nat] :
( ? [Zs_5: list_nat] :
( Xs_100
= ( append_nat @ Ys_7 @ ( cons_nat @ X_4 @ Zs_5 ) ) )
& ( P_18 @ X_4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Ys_7 ) )
=> ~ ( P_18 @ Xa ) ) ) ) ).
thf(fact_555_split__list__first__prop,axiom,
! [P_18: $o > $o,Xs_100: list_o] :
( ? [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_100 ) )
& ( P_18 @ X_4 ) )
=> ? [Ys_7: list_o,X_4: $o] :
( ? [Zs_5: list_o] :
( Xs_100
= ( append_o @ Ys_7 @ ( cons_o @ X_4 @ Zs_5 ) ) )
& ( P_18 @ X_4 )
& ! [Xa: $o] :
( ( member_o @ Xa @ ( set_o @ Ys_7 ) )
=> ~ ( P_18 @ Xa ) ) ) ) ).
thf(fact_556_rev__foldl__cons,axiom,
! [Xs_99: list_nat] :
( ( rev_nat @ Xs_99 )
= ( foldl_list_nat_nat
@ ^ [Xs_23: list_nat,X_4: nat] : ( cons_nat @ X_4 @ Xs_23 )
@ nil_nat
@ Xs_99 ) ) ).
thf(fact_557_rev__foldl__cons,axiom,
! [Xs_99: list_A1402527603le_alt] :
( ( rev_Ar2020079950le_alt @ Xs_99 )
= ( foldl_2120284924le_alt
@ ^ [Xs_23: list_A1402527603le_alt,X_4: arrow_1389031998le_alt] : ( cons_A1142416030le_alt @ X_4 @ Xs_23 )
@ nil_Ar52383470le_alt
@ Xs_99 ) ) ).
thf(fact_558_rev__foldl__cons,axiom,
! [Xs_99: list_o] :
( ( rev_o @ Xs_99 )
= ( foldl_list_o_o
@ ^ [Xs_23: list_o,X_4: $o] : ( cons_o @ X_4 @ Xs_23 )
@ nil_o
@ Xs_99 ) ) ).
thf(fact_559_foldl__absorb0,axiom,
! [X_136: nat,Zs_4: list_nat] :
( ( plus_plus_nat @ X_136 @ ( foldl_nat_nat @ plus_plus_nat @ zero_zero_nat @ Zs_4 ) )
= ( foldl_nat_nat @ plus_plus_nat @ X_136 @ Zs_4 ) ) ).
thf(fact_560_add__right__imp__eq,axiom,
! [B_92: nat,A_114: nat,C_57: nat] :
( ( ( plus_plus_nat @ B_92 @ A_114 )
= ( plus_plus_nat @ C_57 @ A_114 ) )
=> ( B_92 = C_57 ) ) ).
thf(fact_561_add__imp__eq,axiom,
! [A_113: nat,B_91: nat,C_56: nat] :
( ( ( plus_plus_nat @ A_113 @ B_91 )
= ( plus_plus_nat @ A_113 @ C_56 ) )
=> ( B_91 = C_56 ) ) ).
thf(fact_562_add__left__imp__eq,axiom,
! [A_112: nat,B_90: nat,C_55: nat] :
( ( ( plus_plus_nat @ A_112 @ B_90 )
= ( plus_plus_nat @ A_112 @ C_55 ) )
=> ( B_90 = C_55 ) ) ).
thf(fact_563_add__right__cancel,axiom,
! [B_89: nat,A_111: nat,C_54: nat] :
( ( ( plus_plus_nat @ B_89 @ A_111 )
= ( plus_plus_nat @ C_54 @ A_111 ) )
<=> ( B_89 = C_54 ) ) ).
thf(fact_564_add__left__cancel,axiom,
! [A_110: nat,B_88: nat,C_53: nat] :
( ( ( plus_plus_nat @ A_110 @ B_88 )
= ( plus_plus_nat @ A_110 @ C_53 ) )
<=> ( B_88 = C_53 ) ) ).
thf(fact_565_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_109: nat,B_87: nat,C_52: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_109 @ B_87 ) @ C_52 )
= ( plus_plus_nat @ A_109 @ ( plus_plus_nat @ B_87 @ C_52 ) ) ) ).
thf(fact_566_foldl__assoc,axiom,
! [X_135: nat,Y_91: nat,Zs_3: list_nat] :
( ( foldl_nat_nat @ plus_plus_nat @ ( plus_plus_nat @ X_135 @ Y_91 ) @ Zs_3 )
= ( plus_plus_nat @ X_135 @ ( foldl_nat_nat @ plus_plus_nat @ Y_91 @ Zs_3 ) ) ) ).
thf(fact_567_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M = N ) ) ).
thf(fact_568_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N ) )
<=> ( M = N ) ) ).
thf(fact_569_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
thf(fact_570_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_6: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z_6 ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z_6 ) ) ) ).
thf(fact_571_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_572_add_Ocomm__neutral,axiom,
! [A_108: nat] :
( ( plus_plus_nat @ A_108 @ zero_zero_nat )
= A_108 ) ).
thf(fact_573_add__0__right,axiom,
! [A_107: nat] :
( ( plus_plus_nat @ A_107 @ zero_zero_nat )
= A_107 ) ).
thf(fact_574_add__0,axiom,
! [A_106: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_106 )
= A_106 ) ).
thf(fact_575_add__0__left,axiom,
! [A_105: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_105 )
= A_105 ) ).
thf(fact_576_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_577_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_578_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_579_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_580_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_581_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_582_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_583_foldl__Cons,axiom,
! [F_22: nat > nat > nat,A_104: nat,X_134: nat,Xs_98: list_nat] :
( ( foldl_nat_nat @ F_22 @ A_104 @ ( cons_nat @ X_134 @ Xs_98 ) )
= ( foldl_nat_nat @ F_22 @ ( F_22 @ A_104 @ X_134 ) @ Xs_98 ) ) ).
thf(fact_584_foldl__Nil,axiom,
! [F_21: nat > nat > nat,A_103: nat] :
( ( foldl_nat_nat @ F_21 @ A_103 @ nil_nat )
= A_103 ) ).
thf(fact_585_foldl__append,axiom,
! [F_20: nat > nat > nat,A_102: nat,Xs_97: list_nat,Ys_17: list_nat] :
( ( foldl_nat_nat @ F_20 @ A_102 @ ( append_nat @ Xs_97 @ Ys_17 ) )
= ( foldl_nat_nat @ F_20 @ ( foldl_nat_nat @ F_20 @ A_102 @ Xs_97 ) @ Ys_17 ) ) ).
thf(fact_586_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_587_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_588_replicate__add,axiom,
! [N_46: nat,M_11: nat,X_133: arrow_1389031998le_alt] :
( ( replic277728168le_alt @ ( plus_plus_nat @ N_46 @ M_11 ) @ X_133 )
= ( append1092755459le_alt @ ( replic277728168le_alt @ N_46 @ X_133 ) @ ( replic277728168le_alt @ M_11 @ X_133 ) ) ) ).
thf(fact_589_replicate__add,axiom,
! [N_46: nat,M_11: nat,X_133: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N_46 @ M_11 ) @ X_133 )
= ( append_nat @ ( replicate_nat @ N_46 @ X_133 ) @ ( replicate_nat @ M_11 @ X_133 ) ) ) ).
thf(fact_590_foldl__rev,axiom,
! [S_5: nat,Xs_96: list_nat,F_19: nat > nat > nat] :
( ! [X_4: nat,Y_40: nat,S_3: nat] :
( ( F_19 @ ( F_19 @ S_3 @ X_4 ) @ Y_40 )
= ( F_19 @ ( F_19 @ S_3 @ Y_40 ) @ X_4 ) )
=> ( ( foldl_nat_nat @ F_19 @ S_5 @ ( rev_nat @ Xs_96 ) )
= ( foldl_nat_nat @ F_19 @ S_5 @ Xs_96 ) ) ) ).
thf(fact_591_foldl__cong,axiom,
! [F_18: nat > nat > nat,G_1: nat > nat > nat,L_3: list_nat,K_6: list_nat,A_100: nat,B_86: nat] :
( ( A_100 = B_86 )
=> ( ( L_3 = K_6 )
=> ( ! [A_101: nat,X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ L_3 ) )
=> ( ( F_18 @ A_101 @ X_4 )
= ( G_1 @ A_101 @ X_4 ) ) )
=> ( ( foldl_nat_nat @ F_18 @ A_100 @ L_3 )
= ( foldl_nat_nat @ G_1 @ B_86 @ K_6 ) ) ) ) ) ).
thf(fact_592_foldl__weak__invariant,axiom,
! [F_17: nat > nat > nat,Xs_95: list_nat,P_17: nat > $o,S_4: nat] :
( ( P_17 @ S_4 )
=> ( ! [S_3: nat,X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_95 ) )
=> ( ( P_17 @ S_3 )
=> ( P_17 @ ( F_17 @ S_3 @ X_4 ) ) ) )
=> ( P_17 @ ( foldl_nat_nat @ F_17 @ S_4 @ Xs_95 ) ) ) ) ).
thf(fact_593_sort__foldl__insort,axiom,
! [Xs_94: list_nat] :
( ( linord852754176at_nat
@ ^ [X_4: nat] : X_4
@ Xs_94 )
= ( foldl_list_nat_nat
@ ^ [Ys_7: list_nat,X_4: nat] :
( linord1850739451at_nat
@ ^ [Y_40: nat] : Y_40
@ X_4
@ Ys_7 )
@ nil_nat
@ Xs_94 ) ) ).
thf(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_99: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_99 )
= A_99 ) ).
thf(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_98: nat] :
( ( plus_plus_nat @ A_98 @ zero_zero_nat )
= A_98 ) ).
thf(fact_596_add__0__iff,axiom,
! [B_85: nat,A_97: nat] :
( ( B_85
= ( plus_plus_nat @ B_85 @ A_97 ) )
<=> ( A_97 = zero_zero_nat ) ) ).
thf(fact_597_sum__eq__0__conv,axiom,
! [M: nat,Ns: list_nat] :
( ( ( foldl_nat_nat @ plus_plus_nat @ M @ Ns )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Ns ) )
=> ( X_4 = zero_zero_nat ) ) ) ) ).
thf(fact_598_sorted__sort,axiom,
! [Xs_93: list_nat] :
( linorder_sorted_nat
@ ( linord852754176at_nat
@ ^ [X_4: nat] : X_4
@ Xs_93 ) ) ).
thf(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_96: nat,B_84: nat,C_51: nat,D_9: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_96 @ B_84 ) @ ( plus_plus_nat @ C_51 @ D_9 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_96 @ C_51 ) @ ( plus_plus_nat @ B_84 @ D_9 ) ) ) ).
thf(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_95: nat,B_83: nat,C_50: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_95 @ B_83 ) @ C_50 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_95 @ C_50 ) @ B_83 ) ) ).
thf(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_94: nat,B_82: nat,C_49: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_94 @ B_82 ) @ C_49 )
= ( plus_plus_nat @ A_94 @ ( plus_plus_nat @ B_82 @ C_49 ) ) ) ).
thf(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_93: nat,C_48: nat,D_8: nat] :
( ( plus_plus_nat @ A_93 @ ( plus_plus_nat @ C_48 @ D_8 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_93 @ C_48 ) @ D_8 ) ) ).
thf(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_92: nat,C_47: nat,D_7: nat] :
( ( plus_plus_nat @ A_92 @ ( plus_plus_nat @ C_47 @ D_7 ) )
= ( plus_plus_nat @ C_47 @ ( plus_plus_nat @ A_92 @ D_7 ) ) ) ).
thf(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_91: nat,C_46: nat] :
( ( plus_plus_nat @ A_91 @ C_46 )
= ( plus_plus_nat @ C_46 @ A_91 ) ) ).
thf(fact_605_sort__key__simps_I2_J,axiom,
! [F_16: nat > nat,X_132: nat,Xs_92: list_nat] :
( ( linord852754176at_nat @ F_16 @ ( cons_nat @ X_132 @ Xs_92 ) )
= ( linord1850739451at_nat @ F_16 @ X_132 @ ( linord852754176at_nat @ F_16 @ Xs_92 ) ) ) ).
thf(fact_606_foldl__fun__comm,axiom,
! [S_2: nat,Xs_91: list_nat,X_131: nat,F_15: nat > nat > nat] :
( ! [X_4: nat,Y_40: nat,S_3: nat] :
( ( F_15 @ ( F_15 @ S_3 @ X_4 ) @ Y_40 )
= ( F_15 @ ( F_15 @ S_3 @ Y_40 ) @ X_4 ) )
=> ( ( F_15 @ ( foldl_nat_nat @ F_15 @ S_2 @ Xs_91 ) @ X_131 )
= ( foldl_nat_nat @ F_15 @ ( F_15 @ S_2 @ X_131 ) @ Xs_91 ) ) ) ).
thf(fact_607_nat_Osize_I2_J,axiom,
! [Nat: nat] :
( ( nat_size @ ( suc @ Nat ) )
= ( plus_plus_nat @ ( nat_size @ Nat ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_608_foldl__foldr1,axiom,
! [Xs_90: list_nat] :
( ( foldl_nat_nat @ plus_plus_nat @ zero_zero_nat @ Xs_90 )
= ( foldr_nat_nat @ plus_plus_nat @ Xs_90 @ zero_zero_nat ) ) ).
thf(fact_609_foldl__foldr1__lemma,axiom,
! [A_90: nat,Xs_89: list_nat] :
( ( foldl_nat_nat @ plus_plus_nat @ A_90 @ Xs_89 )
= ( plus_plus_nat @ A_90 @ ( foldr_nat_nat @ plus_plus_nat @ Xs_89 @ zero_zero_nat ) ) ) ).
thf(fact_610_foldl__invariant,axiom,
! [F_14: nat > nat > nat,P_16: nat > $o,Xs_88: list_nat,Q_1: nat > $o,X_130: nat] :
( ( Q_1 @ X_130 )
=> ( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_88 ) )
=> ( P_16 @ X_4 ) )
=> ( ! [X_4: nat,Y_40: nat] :
( ( ( P_16 @ X_4 )
& ( Q_1 @ Y_40 ) )
=> ( Q_1 @ ( F_14 @ Y_40 @ X_4 ) ) )
=> ( Q_1 @ ( foldl_nat_nat @ F_14 @ X_130 @ Xs_88 ) ) ) ) ) ).
thf(fact_611_nat_Osize_I1_J,axiom,
( ( nat_size @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_612_foldr__conv__foldl,axiom,
! [Xs_87: list_nat,A_89: nat] :
( ( foldr_nat_nat @ plus_plus_nat @ Xs_87 @ A_89 )
= ( foldl_nat_nat @ plus_plus_nat @ A_89 @ Xs_87 ) ) ).
thf(fact_613_foldr__foldl,axiom,
! [F_13: nat > nat > nat,Xs_86: list_nat,A_88: nat] :
( ( foldr_nat_nat @ F_13 @ Xs_86 @ A_88 )
= ( foldl_nat_nat
@ ^ [X_4: nat,Y_40: nat] : ( F_13 @ Y_40 @ X_4 )
@ A_88
@ ( rev_nat @ Xs_86 ) ) ) ).
thf(fact_614_foldl__foldr,axiom,
! [F_12: nat > nat > nat,A_87: nat,Xs_85: list_nat] :
( ( foldl_nat_nat @ F_12 @ A_87 @ Xs_85 )
= ( foldr_nat_nat
@ ^ [X_4: nat,Y_40: nat] : ( F_12 @ Y_40 @ X_4 )
@ ( rev_nat @ Xs_85 )
@ A_87 ) ) ).
thf(fact_615_nat_Osize_I4_J,axiom,
! [Nat: nat] :
( ( size_size_nat @ ( suc @ Nat ) )
= ( plus_plus_nat @ ( size_size_nat @ Nat ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_616_foldr__max__sorted,axiom,
! [Y_90: nat,Xs_84: list_nat] :
( ( linorder_sorted_nat @ ( rev_nat @ Xs_84 ) )
=> ( ( ( Xs_84 = nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs_84 @ Y_90 )
= Y_90 ) )
& ( ( Xs_84 != nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs_84 @ Y_90 )
= ( ord_max_nat @ ( nth_nat @ Xs_84 @ zero_zero_nat ) @ Y_90 ) ) ) ) ) ).
thf(fact_617_elem__le__sum,axiom,
! [N: nat,Ns: list_nat] :
( ( member_nat @ N @ ( set_nat @ Ns ) )
=> ( ord_less_eq_nat @ N @ ( foldl_nat_nat @ plus_plus_nat @ zero_zero_nat @ Ns ) ) ) ).
thf(fact_618_bool_Osize_I1_J,axiom,
( ( bool_size @ $true )
= zero_zero_nat ) ).
thf(fact_619_bool_Osize_I2_J,axiom,
( ( bool_size @ $false )
= zero_zero_nat ) ).
thf(fact_620_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_621_nat__size,axiom,
! [N: nat] :
( ( size_size_nat @ N )
= N ) ).
thf(fact_622_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_623_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_624_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_625_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
thf(fact_626_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_627_max__add__distrib__left,axiom,
! [X_129: nat,Y_89: nat,Z_33: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X_129 @ Y_89 ) @ Z_33 )
= ( ord_max_nat @ ( plus_plus_nat @ X_129 @ Z_33 ) @ ( plus_plus_nat @ Y_89 @ Z_33 ) ) ) ).
thf(fact_628_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
thf(fact_629_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_630_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
thf(fact_631_add__le__cancel__right,axiom,
! [A_86: nat,C_45: nat,B_81: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_86 @ C_45 ) @ ( plus_plus_nat @ B_81 @ C_45 ) )
<=> ( ord_less_eq_nat @ A_86 @ B_81 ) ) ).
thf(fact_632_add__le__cancel__left,axiom,
! [C_44: nat,A_85: nat,B_80: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_44 @ A_85 ) @ ( plus_plus_nat @ C_44 @ B_80 ) )
<=> ( ord_less_eq_nat @ A_85 @ B_80 ) ) ).
thf(fact_633_add__right__mono,axiom,
! [C_43: nat,A_84: nat,B_79: nat] :
( ( ord_less_eq_nat @ A_84 @ B_79 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_84 @ C_43 ) @ ( plus_plus_nat @ B_79 @ C_43 ) ) ) ).
thf(fact_634_add__left__mono,axiom,
! [C_42: nat,A_83: nat,B_78: nat] :
( ( ord_less_eq_nat @ A_83 @ B_78 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_42 @ A_83 ) @ ( plus_plus_nat @ C_42 @ B_78 ) ) ) ).
thf(fact_635_add__mono,axiom,
! [C_41: nat,D_6: nat,A_82: nat,B_77: nat] :
( ( ord_less_eq_nat @ A_82 @ B_77 )
=> ( ( ord_less_eq_nat @ C_41 @ D_6 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_82 @ C_41 ) @ ( plus_plus_nat @ B_77 @ D_6 ) ) ) ) ).
thf(fact_636_add__le__imp__le__right,axiom,
! [A_81: nat,C_40: nat,B_76: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_81 @ C_40 ) @ ( plus_plus_nat @ B_76 @ C_40 ) )
=> ( ord_less_eq_nat @ A_81 @ B_76 ) ) ).
thf(fact_637_add__le__imp__le__left,axiom,
! [C_39: nat,A_80: nat,B_75: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_39 @ A_80 ) @ ( plus_plus_nat @ C_39 @ B_75 ) )
=> ( ord_less_eq_nat @ A_80 @ B_75 ) ) ).
thf(fact_638_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_639_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_640_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_641_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_642_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
thf(fact_643_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_644_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_645_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
thf(fact_646_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_647_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_648_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_649_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K_2: nat] :
( N
= ( plus_plus_nat @ M @ K_2 ) ) ) ).
thf(fact_650_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_651_trans__le__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_652_trans__le__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_653_add__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_654_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_655_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
thf(fact_656_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_657_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_658_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( nat_case_nat @ ( suc @ N )
@ ^ [M_7: nat] : ( suc @ ( ord_max_nat @ N @ M_7 ) )
@ M ) ) ).
thf(fact_659_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ ( suc @ N )
@ ^ [M_7: nat] : ( suc @ ( ord_max_nat @ M_7 @ N ) )
@ M ) ) ).
thf(fact_660_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
thf(fact_661_add__nonpos__nonpos,axiom,
! [B_74: nat,A_79: nat] :
( ( ord_less_eq_nat @ A_79 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_74 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_79 @ B_74 ) @ zero_zero_nat ) ) ) ).
thf(fact_662_add__increasing2,axiom,
! [B_73: nat,A_78: nat,C_38: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C_38 )
=> ( ( ord_less_eq_nat @ B_73 @ A_78 )
=> ( ord_less_eq_nat @ B_73 @ ( plus_plus_nat @ A_78 @ C_38 ) ) ) ) ).
thf(fact_663_add__increasing,axiom,
! [B_72: nat,C_37: nat,A_77: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_77 )
=> ( ( ord_less_eq_nat @ B_72 @ C_37 )
=> ( ord_less_eq_nat @ B_72 @ ( plus_plus_nat @ A_77 @ C_37 ) ) ) ) ).
thf(fact_664_add__nonneg__eq__0__iff,axiom,
! [Y_88: nat,X_128: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X_128 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_88 )
=> ( ( ( plus_plus_nat @ X_128 @ Y_88 )
= zero_zero_nat )
<=> ( ( X_128 = zero_zero_nat )
& ( Y_88 = zero_zero_nat ) ) ) ) ) ).
thf(fact_665_add__nonneg__nonneg,axiom,
! [B_71: nat,A_76: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_76 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_71 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A_76 @ B_71 ) ) ) ) ).
thf(fact_666_sorted__many,axiom,
! [Zs_2: list_nat,X_127: nat,Y_87: nat] :
( ( ord_less_eq_nat @ X_127 @ Y_87 )
=> ( ( linorder_sorted_nat @ ( cons_nat @ Y_87 @ Zs_2 ) )
=> ( linorder_sorted_nat @ ( cons_nat @ X_127 @ ( cons_nat @ Y_87 @ Zs_2 ) ) ) ) ) ).
thf(fact_667_sorted__many__eq,axiom,
! [X_126: nat,Y_86: nat,Zs_1: list_nat] :
( ( linorder_sorted_nat @ ( cons_nat @ X_126 @ ( cons_nat @ Y_86 @ Zs_1 ) ) )
<=> ( ( ord_less_eq_nat @ X_126 @ Y_86 )
& ( linorder_sorted_nat @ ( cons_nat @ Y_86 @ Zs_1 ) ) ) ) ).
thf(fact_668_insort__key_Osimps_I2_J,axiom,
! [Ys_16: list_nat,F_11: nat > nat,X_125: nat,Y_85: nat] :
( ( ( ord_less_eq_nat @ ( F_11 @ X_125 ) @ ( F_11 @ Y_85 ) )
=> ( ( linord1850739451at_nat @ F_11 @ X_125 @ ( cons_nat @ Y_85 @ Ys_16 ) )
= ( cons_nat @ X_125 @ ( cons_nat @ Y_85 @ Ys_16 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( F_11 @ X_125 ) @ ( F_11 @ Y_85 ) )
=> ( ( linord1850739451at_nat @ F_11 @ X_125 @ ( cons_nat @ Y_85 @ Ys_16 ) )
= ( cons_nat @ Y_85 @ ( linord1850739451at_nat @ F_11 @ X_125 @ Ys_16 ) ) ) ) ) ).
thf(fact_669_list__size__estimation_H,axiom,
! [Y_84: nat,F_10: arrow_1389031998le_alt > nat,X_124: arrow_1389031998le_alt,Xs_83: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_124 @ ( set_Ar1491127311le_alt @ Xs_83 ) )
=> ( ( ord_less_eq_nat @ Y_84 @ ( F_10 @ X_124 ) )
=> ( ord_less_eq_nat @ Y_84 @ ( list_s1163968880le_alt @ F_10 @ Xs_83 ) ) ) ) ).
thf(fact_670_list__size__estimation_H,axiom,
! [Y_84: nat,F_10: nat > nat,X_124: nat,Xs_83: list_nat] :
( ( member_nat @ X_124 @ ( set_nat @ Xs_83 ) )
=> ( ( ord_less_eq_nat @ Y_84 @ ( F_10 @ X_124 ) )
=> ( ord_less_eq_nat @ Y_84 @ ( list_size_nat @ F_10 @ Xs_83 ) ) ) ) ).
thf(fact_671_nat_Osize_I3_J,axiom,
( ( size_size_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_672_start__le__sum,axiom,
! [Ns: list_nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( foldl_nat_nat @ plus_plus_nat @ N @ Ns ) ) ) ).
thf(fact_673_sorted__Cons,axiom,
! [X_123: nat,Xs_82: list_nat] :
( ( linorder_sorted_nat @ ( cons_nat @ X_123 @ Xs_82 ) )
<=> ( ( linorder_sorted_nat @ Xs_82 )
& ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_82 ) )
=> ( ord_less_eq_nat @ X_123 @ X_4 ) ) ) ) ).
thf(fact_674_sorted__append,axiom,
! [Xs_81: list_nat,Ys_15: list_nat] :
( ( linorder_sorted_nat @ ( append_nat @ Xs_81 @ Ys_15 ) )
<=> ( ( linorder_sorted_nat @ Xs_81 )
& ( linorder_sorted_nat @ Ys_15 )
& ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_81 ) )
=> ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Ys_15 ) )
=> ( ord_less_eq_nat @ X_4 @ Xa ) ) ) ) ) ).
thf(fact_675_sorted_Osimps,axiom,
! [A_75: list_nat] :
( ( linorder_sorted_nat @ A_75 )
<=> ( ( A_75 = nil_nat )
| ? [Xs_23: list_nat,X_4: nat] :
( ( A_75
= ( cons_nat @ X_4 @ Xs_23 ) )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat @ Xs_23 ) )
=> ( ord_less_eq_nat @ X_4 @ Xa ) )
& ( linorder_sorted_nat @ Xs_23 ) ) ) ) ).
thf(fact_676_insort__is__Cons,axiom,
! [F_9: nat > nat,A_74: nat,Xs_80: list_nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_80 ) )
=> ( ord_less_eq_nat @ ( F_9 @ A_74 ) @ ( F_9 @ X_4 ) ) )
=> ( ( linord1850739451at_nat @ F_9 @ A_74 @ Xs_80 )
= ( cons_nat @ A_74 @ Xs_80 ) ) ) ).
thf(fact_677_sorted_OCons,axiom,
! [X_122: nat,Xs_79: list_nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_79 ) )
=> ( ord_less_eq_nat @ X_122 @ X_4 ) )
=> ( ( linorder_sorted_nat @ Xs_79 )
=> ( linorder_sorted_nat @ ( cons_nat @ X_122 @ Xs_79 ) ) ) ) ).
thf(fact_678_lift__Suc__mono__le,axiom,
! [N_45: nat,N_44: nat,F_8: nat > nat] :
( ! [N_23: nat] : ( ord_less_eq_nat @ ( F_8 @ N_23 ) @ ( F_8 @ ( suc @ N_23 ) ) )
=> ( ( ord_less_eq_nat @ N_45 @ N_44 )
=> ( ord_less_eq_nat @ ( F_8 @ N_45 ) @ ( F_8 @ N_44 ) ) ) ) ).
thf(fact_679_Suc__le__D,axiom,
! [N: nat,M_10: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M_10 )
=> ? [M_6: nat] :
( M_10
= ( suc @ M_6 ) ) ) ).
thf(fact_680_order__refl,axiom,
! [X_121: nat] : ( ord_less_eq_nat @ X_121 @ X_121 ) ).
thf(fact_681_set__subset__Cons,axiom,
! [Xs_78: list_A1402527603le_alt,X_120: arrow_1389031998le_alt] : ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ Xs_78 ) @ ( set_Ar1491127311le_alt @ ( cons_A1142416030le_alt @ X_120 @ Xs_78 ) ) ) ).
thf(fact_682_set__subset__Cons,axiom,
! [Xs_78: list_nat,X_120: nat] : ( ord_less_eq_nat_o @ ( set_nat @ Xs_78 ) @ ( set_nat @ ( cons_nat @ X_120 @ Xs_78 ) ) ) ).
thf(fact_683_set__subset__Cons,axiom,
! [Xs_78: list_o,X_120: $o] : ( ord_less_eq_o_o @ ( set_o @ Xs_78 ) @ ( set_o @ ( cons_o @ X_120 @ Xs_78 ) ) ) ).
thf(fact_684_set__take__subset,axiom,
! [N_43: nat,Xs_77: list_A1402527603le_alt] : ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ ( take_A1527720662le_alt @ N_43 @ Xs_77 ) ) @ ( set_Ar1491127311le_alt @ Xs_77 ) ) ).
thf(fact_685_set__take__subset,axiom,
! [N_43: nat,Xs_77: list_nat] : ( ord_less_eq_nat_o @ ( set_nat @ ( take_nat @ N_43 @ Xs_77 ) ) @ ( set_nat @ Xs_77 ) ) ).
thf(fact_686_set__remove1__subset,axiom,
! [X_119: arrow_1389031998le_alt,Xs_76: list_A1402527603le_alt] : ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ ( remove356532612le_alt @ X_119 @ Xs_76 ) ) @ ( set_Ar1491127311le_alt @ Xs_76 ) ) ).
thf(fact_687_set__remove1__subset,axiom,
! [X_119: nat,Xs_76: list_nat] : ( ord_less_eq_nat_o @ ( set_nat @ ( remove1_nat @ X_119 @ Xs_76 ) ) @ ( set_nat @ Xs_76 ) ) ).
thf(fact_688_set__sublist__subset,axiom,
! [Xs_75: list_A1402527603le_alt,I_11: nat > $o] : ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ ( sublis814142963le_alt @ Xs_75 @ I_11 ) ) @ ( set_Ar1491127311le_alt @ Xs_75 ) ) ).
thf(fact_689_set__sublist__subset,axiom,
! [Xs_75: list_nat,I_11: nat > $o] : ( ord_less_eq_nat_o @ ( set_nat @ ( sublist_nat @ Xs_75 @ I_11 ) ) @ ( set_nat @ Xs_75 ) ) ).
thf(fact_690_linorder__linear,axiom,
! [X_118: nat,Y_83: nat] :
( ( ord_less_eq_nat @ X_118 @ Y_83 )
| ( ord_less_eq_nat @ Y_83 @ X_118 ) ) ).
thf(fact_691_order__eq__iff,axiom,
! [X_117: nat,Y_82: nat] :
( ( X_117 = Y_82 )
<=> ( ( ord_less_eq_nat @ X_117 @ Y_82 )
& ( ord_less_eq_nat @ Y_82 @ X_117 ) ) ) ).
thf(fact_692_order__eq__refl,axiom,
! [X_116: nat,Y_81: nat] :
( ( X_116 = Y_81 )
=> ( ord_less_eq_nat @ X_116 @ Y_81 ) ) ).
thf(fact_693_order__antisym__conv,axiom,
! [Y_80: nat,X_115: nat] :
( ( ord_less_eq_nat @ Y_80 @ X_115 )
=> ( ( ord_less_eq_nat @ X_115 @ Y_80 )
<=> ( X_115 = Y_80 ) ) ) ).
thf(fact_694_ord__eq__le__trans,axiom,
! [C_36: nat,A_73: nat,B_70: nat] :
( ( A_73 = B_70 )
=> ( ( ord_less_eq_nat @ B_70 @ C_36 )
=> ( ord_less_eq_nat @ A_73 @ C_36 ) ) ) ).
thf(fact_695_xt1_I3_J,axiom,
! [C_35: nat,A_72: nat,B_69: nat] :
( ( A_72 = B_69 )
=> ( ( ord_less_eq_nat @ C_35 @ B_69 )
=> ( ord_less_eq_nat @ C_35 @ A_72 ) ) ) ).
thf(fact_696_ord__le__eq__trans,axiom,
! [C_34: nat,A_71: nat,B_68: nat] :
( ( ord_less_eq_nat @ A_71 @ B_68 )
=> ( ( B_68 = C_34 )
=> ( ord_less_eq_nat @ A_71 @ C_34 ) ) ) ).
thf(fact_697_xt1_I4_J,axiom,
! [C_33: nat,B_67: nat,A_70: nat] :
( ( ord_less_eq_nat @ B_67 @ A_70 )
=> ( ( B_67 = C_33 )
=> ( ord_less_eq_nat @ C_33 @ A_70 ) ) ) ).
thf(fact_698_order__antisym,axiom,
! [X_114: nat,Y_79: nat] :
( ( ord_less_eq_nat @ X_114 @ Y_79 )
=> ( ( ord_less_eq_nat @ Y_79 @ X_114 )
=> ( X_114 = Y_79 ) ) ) ).
thf(fact_699_order__trans,axiom,
! [Z_32: nat,X_113: nat,Y_78: nat] :
( ( ord_less_eq_nat @ X_113 @ Y_78 )
=> ( ( ord_less_eq_nat @ Y_78 @ Z_32 )
=> ( ord_less_eq_nat @ X_113 @ Z_32 ) ) ) ).
thf(fact_700_xt1_I5_J,axiom,
! [Y_77: nat,X_112: nat] :
( ( ord_less_eq_nat @ Y_77 @ X_112 )
=> ( ( ord_less_eq_nat @ X_112 @ Y_77 )
=> ( X_112 = Y_77 ) ) ) ).
thf(fact_701_xt1_I6_J,axiom,
! [Z_31: nat,Y_76: nat,X_111: nat] :
( ( ord_less_eq_nat @ Y_76 @ X_111 )
=> ( ( ord_less_eq_nat @ Z_31 @ Y_76 )
=> ( ord_less_eq_nat @ Z_31 @ X_111 ) ) ) ).
thf(fact_702_linorder__le__cases,axiom,
! [X_110: nat,Y_75: nat] :
( ~ ( ord_less_eq_nat @ X_110 @ Y_75 )
=> ( ord_less_eq_nat @ Y_75 @ X_110 ) ) ).
thf(fact_703_set__take__subset__set__take,axiom,
! [Xs_74: list_A1402527603le_alt,M_9: nat,N_42: nat] :
( ( ord_less_eq_nat @ M_9 @ N_42 )
=> ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ ( take_A1527720662le_alt @ M_9 @ Xs_74 ) ) @ ( set_Ar1491127311le_alt @ ( take_A1527720662le_alt @ N_42 @ Xs_74 ) ) ) ) ).
thf(fact_704_set__take__subset__set__take,axiom,
! [Xs_74: list_nat,M_9: nat,N_42: nat] :
( ( ord_less_eq_nat @ M_9 @ N_42 )
=> ( ord_less_eq_nat_o @ ( set_nat @ ( take_nat @ M_9 @ Xs_74 ) ) @ ( set_nat @ ( take_nat @ N_42 @ Xs_74 ) ) ) ) ).
thf(fact_705_le__max__iff__disj,axiom,
! [Z_30: nat,X_109: nat,Y_74: nat] :
( ( ord_less_eq_nat @ Z_30 @ ( ord_max_nat @ X_109 @ Y_74 ) )
<=> ( ( ord_less_eq_nat @ Z_30 @ X_109 )
| ( ord_less_eq_nat @ Z_30 @ Y_74 ) ) ) ).
thf(fact_706_max__leastR,axiom,
! [X_108: nat,Least_3: nat] :
( ( all @ ( ord_less_eq_nat @ Least_3 ) )
=> ( ( ord_max_nat @ X_108 @ Least_3 )
= X_108 ) ) ).
thf(fact_707_max__leastL,axiom,
! [X_107: nat,Least_2: nat] :
( ( all @ ( ord_less_eq_nat @ Least_2 ) )
=> ( ( ord_max_nat @ Least_2 @ X_107 )
= X_107 ) ) ).
thf(fact_708_max__ord__max,axiom,
( ord_max_nat
= ( max_nat @ ord_less_eq_nat ) ) ).
thf(fact_709_termination__basic__simps_I3_J,axiom,
! [Z_6: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z_6 ) ) ) ).
thf(fact_710_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z_6: nat] :
( ( ord_less_eq_nat @ X @ Z_6 )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z_6 ) ) ) ).
thf(fact_711_subsetD,axiom,
! [C_32: arrow_1389031998le_alt,A_69: arrow_1389031998le_alt > $o,B_66: arrow_1389031998le_alt > $o] :
( ( ord_le1423491573_alt_o @ A_69 @ B_66 )
=> ( ( member998036369le_alt @ C_32 @ A_69 )
=> ( member998036369le_alt @ C_32 @ B_66 ) ) ) ).
thf(fact_712_subsetD,axiom,
! [C_32: nat,A_69: nat > $o,B_66: nat > $o] :
( ( ord_less_eq_nat_o @ A_69 @ B_66 )
=> ( ( member_nat @ C_32 @ A_69 )
=> ( member_nat @ C_32 @ B_66 ) ) ) ).
thf(fact_713_min__max_Ole__supE,axiom,
! [A_68: nat,B_65: nat,X_106: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ A_68 @ B_65 ) @ X_106 )
=> ~ ( ( ord_less_eq_nat @ A_68 @ X_106 )
=> ~ ( ord_less_eq_nat @ B_65 @ X_106 ) ) ) ).
thf(fact_714_min__max_Osup__mono,axiom,
! [B_64: nat,D_5: nat,A_67: nat,C_31: nat] :
( ( ord_less_eq_nat @ A_67 @ C_31 )
=> ( ( ord_less_eq_nat @ B_64 @ D_5 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ A_67 @ B_64 ) @ ( ord_max_nat @ C_31 @ D_5 ) ) ) ) ).
thf(fact_715_in__mono,axiom,
! [X_105: arrow_1389031998le_alt,A_66: arrow_1389031998le_alt > $o,B_63: arrow_1389031998le_alt > $o] :
( ( ord_le1423491573_alt_o @ A_66 @ B_63 )
=> ( ( member998036369le_alt @ X_105 @ A_66 )
=> ( member998036369le_alt @ X_105 @ B_63 ) ) ) ).
thf(fact_716_in__mono,axiom,
! [X_105: nat,A_66: nat > $o,B_63: nat > $o] :
( ( ord_less_eq_nat_o @ A_66 @ B_63 )
=> ( ( member_nat @ X_105 @ A_66 )
=> ( member_nat @ X_105 @ B_63 ) ) ) ).
thf(fact_717_set__rev__mp,axiom,
! [B_62: arrow_1389031998le_alt > $o,X_104: arrow_1389031998le_alt,A_65: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ X_104 @ A_65 )
=> ( ( ord_le1423491573_alt_o @ A_65 @ B_62 )
=> ( member998036369le_alt @ X_104 @ B_62 ) ) ) ).
thf(fact_718_set__rev__mp,axiom,
! [B_62: nat > $o,X_104: nat,A_65: nat > $o] :
( ( member_nat @ X_104 @ A_65 )
=> ( ( ord_less_eq_nat_o @ A_65 @ B_62 )
=> ( member_nat @ X_104 @ B_62 ) ) ) ).
thf(fact_719_set__mp,axiom,
! [X_103: arrow_1389031998le_alt,A_64: arrow_1389031998le_alt > $o,B_61: arrow_1389031998le_alt > $o] :
( ( ord_le1423491573_alt_o @ A_64 @ B_61 )
=> ( ( member998036369le_alt @ X_103 @ A_64 )
=> ( member998036369le_alt @ X_103 @ B_61 ) ) ) ).
thf(fact_720_set__mp,axiom,
! [X_103: nat,A_64: nat > $o,B_61: nat > $o] :
( ( ord_less_eq_nat_o @ A_64 @ B_61 )
=> ( ( member_nat @ X_103 @ A_64 )
=> ( member_nat @ X_103 @ B_61 ) ) ) ).
thf(fact_721_min__max_Osup_Oidem,axiom,
! [A_63: nat] :
( ( ord_max_nat @ A_63 @ A_63 )
= A_63 ) ).
thf(fact_722_min__max_Osup_Ocommute,axiom,
! [A_62: nat,B_60: nat] :
( ( ord_max_nat @ A_62 @ B_60 )
= ( ord_max_nat @ B_60 @ A_62 ) ) ).
thf(fact_723_min__max_Osup__commute,axiom,
! [X_102: nat,Y_73: nat] :
( ( ord_max_nat @ X_102 @ Y_73 )
= ( ord_max_nat @ Y_73 @ X_102 ) ) ).
thf(fact_724_min__max_Osup_Oleft__idem,axiom,
! [A_61: nat,B_59: nat] :
( ( ord_max_nat @ A_61 @ ( ord_max_nat @ A_61 @ B_59 ) )
= ( ord_max_nat @ A_61 @ B_59 ) ) ).
thf(fact_725_min__max_Osup__left__idem,axiom,
! [X_101: nat,Y_72: nat] :
( ( ord_max_nat @ X_101 @ ( ord_max_nat @ X_101 @ Y_72 ) )
= ( ord_max_nat @ X_101 @ Y_72 ) ) ).
thf(fact_726_min__max_Osup_Oleft__commute,axiom,
! [B_58: nat,A_60: nat,C_30: nat] :
( ( ord_max_nat @ B_58 @ ( ord_max_nat @ A_60 @ C_30 ) )
= ( ord_max_nat @ A_60 @ ( ord_max_nat @ B_58 @ C_30 ) ) ) ).
thf(fact_727_min__max_Osup__left__commute,axiom,
! [X_100: nat,Y_71: nat,Z_29: nat] :
( ( ord_max_nat @ X_100 @ ( ord_max_nat @ Y_71 @ Z_29 ) )
= ( ord_max_nat @ Y_71 @ ( ord_max_nat @ X_100 @ Z_29 ) ) ) ).
thf(fact_728_min__max_Osup_Oassoc,axiom,
! [A_59: nat,B_57: nat,C_29: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A_59 @ B_57 ) @ C_29 )
= ( ord_max_nat @ A_59 @ ( ord_max_nat @ B_57 @ C_29 ) ) ) ).
thf(fact_729_min__max_Osup__assoc,axiom,
! [X_99: nat,Y_70: nat,Z_28: nat] :
( ( ord_max_nat @ ( ord_max_nat @ X_99 @ Y_70 ) @ Z_28 )
= ( ord_max_nat @ X_99 @ ( ord_max_nat @ Y_70 @ Z_28 ) ) ) ).
thf(fact_730_le__maxI1,axiom,
! [X_98: nat,Y_69: nat] : ( ord_less_eq_nat @ X_98 @ ( ord_max_nat @ X_98 @ Y_69 ) ) ).
thf(fact_731_le__maxI2,axiom,
! [Y_68: nat,X_97: nat] : ( ord_less_eq_nat @ Y_68 @ ( ord_max_nat @ X_97 @ Y_68 ) ) ).
thf(fact_732_min__max_Ole__iff__sup,axiom,
! [X_96: nat,Y_67: nat] :
( ( ord_less_eq_nat @ X_96 @ Y_67 )
<=> ( ( ord_max_nat @ X_96 @ Y_67 )
= Y_67 ) ) ).
thf(fact_733_min__max_Ole__sup__iff,axiom,
! [X_95: nat,Y_66: nat,Z_27: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ X_95 @ Y_66 ) @ Z_27 )
<=> ( ( ord_less_eq_nat @ X_95 @ Z_27 )
& ( ord_less_eq_nat @ Y_66 @ Z_27 ) ) ) ).
thf(fact_734_min__max_Ole__supI1,axiom,
! [B_56: nat,X_94: nat,A_58: nat] :
( ( ord_less_eq_nat @ X_94 @ A_58 )
=> ( ord_less_eq_nat @ X_94 @ ( ord_max_nat @ A_58 @ B_56 ) ) ) ).
thf(fact_735_min__max_Ole__supI2,axiom,
! [A_57: nat,X_93: nat,B_55: nat] :
( ( ord_less_eq_nat @ X_93 @ B_55 )
=> ( ord_less_eq_nat @ X_93 @ ( ord_max_nat @ A_57 @ B_55 ) ) ) ).
thf(fact_736_min__max_Osup__absorb2,axiom,
! [X_92: nat,Y_65: nat] :
( ( ord_less_eq_nat @ X_92 @ Y_65 )
=> ( ( ord_max_nat @ X_92 @ Y_65 )
= Y_65 ) ) ).
thf(fact_737_min__max_Osup__absorb1,axiom,
! [Y_64: nat,X_91: nat] :
( ( ord_less_eq_nat @ Y_64 @ X_91 )
=> ( ( ord_max_nat @ X_91 @ Y_64 )
= X_91 ) ) ).
thf(fact_738_min__max_Ole__supI,axiom,
! [B_54: nat,A_56: nat,X_90: nat] :
( ( ord_less_eq_nat @ A_56 @ X_90 )
=> ( ( ord_less_eq_nat @ B_54 @ X_90 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ A_56 @ B_54 ) @ X_90 ) ) ) ).
thf(fact_739_min__max_Osup__least,axiom,
! [Z_26: nat,Y_63: nat,X_89: nat] :
( ( ord_less_eq_nat @ Y_63 @ X_89 )
=> ( ( ord_less_eq_nat @ Z_26 @ X_89 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ Y_63 @ Z_26 ) @ X_89 ) ) ) ).
thf(fact_740_subsetI,axiom,
! [B_53: arrow_1389031998le_alt > $o,A_55: arrow_1389031998le_alt > $o] :
( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ A_55 )
=> ( member998036369le_alt @ X_4 @ B_53 ) )
=> ( ord_le1423491573_alt_o @ A_55 @ B_53 ) ) ).
thf(fact_741_subsetI,axiom,
! [B_53: nat > $o,A_55: nat > $o] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ A_55 )
=> ( member_nat @ X_4 @ B_53 ) )
=> ( ord_less_eq_nat_o @ A_55 @ B_53 ) ) ).
thf(fact_742_weak__decr__stable,axiom,
! [F: nat > nat] :
( ! [I_3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ I_3 ) ) @ ( F @ I_3 ) )
=> ? [I_3: nat] :
! [K_2: nat] :
( ( F @ ( plus_plus_nat @ I_3 @ K_2 ) )
= ( F @ I_3 ) ) ) ).
thf(fact_743_Max__fin__set__fold,axiom,
! [X_88: nat,Xs_73: list_nat] :
( ( big_linorder_Max_nat @ ( set_nat @ ( cons_nat @ X_88 @ Xs_73 ) ) )
= ( foldl_nat_nat @ ord_max_nat @ X_88 @ Xs_73 ) ) ).
thf(fact_744_Collect__mono,axiom,
! [Q: nat > $o,P_15: nat > $o] :
( ! [X_4: nat] :
( ( P_15 @ X_4 )
=> ( Q @ X_4 ) )
=> ( ord_less_eq_nat_o @ ( collect_nat @ P_15 ) @ ( collect_nat @ Q ) ) ) ).
thf(fact_745_pred__subset__eq,axiom,
! [R: arrow_1389031998le_alt > $o,S_1: arrow_1389031998le_alt > $o] :
( ( ord_le1423491573_alt_o
@ ^ [X_4: arrow_1389031998le_alt] : ( member998036369le_alt @ X_4 @ R )
@ ^ [X_4: arrow_1389031998le_alt] : ( member998036369le_alt @ X_4 @ S_1 ) )
<=> ( ord_le1423491573_alt_o @ R @ S_1 ) ) ).
thf(fact_746_pred__subset__eq,axiom,
! [R: nat > $o,S_1: nat > $o] :
( ( ord_less_eq_nat_o
@ ^ [X_4: nat] : ( member_nat @ X_4 @ R )
@ ^ [X_4: nat] : ( member_nat @ X_4 @ S_1 ) )
<=> ( ord_less_eq_nat_o @ R @ S_1 ) ) ).
thf(fact_747_min__max_OSup__fin__set__fold,axiom,
! [X_87: nat,Xs_72: list_nat] :
( ( big_Sup_fin_nat @ ord_max_nat @ ( set_nat @ ( cons_nat @ X_87 @ Xs_72 ) ) )
= ( foldl_nat_nat @ ord_max_nat @ X_87 @ Xs_72 ) ) ).
thf(fact_748_xt1_I15_J,axiom,
! [C_28: nat,A_54: nat,F_7: nat > nat,B_52: nat] :
( ( A_54
= ( F_7 @ B_52 ) )
=> ( ( ord_less_eq_nat @ C_28 @ B_52 )
=> ( ! [X_4: nat,Y_40: nat] :
( ( ord_less_eq_nat @ Y_40 @ X_4 )
=> ( ord_less_eq_nat @ ( F_7 @ Y_40 ) @ ( F_7 @ X_4 ) ) )
=> ( ord_less_eq_nat @ ( F_7 @ C_28 ) @ A_54 ) ) ) ) ).
thf(fact_749_xt1_I16_J,axiom,
! [F_6: nat > nat,C_27: nat,B_51: nat,A_53: nat] :
( ( ord_less_eq_nat @ B_51 @ A_53 )
=> ( ( ( F_6 @ B_51 )
= C_27 )
=> ( ! [X_4: nat,Y_40: nat] :
( ( ord_less_eq_nat @ Y_40 @ X_4 )
=> ( ord_less_eq_nat @ ( F_6 @ Y_40 ) @ ( F_6 @ X_4 ) ) )
=> ( ord_less_eq_nat @ C_27 @ ( F_6 @ A_53 ) ) ) ) ) ).
thf(fact_750_full__nat__induct,axiom,
! [N: nat,P: nat > $o] :
( ! [N_23: nat] :
( ! [M_6: nat] :
( ( ord_less_eq_nat @ ( suc @ M_6 ) @ N_23 )
=> ( P @ M_6 ) )
=> ( P @ N_23 ) )
=> ( P @ N ) ) ).
thf(fact_751_dual__min,axiom,
( ( min_nat
@ ^ [X_4: nat,Y_40: nat] : ( ord_less_eq_nat @ Y_40 @ X_4 ) )
= ord_max_nat ) ).
thf(fact_752_list_Osize_I4_J,axiom,
! [A_52: $o,List: list_o] :
( ( size_size_list_o @ ( cons_o @ A_52 @ List ) )
= ( plus_plus_nat @ ( size_size_list_o @ List ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_753_list_Osize_I4_J,axiom,
! [A_52: nat,List: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ A_52 @ List ) )
= ( plus_plus_nat @ ( size_size_list_nat @ List ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_754_list_Osize_I4_J,axiom,
! [A_52: arrow_1389031998le_alt,List: list_A1402527603le_alt] :
( ( size_s1146069981le_alt @ ( cons_A1142416030le_alt @ A_52 @ List ) )
= ( plus_plus_nat @ ( size_s1146069981le_alt @ List ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_755_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K_2: nat] : K_2
@ ( minus_minus_nat @ M @ N ) ) ) ).
thf(fact_756_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [M_7: nat] : ( suc @ ( ord_min_nat @ M_7 @ N ) )
@ M ) ) ).
thf(fact_757_min__max_Ole__infE,axiom,
! [X_86: nat,A_51: nat,B_50: nat] :
( ( ord_less_eq_nat @ X_86 @ ( ord_min_nat @ A_51 @ B_50 ) )
=> ~ ( ( ord_less_eq_nat @ X_86 @ A_51 )
=> ~ ( ord_less_eq_nat @ X_86 @ B_50 ) ) ) ).
thf(fact_758_min__max_Oinf__mono,axiom,
! [B_49: nat,D_4: nat,A_50: nat,C_26: nat] :
( ( ord_less_eq_nat @ A_50 @ C_26 )
=> ( ( ord_less_eq_nat @ B_49 @ D_4 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A_50 @ B_49 ) @ ( ord_min_nat @ C_26 @ D_4 ) ) ) ) ).
thf(fact_759_min__max_Oinf__greatest,axiom,
! [Z_25: nat,X_85: nat,Y_62: nat] :
( ( ord_less_eq_nat @ X_85 @ Y_62 )
=> ( ( ord_less_eq_nat @ X_85 @ Z_25 )
=> ( ord_less_eq_nat @ X_85 @ ( ord_min_nat @ Y_62 @ Z_25 ) ) ) ) ).
thf(fact_760_min__max_Ole__infI,axiom,
! [B_48: nat,X_84: nat,A_49: nat] :
( ( ord_less_eq_nat @ X_84 @ A_49 )
=> ( ( ord_less_eq_nat @ X_84 @ B_48 )
=> ( ord_less_eq_nat @ X_84 @ ( ord_min_nat @ A_49 @ B_48 ) ) ) ) ).
thf(fact_761_min__max_Oinf__absorb2,axiom,
! [Y_61: nat,X_83: nat] :
( ( ord_less_eq_nat @ Y_61 @ X_83 )
=> ( ( ord_min_nat @ X_83 @ Y_61 )
= Y_61 ) ) ).
thf(fact_762_min__max_Oinf__absorb1,axiom,
! [X_82: nat,Y_60: nat] :
( ( ord_less_eq_nat @ X_82 @ Y_60 )
=> ( ( ord_min_nat @ X_82 @ Y_60 )
= X_82 ) ) ).
thf(fact_763_min__max_Ole__infI2,axiom,
! [A_48: nat,B_47: nat,X_81: nat] :
( ( ord_less_eq_nat @ B_47 @ X_81 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A_48 @ B_47 ) @ X_81 ) ) ).
thf(fact_764_min__max_Ole__infI1,axiom,
! [B_46: nat,A_47: nat,X_80: nat] :
( ( ord_less_eq_nat @ A_47 @ X_80 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A_47 @ B_46 ) @ X_80 ) ) ).
thf(fact_765_min__max_Ole__inf__iff,axiom,
! [X_79: nat,Y_59: nat,Z_24: nat] :
( ( ord_less_eq_nat @ X_79 @ ( ord_min_nat @ Y_59 @ Z_24 ) )
<=> ( ( ord_less_eq_nat @ X_79 @ Y_59 )
& ( ord_less_eq_nat @ X_79 @ Z_24 ) ) ) ).
thf(fact_766_min__max_Ole__iff__inf,axiom,
! [X_78: nat,Y_58: nat] :
( ( ord_less_eq_nat @ X_78 @ Y_58 )
<=> ( ( ord_min_nat @ X_78 @ Y_58 )
= X_78 ) ) ).
thf(fact_767_min__max_Oinf__le2,axiom,
! [X_77: nat,Y_57: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ X_77 @ Y_57 ) @ Y_57 ) ).
thf(fact_768_min__max_Oinf__le1,axiom,
! [X_76: nat,Y_56: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ X_76 @ Y_56 ) @ X_76 ) ).
thf(fact_769_min__max_Oinf_Oidem,axiom,
! [A_46: nat] :
( ( ord_min_nat @ A_46 @ A_46 )
= A_46 ) ).
thf(fact_770_min__max_Oinf_Ocommute,axiom,
! [A_45: nat,B_45: nat] :
( ( ord_min_nat @ A_45 @ B_45 )
= ( ord_min_nat @ B_45 @ A_45 ) ) ).
thf(fact_771_min__max_Oinf__commute,axiom,
! [X_75: nat,Y_55: nat] :
( ( ord_min_nat @ X_75 @ Y_55 )
= ( ord_min_nat @ Y_55 @ X_75 ) ) ).
thf(fact_772_min__max_Oinf_Oleft__idem,axiom,
! [A_44: nat,B_44: nat] :
( ( ord_min_nat @ A_44 @ ( ord_min_nat @ A_44 @ B_44 ) )
= ( ord_min_nat @ A_44 @ B_44 ) ) ).
thf(fact_773_min__max_Oinf__left__idem,axiom,
! [X_74: nat,Y_54: nat] :
( ( ord_min_nat @ X_74 @ ( ord_min_nat @ X_74 @ Y_54 ) )
= ( ord_min_nat @ X_74 @ Y_54 ) ) ).
thf(fact_774_min__max_Oinf_Oleft__commute,axiom,
! [B_43: nat,A_43: nat,C_25: nat] :
( ( ord_min_nat @ B_43 @ ( ord_min_nat @ A_43 @ C_25 ) )
= ( ord_min_nat @ A_43 @ ( ord_min_nat @ B_43 @ C_25 ) ) ) ).
thf(fact_775_min__max_Oinf__left__commute,axiom,
! [X_73: nat,Y_53: nat,Z_23: nat] :
( ( ord_min_nat @ X_73 @ ( ord_min_nat @ Y_53 @ Z_23 ) )
= ( ord_min_nat @ Y_53 @ ( ord_min_nat @ X_73 @ Z_23 ) ) ) ).
thf(fact_776_min__max_Oinf_Oassoc,axiom,
! [A_42: nat,B_42: nat,C_24: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A_42 @ B_42 ) @ C_24 )
= ( ord_min_nat @ A_42 @ ( ord_min_nat @ B_42 @ C_24 ) ) ) ).
thf(fact_777_min__max_Oinf__assoc,axiom,
! [X_72: nat,Y_52: nat,Z_22: nat] :
( ( ord_min_nat @ ( ord_min_nat @ X_72 @ Y_52 ) @ Z_22 )
= ( ord_min_nat @ X_72 @ ( ord_min_nat @ Y_52 @ Z_22 ) ) ) ).
thf(fact_778_min__max_Osup__inf__absorb,axiom,
! [X_71: nat,Y_51: nat] :
( ( ord_max_nat @ X_71 @ ( ord_min_nat @ X_71 @ Y_51 ) )
= X_71 ) ).
thf(fact_779_min__max_Oinf__sup__absorb,axiom,
! [X_70: nat,Y_50: nat] :
( ( ord_min_nat @ X_70 @ ( ord_max_nat @ X_70 @ Y_50 ) )
= X_70 ) ).
thf(fact_780_min__max_Osup__inf__distrib1,axiom,
! [X_69: nat,Y_49: nat,Z_21: nat] :
( ( ord_max_nat @ X_69 @ ( ord_min_nat @ Y_49 @ Z_21 ) )
= ( ord_min_nat @ ( ord_max_nat @ X_69 @ Y_49 ) @ ( ord_max_nat @ X_69 @ Z_21 ) ) ) ).
thf(fact_781_min__max_Oinf__sup__distrib1,axiom,
! [X_68: nat,Y_48: nat,Z_20: nat] :
( ( ord_min_nat @ X_68 @ ( ord_max_nat @ Y_48 @ Z_20 ) )
= ( ord_max_nat @ ( ord_min_nat @ X_68 @ Y_48 ) @ ( ord_min_nat @ X_68 @ Z_20 ) ) ) ).
thf(fact_782_min__max_Osup__inf__distrib2,axiom,
! [Y_47: nat,Z_19: nat,X_67: nat] :
( ( ord_max_nat @ ( ord_min_nat @ Y_47 @ Z_19 ) @ X_67 )
= ( ord_min_nat @ ( ord_max_nat @ Y_47 @ X_67 ) @ ( ord_max_nat @ Z_19 @ X_67 ) ) ) ).
thf(fact_783_min__max_Oinf__sup__distrib2,axiom,
! [Y_46: nat,Z_18: nat,X_66: nat] :
( ( ord_min_nat @ ( ord_max_nat @ Y_46 @ Z_18 ) @ X_66 )
= ( ord_max_nat @ ( ord_min_nat @ Y_46 @ X_66 ) @ ( ord_min_nat @ Z_18 @ X_66 ) ) ) ).
thf(fact_784_length__rev,axiom,
! [Xs_71: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs_71 ) )
= ( size_size_list_nat @ Xs_71 ) ) ).
thf(fact_785_length__rev,axiom,
! [Xs_71: list_A1402527603le_alt] :
( ( size_s1146069981le_alt @ ( rev_Ar2020079950le_alt @ Xs_71 ) )
= ( size_s1146069981le_alt @ Xs_71 ) ) ).
thf(fact_786_take__append,axiom,
! [N_41: nat,Xs_70: list_A1402527603le_alt,Ys_14: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ N_41 @ ( append1092755459le_alt @ Xs_70 @ Ys_14 ) )
= ( append1092755459le_alt @ ( take_A1527720662le_alt @ N_41 @ Xs_70 ) @ ( take_A1527720662le_alt @ ( minus_minus_nat @ N_41 @ ( size_s1146069981le_alt @ Xs_70 ) ) @ Ys_14 ) ) ) ).
thf(fact_787_take__append,axiom,
! [N_41: nat,Xs_70: list_nat,Ys_14: list_nat] :
( ( take_nat @ N_41 @ ( append_nat @ Xs_70 @ Ys_14 ) )
= ( append_nat @ ( take_nat @ N_41 @ Xs_70 ) @ ( take_nat @ ( minus_minus_nat @ N_41 @ ( size_size_list_nat @ Xs_70 ) ) @ Ys_14 ) ) ) ).
thf(fact_788_min__add__distrib__left,axiom,
! [X_65: nat,Y_45: nat,Z_17: nat] :
( ( plus_plus_nat @ ( ord_min_nat @ X_65 @ Y_45 ) @ Z_17 )
= ( ord_min_nat @ ( plus_plus_nat @ X_65 @ Z_17 ) @ ( plus_plus_nat @ Y_45 @ Z_17 ) ) ) ).
thf(fact_789_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_790_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_791_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
thf(fact_792_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_793_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_794_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_795_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
thf(fact_796_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
thf(fact_797_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_798_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_799_diff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_800_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_801_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_802_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_803_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
thf(fact_804_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
thf(fact_805_Min_Oidem,axiom,
! [X_64: nat] :
( ( ord_min_nat @ X_64 @ X_64 )
= X_64 ) ).
thf(fact_806_length__take,axiom,
! [N_40: nat,Xs_69: list_A1402527603le_alt] :
( ( size_s1146069981le_alt @ ( take_A1527720662le_alt @ N_40 @ Xs_69 ) )
= ( ord_min_nat @ ( size_s1146069981le_alt @ Xs_69 ) @ N_40 ) ) ).
thf(fact_807_length__take,axiom,
! [N_40: nat,Xs_69: list_nat] :
( ( size_size_list_nat @ ( take_nat @ N_40 @ Xs_69 ) )
= ( ord_min_nat @ ( size_size_list_nat @ Xs_69 ) @ N_40 ) ) ).
thf(fact_808_neq__if__length__neq,axiom,
! [Xs_68: list_nat,Ys_13: list_nat] :
( ( ( size_size_list_nat @ Xs_68 )
!= ( size_size_list_nat @ Ys_13 ) )
=> ( Xs_68 != Ys_13 ) ) ).
thf(fact_809_take__take,axiom,
! [N_39: nat,M_8: nat,Xs_67: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ N_39 @ ( take_A1527720662le_alt @ M_8 @ Xs_67 ) )
= ( take_A1527720662le_alt @ ( ord_min_nat @ N_39 @ M_8 ) @ Xs_67 ) ) ).
thf(fact_810_take__take,axiom,
! [N_39: nat,M_8: nat,Xs_67: list_nat] :
( ( take_nat @ N_39 @ ( take_nat @ M_8 @ Xs_67 ) )
= ( take_nat @ ( ord_min_nat @ N_39 @ M_8 ) @ Xs_67 ) ) ).
thf(fact_811_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_812_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_813_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
thf(fact_814_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
thf(fact_815_eq__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
<=> ( M = N ) ) ) ) ).
thf(fact_816_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_817_le__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_818_min__ord__min,axiom,
( ord_min_nat
= ( min_nat @ ord_less_eq_nat ) ) ).
thf(fact_819_min__le__iff__disj,axiom,
! [X_63: nat,Y_44: nat,Z_16: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X_63 @ Y_44 ) @ Z_16 )
<=> ( ( ord_less_eq_nat @ X_63 @ Z_16 )
| ( ord_less_eq_nat @ Y_44 @ Z_16 ) ) ) ).
thf(fact_820_length__replicate,axiom,
! [N_38: nat,X_62: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N_38 @ X_62 ) )
= N_38 ) ).
thf(fact_821_length__replicate,axiom,
! [N_38: nat,X_62: arrow_1389031998le_alt] :
( ( size_s1146069981le_alt @ ( replic277728168le_alt @ N_38 @ X_62 ) )
= N_38 ) ).
thf(fact_822_length__rotate1,axiom,
! [Xs_66: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs_66 ) )
= ( size_size_list_nat @ Xs_66 ) ) ).
thf(fact_823_length__rotate1,axiom,
! [Xs_66: list_A1402527603le_alt] :
( ( size_s1146069981le_alt @ ( rotate1249022267le_alt @ Xs_66 ) )
= ( size_s1146069981le_alt @ Xs_66 ) ) ).
thf(fact_824_min__max_Odistrib__inf__le,axiom,
! [X_61: nat,Y_43: nat,Z_15: nat] : ( ord_less_eq_nat @ ( ord_max_nat @ ( ord_min_nat @ X_61 @ Y_43 ) @ ( ord_min_nat @ X_61 @ Z_15 ) ) @ ( ord_min_nat @ X_61 @ ( ord_max_nat @ Y_43 @ Z_15 ) ) ) ).
thf(fact_825_min__max_Odistrib__sup__le,axiom,
! [X_60: nat,Y_42: nat,Z_14: nat] : ( ord_less_eq_nat @ ( ord_max_nat @ X_60 @ ( ord_min_nat @ Y_42 @ Z_14 ) ) @ ( ord_min_nat @ ( ord_max_nat @ X_60 @ Y_42 ) @ ( ord_max_nat @ X_60 @ Z_14 ) ) ) ).
thf(fact_826_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_827_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
thf(fact_828_list_Osize_I3_J,axiom,
( ( size_s1146069981le_alt @ nil_Ar52383470le_alt )
= zero_zero_nat ) ).
thf(fact_829_length__0__conv,axiom,
! [Xs_65: list_nat] :
( ( ( size_size_list_nat @ Xs_65 )
= zero_zero_nat )
<=> ( Xs_65 = nil_nat ) ) ).
thf(fact_830_length__0__conv,axiom,
! [Xs_65: list_A1402527603le_alt] :
( ( ( size_s1146069981le_alt @ Xs_65 )
= zero_zero_nat )
<=> ( Xs_65 = nil_Ar52383470le_alt ) ) ).
thf(fact_831_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_832_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_833_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_834_impossible__Cons,axiom,
! [X_59: $o,Xs_64: list_o,Ys_12: list_o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs_64 ) @ ( size_size_list_o @ Ys_12 ) )
=> ( Xs_64
!= ( cons_o @ X_59 @ Ys_12 ) ) ) ).
thf(fact_835_impossible__Cons,axiom,
! [X_59: nat,Xs_64: list_nat,Ys_12: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_64 ) @ ( size_size_list_nat @ Ys_12 ) )
=> ( Xs_64
!= ( cons_nat @ X_59 @ Ys_12 ) ) ) ).
thf(fact_836_impossible__Cons,axiom,
! [X_59: arrow_1389031998le_alt,Xs_64: list_A1402527603le_alt,Ys_12: list_A1402527603le_alt] :
( ( ord_less_eq_nat @ ( size_s1146069981le_alt @ Xs_64 ) @ ( size_s1146069981le_alt @ Ys_12 ) )
=> ( Xs_64
!= ( cons_A1142416030le_alt @ X_59 @ Ys_12 ) ) ) ).
thf(fact_837_diff__diff__right,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_838_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
thf(fact_839_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
thf(fact_840_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_841_add__diff__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
thf(fact_842_le__diff__conv2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_843_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
= M ) ) ).
thf(fact_844_le__imp__diff__is__add,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
thf(fact_845_diff__add__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_846_add__diff__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
thf(fact_847_diff__add__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
thf(fact_848_length__append,axiom,
! [Xs_63: list_A1402527603le_alt,Ys_11: list_A1402527603le_alt] :
( ( size_s1146069981le_alt @ ( append1092755459le_alt @ Xs_63 @ Ys_11 ) )
= ( plus_plus_nat @ ( size_s1146069981le_alt @ Xs_63 ) @ ( size_s1146069981le_alt @ Ys_11 ) ) ) ).
thf(fact_849_length__append,axiom,
! [Xs_63: list_nat,Ys_11: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs_63 @ Ys_11 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs_63 ) @ ( size_size_list_nat @ Ys_11 ) ) ) ).
thf(fact_850_take__all,axiom,
! [Xs_62: list_A1402527603le_alt,N_37: nat] :
( ( ord_less_eq_nat @ ( size_s1146069981le_alt @ Xs_62 ) @ N_37 )
=> ( ( take_A1527720662le_alt @ N_37 @ Xs_62 )
= Xs_62 ) ) ).
thf(fact_851_take__all,axiom,
! [Xs_62: list_nat,N_37: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_62 ) @ N_37 )
=> ( ( take_nat @ N_37 @ Xs_62 )
= Xs_62 ) ) ).
thf(fact_852_take__replicate,axiom,
! [I_10: nat,K_5: nat,X_58: arrow_1389031998le_alt] :
( ( take_A1527720662le_alt @ I_10 @ ( replic277728168le_alt @ K_5 @ X_58 ) )
= ( replic277728168le_alt @ ( ord_min_nat @ I_10 @ K_5 ) @ X_58 ) ) ).
thf(fact_853_take__replicate,axiom,
! [I_10: nat,K_5: nat,X_58: nat] :
( ( take_nat @ I_10 @ ( replicate_nat @ K_5 @ X_58 ) )
= ( replicate_nat @ ( ord_min_nat @ I_10 @ K_5 ) @ X_58 ) ) ).
thf(fact_854_length__takeWhile__le,axiom,
! [P_14: nat > $o,Xs_61: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P_14 @ Xs_61 ) ) @ ( size_size_list_nat @ Xs_61 ) ) ).
thf(fact_855_length__takeWhile__le,axiom,
! [P_14: arrow_1389031998le_alt > $o,Xs_61: list_A1402527603le_alt] : ( ord_less_eq_nat @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_14 @ Xs_61 ) ) @ ( size_s1146069981le_alt @ Xs_61 ) ) ).
thf(fact_856_length__insort,axiom,
! [F_5: nat > nat,X_57: nat,Xs_60: list_nat] :
( ( size_size_list_nat @ ( linord1850739451at_nat @ F_5 @ X_57 @ Xs_60 ) )
= ( suc @ ( size_size_list_nat @ Xs_60 ) ) ) ).
thf(fact_857_length__dropWhile__le,axiom,
! [P_13: nat > $o,Xs_59: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( dropWhile_nat @ P_13 @ Xs_59 ) ) @ ( size_size_list_nat @ Xs_59 ) ) ).
thf(fact_858_length__dropWhile__le,axiom,
! [P_13: arrow_1389031998le_alt > $o,Xs_59: list_A1402527603le_alt] : ( ord_less_eq_nat @ ( size_s1146069981le_alt @ ( dropWh82971279le_alt @ P_13 @ Xs_59 ) ) @ ( size_s1146069981le_alt @ Xs_59 ) ) ).
thf(fact_859_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
thf(fact_860_takeWhile__eq__take,axiom,
! [P_12: arrow_1389031998le_alt > $o,Xs_58: list_A1402527603le_alt] :
( ( takeWh462480871le_alt @ P_12 @ Xs_58 )
= ( take_A1527720662le_alt @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_12 @ Xs_58 ) ) @ Xs_58 ) ) ).
thf(fact_861_takeWhile__eq__take,axiom,
! [P_12: nat > $o,Xs_58: list_nat] :
( ( takeWhile_nat @ P_12 @ Xs_58 )
= ( take_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P_12 @ Xs_58 ) ) @ Xs_58 ) ) ).
thf(fact_862_dual__max,axiom,
( ( max_nat
@ ^ [X_4: nat,Y_40: nat] : ( ord_less_eq_nat @ Y_40 @ X_4 ) )
= ord_min_nat ) ).
thf(fact_863_length__splice,axiom,
! [Xs_57: list_nat,Ys_10: list_nat] :
( ( size_size_list_nat @ ( splice_nat @ Xs_57 @ Ys_10 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs_57 ) @ ( size_size_list_nat @ Ys_10 ) ) ) ).
thf(fact_864_length__splice,axiom,
! [Xs_57: list_A1402527603le_alt,Ys_10: list_A1402527603le_alt] :
( ( size_s1146069981le_alt @ ( splice287087809le_alt @ Xs_57 @ Ys_10 ) )
= ( plus_plus_nat @ ( size_s1146069981le_alt @ Xs_57 ) @ ( size_s1146069981le_alt @ Ys_10 ) ) ) ).
thf(fact_865_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
thf(fact_866_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
thf(fact_867_nth__append__length,axiom,
! [Xs_56: list_o,X_56: $o,Ys_9: list_o] :
( ( nth_o @ ( append_o @ Xs_56 @ ( cons_o @ X_56 @ Ys_9 ) ) @ ( size_size_list_o @ Xs_56 ) )
<=> X_56 ) ).
thf(fact_868_nth__append__length,axiom,
! [Xs_56: list_A1402527603le_alt,X_56: arrow_1389031998le_alt,Ys_9: list_A1402527603le_alt] :
( ( nth_Ar1039551983le_alt @ ( append1092755459le_alt @ Xs_56 @ ( cons_A1142416030le_alt @ X_56 @ Ys_9 ) ) @ ( size_s1146069981le_alt @ Xs_56 ) )
= X_56 ) ).
thf(fact_869_nth__append__length,axiom,
! [Xs_56: list_nat,X_56: nat,Ys_9: list_nat] :
( ( nth_nat @ ( append_nat @ Xs_56 @ ( cons_nat @ X_56 @ Ys_9 ) ) @ ( size_size_list_nat @ Xs_56 ) )
= X_56 ) ).
thf(fact_870_nth__append__length__plus,axiom,
! [Xs_55: list_o,Ys_8: list_o,N_36: nat] :
( ( nth_o @ ( append_o @ Xs_55 @ Ys_8 ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs_55 ) @ N_36 ) )
<=> ( nth_o @ Ys_8 @ N_36 ) ) ).
thf(fact_871_nth__append__length__plus,axiom,
! [Xs_55: list_A1402527603le_alt,Ys_8: list_A1402527603le_alt,N_36: nat] :
( ( nth_Ar1039551983le_alt @ ( append1092755459le_alt @ Xs_55 @ Ys_8 ) @ ( plus_plus_nat @ ( size_s1146069981le_alt @ Xs_55 ) @ N_36 ) )
= ( nth_Ar1039551983le_alt @ Ys_8 @ N_36 ) ) ).
thf(fact_872_nth__append__length__plus,axiom,
! [Xs_55: list_nat,Ys_8: list_nat,N_36: nat] :
( ( nth_nat @ ( append_nat @ Xs_55 @ Ys_8 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs_55 ) @ N_36 ) )
= ( nth_nat @ Ys_8 @ N_36 ) ) ).
thf(fact_873_Max_Oidem,axiom,
! [X_55: nat] :
( ( ord_max_nat @ X_55 @ X_55 )
= X_55 ) ).
thf(fact_874_sublist__append,axiom,
! [L_2: list_A1402527603le_alt,L_1: list_A1402527603le_alt,A_41: nat > $o] :
( ( sublis814142963le_alt @ ( append1092755459le_alt @ L_2 @ L_1 ) @ A_41 )
= ( append1092755459le_alt @ ( sublis814142963le_alt @ L_2 @ A_41 )
@ ( sublis814142963le_alt @ L_1
@ ( collect_nat
@ ^ [J_1: nat] : ( member_nat @ ( plus_plus_nat @ J_1 @ ( size_s1146069981le_alt @ L_2 ) ) @ A_41 ) ) ) ) ) ).
thf(fact_875_sublist__append,axiom,
! [L_2: list_nat,L_1: list_nat,A_41: nat > $o] :
( ( sublist_nat @ ( append_nat @ L_2 @ L_1 ) @ A_41 )
= ( append_nat @ ( sublist_nat @ L_2 @ A_41 )
@ ( sublist_nat @ L_1
@ ( collect_nat
@ ^ [J_1: nat] : ( member_nat @ ( plus_plus_nat @ J_1 @ ( size_size_list_nat @ L_2 ) ) @ A_41 ) ) ) ) ) ).
thf(fact_876_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( nat_case_nat @ zero_zero_nat
@ ^ [M_7: nat] : ( suc @ ( ord_min_nat @ N @ M_7 ) )
@ M ) ) ).
thf(fact_877_Suc__length__conv,axiom,
! [N_35: nat,Xs_54: list_o] :
( ( ( suc @ N_35 )
= ( size_size_list_o @ Xs_54 ) )
<=> ? [Y_40: $o,Ys_7: list_o] :
( ( Xs_54
= ( cons_o @ Y_40 @ Ys_7 ) )
& ( ( size_size_list_o @ Ys_7 )
= N_35 ) ) ) ).
thf(fact_878_Suc__length__conv,axiom,
! [N_35: nat,Xs_54: list_nat] :
( ( ( suc @ N_35 )
= ( size_size_list_nat @ Xs_54 ) )
<=> ? [Y_40: nat,Ys_7: list_nat] :
( ( Xs_54
= ( cons_nat @ Y_40 @ Ys_7 ) )
& ( ( size_size_list_nat @ Ys_7 )
= N_35 ) ) ) ).
thf(fact_879_Suc__length__conv,axiom,
! [N_35: nat,Xs_54: list_A1402527603le_alt] :
( ( ( suc @ N_35 )
= ( size_s1146069981le_alt @ Xs_54 ) )
<=> ? [Y_40: arrow_1389031998le_alt,Ys_7: list_A1402527603le_alt] :
( ( Xs_54
= ( cons_A1142416030le_alt @ Y_40 @ Ys_7 ) )
& ( ( size_s1146069981le_alt @ Ys_7 )
= N_35 ) ) ) ).
thf(fact_880_length__Suc__conv,axiom,
! [Xs_53: list_o,N_34: nat] :
( ( ( size_size_list_o @ Xs_53 )
= ( suc @ N_34 ) )
<=> ? [Y_40: $o,Ys_7: list_o] :
( ( Xs_53
= ( cons_o @ Y_40 @ Ys_7 ) )
& ( ( size_size_list_o @ Ys_7 )
= N_34 ) ) ) ).
thf(fact_881_length__Suc__conv,axiom,
! [Xs_53: list_nat,N_34: nat] :
( ( ( size_size_list_nat @ Xs_53 )
= ( suc @ N_34 ) )
<=> ? [Y_40: nat,Ys_7: list_nat] :
( ( Xs_53
= ( cons_nat @ Y_40 @ Ys_7 ) )
& ( ( size_size_list_nat @ Ys_7 )
= N_34 ) ) ) ).
thf(fact_882_length__Suc__conv,axiom,
! [Xs_53: list_A1402527603le_alt,N_34: nat] :
( ( ( size_s1146069981le_alt @ Xs_53 )
= ( suc @ N_34 ) )
<=> ? [Y_40: arrow_1389031998le_alt,Ys_7: list_A1402527603le_alt] :
( ( Xs_53
= ( cons_A1142416030le_alt @ Y_40 @ Ys_7 ) )
& ( ( size_s1146069981le_alt @ Ys_7 )
= N_34 ) ) ) ).
thf(fact_883_min__max_Odistrib__imp2,axiom,
! [X_54: nat,Y_41: nat,Z_13: nat] :
( ! [X_4: nat,Y_40: nat,Z_12: nat] :
( ( ord_max_nat @ X_4 @ ( ord_min_nat @ Y_40 @ Z_12 ) )
= ( ord_min_nat @ ( ord_max_nat @ X_4 @ Y_40 ) @ ( ord_max_nat @ X_4 @ Z_12 ) ) )
=> ( ( ord_min_nat @ X_54 @ ( ord_max_nat @ Y_41 @ Z_13 ) )
= ( ord_max_nat @ ( ord_min_nat @ X_54 @ Y_41 ) @ ( ord_min_nat @ X_54 @ Z_13 ) ) ) ) ).
thf(fact_884_min__max_Odistrib__imp1,axiom,
! [X_53: nat,Y_39: nat,Z_11: nat] :
( ! [X_4: nat,Y_40: nat,Z_12: nat] :
( ( ord_min_nat @ X_4 @ ( ord_max_nat @ Y_40 @ Z_12 ) )
= ( ord_max_nat @ ( ord_min_nat @ X_4 @ Y_40 ) @ ( ord_min_nat @ X_4 @ Z_12 ) ) )
=> ( ( ord_max_nat @ X_53 @ ( ord_min_nat @ Y_39 @ Z_11 ) )
= ( ord_min_nat @ ( ord_max_nat @ X_53 @ Y_39 ) @ ( ord_max_nat @ X_53 @ Z_11 ) ) ) ) ).
thf(fact_885_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_23: nat] :
( ( P @ ( suc @ N_23 ) )
=> ( P @ N_23 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
thf(fact_886_min__leastL,axiom,
! [X_52: nat,Least_1: nat] :
( ( all @ ( ord_less_eq_nat @ Least_1 ) )
=> ( ( ord_min_nat @ Least_1 @ X_52 )
= Least_1 ) ) ).
thf(fact_887_DiffI,axiom,
! [B_41: arrow_1389031998le_alt > $o,C_23: arrow_1389031998le_alt,A_40: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ C_23 @ A_40 )
=> ( ~ ( member998036369le_alt @ C_23 @ B_41 )
=> ( member998036369le_alt @ C_23 @ ( minus_1594223470_alt_o @ A_40 @ B_41 ) ) ) ) ).
thf(fact_888_DiffI,axiom,
! [B_41: nat > $o,C_23: nat,A_40: nat > $o] :
( ( member_nat @ C_23 @ A_40 )
=> ( ~ ( member_nat @ C_23 @ B_41 )
=> ( member_nat @ C_23 @ ( minus_minus_nat_o @ A_40 @ B_41 ) ) ) ) ).
thf(fact_889_DiffE,axiom,
! [C_22: arrow_1389031998le_alt,A_39: arrow_1389031998le_alt > $o,B_40: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ C_22 @ ( minus_1594223470_alt_o @ A_39 @ B_40 ) )
=> ~ ( ( member998036369le_alt @ C_22 @ A_39 )
=> ( member998036369le_alt @ C_22 @ B_40 ) ) ) ).
thf(fact_890_DiffE,axiom,
! [C_22: nat,A_39: nat > $o,B_40: nat > $o] :
( ( member_nat @ C_22 @ ( minus_minus_nat_o @ A_39 @ B_40 ) )
=> ~ ( ( member_nat @ C_22 @ A_39 )
=> ( member_nat @ C_22 @ B_40 ) ) ) ).
thf(fact_891_set__diff__eq,axiom,
! [A_38: arrow_1389031998le_alt > $o,B_39: arrow_1389031998le_alt > $o] :
( ( minus_1594223470_alt_o @ A_38 @ B_39 )
= ( collec1655747795le_alt
@ ^ [X_4: arrow_1389031998le_alt] : ( (&) @ ( member998036369le_alt @ X_4 @ A_38 ) @ ( (~) @ ( member998036369le_alt @ X_4 @ B_39 ) ) ) ) ) ).
thf(fact_892_set__diff__eq,axiom,
! [A_38: nat > $o,B_39: nat > $o] :
( ( minus_minus_nat_o @ A_38 @ B_39 )
= ( collect_nat
@ ^ [X_4: nat] : ( (&) @ ( member_nat @ X_4 @ A_38 ) @ ( (~) @ ( member_nat @ X_4 @ B_39 ) ) ) ) ) ).
thf(fact_893_Diff__iff,axiom,
! [C_21: arrow_1389031998le_alt,A_37: arrow_1389031998le_alt > $o,B_38: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ C_21 @ ( minus_1594223470_alt_o @ A_37 @ B_38 ) )
<=> ( ( member998036369le_alt @ C_21 @ A_37 )
& ~ ( member998036369le_alt @ C_21 @ B_38 ) ) ) ).
thf(fact_894_Diff__iff,axiom,
! [C_21: nat,A_37: nat > $o,B_38: nat > $o] :
( ( member_nat @ C_21 @ ( minus_minus_nat_o @ A_37 @ B_38 ) )
<=> ( ( member_nat @ C_21 @ A_37 )
& ~ ( member_nat @ C_21 @ B_38 ) ) ) ).
thf(fact_895_DiffD1,axiom,
! [C_20: arrow_1389031998le_alt,A_36: arrow_1389031998le_alt > $o,B_37: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ C_20 @ ( minus_1594223470_alt_o @ A_36 @ B_37 ) )
=> ( member998036369le_alt @ C_20 @ A_36 ) ) ).
thf(fact_896_DiffD1,axiom,
! [C_20: nat,A_36: nat > $o,B_37: nat > $o] :
( ( member_nat @ C_20 @ ( minus_minus_nat_o @ A_36 @ B_37 ) )
=> ( member_nat @ C_20 @ A_36 ) ) ).
thf(fact_897_DiffD2,axiom,
! [C_19: arrow_1389031998le_alt,A_35: arrow_1389031998le_alt > $o,B_36: arrow_1389031998le_alt > $o] :
( ( member998036369le_alt @ C_19 @ ( minus_1594223470_alt_o @ A_35 @ B_36 ) )
=> ~ ( member998036369le_alt @ C_19 @ B_36 ) ) ).
thf(fact_898_DiffD2,axiom,
! [C_19: nat,A_35: nat > $o,B_36: nat > $o] :
( ( member_nat @ C_19 @ ( minus_minus_nat_o @ A_35 @ B_36 ) )
=> ~ ( member_nat @ C_19 @ B_36 ) ) ).
thf(fact_899_min__leastR,axiom,
! [X_51: nat,Least: nat] :
( ( all @ ( ord_less_eq_nat @ Least ) )
=> ( ( ord_min_nat @ X_51 @ Least )
= Least ) ) ).
thf(fact_900_take__Suc__conv__app__nth,axiom,
! [I_9: nat,Xs_52: list_o] :
( ( ord_less_nat @ I_9 @ ( size_size_list_o @ Xs_52 ) )
=> ( ( take_o @ ( suc @ I_9 ) @ Xs_52 )
= ( append_o @ ( take_o @ I_9 @ Xs_52 ) @ ( cons_o @ ( nth_o @ Xs_52 @ I_9 ) @ nil_o ) ) ) ) ).
thf(fact_901_take__Suc__conv__app__nth,axiom,
! [I_9: nat,Xs_52: list_A1402527603le_alt] :
( ( ord_less_nat @ I_9 @ ( size_s1146069981le_alt @ Xs_52 ) )
=> ( ( take_A1527720662le_alt @ ( suc @ I_9 ) @ Xs_52 )
= ( append1092755459le_alt @ ( take_A1527720662le_alt @ I_9 @ Xs_52 ) @ ( cons_A1142416030le_alt @ ( nth_Ar1039551983le_alt @ Xs_52 @ I_9 ) @ nil_Ar52383470le_alt ) ) ) ) ).
thf(fact_902_take__Suc__conv__app__nth,axiom,
! [I_9: nat,Xs_52: list_nat] :
( ( ord_less_nat @ I_9 @ ( size_size_list_nat @ Xs_52 ) )
=> ( ( take_nat @ ( suc @ I_9 ) @ Xs_52 )
= ( append_nat @ ( take_nat @ I_9 @ Xs_52 ) @ ( cons_nat @ ( nth_nat @ Xs_52 @ I_9 ) @ nil_nat ) ) ) ) ).
thf(fact_903_sorted__rev__nth__mono,axiom,
! [I_8: nat,J_7: nat,Xs_51: list_nat] :
( ( linorder_sorted_nat @ ( rev_nat @ Xs_51 ) )
=> ( ( ord_less_eq_nat @ I_8 @ J_7 )
=> ( ( ord_less_nat @ J_7 @ ( size_size_list_nat @ Xs_51 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs_51 @ J_7 ) @ ( nth_nat @ Xs_51 @ I_8 ) ) ) ) ) ).
thf(fact_904_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_905_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_906_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_907_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_908_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
thf(fact_909_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_910_xt1_I8_J,axiom,
! [Z_10: nat,Y_38: nat,X_50: nat] :
( ( ord_less_eq_nat @ Y_38 @ X_50 )
=> ( ( ord_less_nat @ Z_10 @ Y_38 )
=> ( ord_less_nat @ Z_10 @ X_50 ) ) ) ).
thf(fact_911_order__le__less__trans,axiom,
! [Z_9: nat,X_49: nat,Y_37: nat] :
( ( ord_less_eq_nat @ X_49 @ Y_37 )
=> ( ( ord_less_nat @ Y_37 @ Z_9 )
=> ( ord_less_nat @ X_49 @ Z_9 ) ) ) ).
thf(fact_912_xt1_I7_J,axiom,
! [Z_8: nat,Y_36: nat,X_48: nat] :
( ( ord_less_nat @ Y_36 @ X_48 )
=> ( ( ord_less_eq_nat @ Z_8 @ Y_36 )
=> ( ord_less_nat @ Z_8 @ X_48 ) ) ) ).
thf(fact_913_order__less__le__trans,axiom,
! [Z_7: nat,X_47: nat,Y_35: nat] :
( ( ord_less_nat @ X_47 @ Y_35 )
=> ( ( ord_less_eq_nat @ Y_35 @ Z_7 )
=> ( ord_less_nat @ X_47 @ Z_7 ) ) ) ).
thf(fact_914_xt1_I11_J,axiom,
! [B_35: nat,A_34: nat] :
( ( ord_less_eq_nat @ B_35 @ A_34 )
=> ( ( A_34 != B_35 )
=> ( ord_less_nat @ B_35 @ A_34 ) ) ) ).
thf(fact_915_order__le__neq__trans,axiom,
! [A_33: nat,B_34: nat] :
( ( ord_less_eq_nat @ A_33 @ B_34 )
=> ( ( A_33 != B_34 )
=> ( ord_less_nat @ A_33 @ B_34 ) ) ) ).
thf(fact_916_order__le__imp__less__or__eq,axiom,
! [X_46: nat,Y_34: nat] :
( ( ord_less_eq_nat @ X_46 @ Y_34 )
=> ( ( ord_less_nat @ X_46 @ Y_34 )
| ( X_46 = Y_34 ) ) ) ).
thf(fact_917_linorder__antisym__conv2,axiom,
! [X_45: nat,Y_33: nat] :
( ( ord_less_eq_nat @ X_45 @ Y_33 )
=> ( ~ ( ord_less_nat @ X_45 @ Y_33 )
<=> ( X_45 = Y_33 ) ) ) ).
thf(fact_918_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_919_order__less__imp__le,axiom,
! [X_44: nat,Y_32: nat] :
( ( ord_less_nat @ X_44 @ Y_32 )
=> ( ord_less_eq_nat @ X_44 @ Y_32 ) ) ).
thf(fact_920_leD,axiom,
! [Y_31: nat,X_43: nat] :
( ( ord_less_eq_nat @ Y_31 @ X_43 )
=> ~ ( ord_less_nat @ X_43 @ Y_31 ) ) ).
thf(fact_921_xt1_I12_J,axiom,
! [A_32: nat,B_33: nat] :
( ( A_32 != B_33 )
=> ( ( ord_less_eq_nat @ B_33 @ A_32 )
=> ( ord_less_nat @ B_33 @ A_32 ) ) ) ).
thf(fact_922_order__neq__le__trans,axiom,
! [A_31: nat,B_32: nat] :
( ( A_31 != B_32 )
=> ( ( ord_less_eq_nat @ A_31 @ B_32 )
=> ( ord_less_nat @ A_31 @ B_32 ) ) ) ).
thf(fact_923_linorder__antisym__conv1,axiom,
! [X_42: nat,Y_30: nat] :
( ~ ( ord_less_nat @ X_42 @ Y_30 )
=> ( ( ord_less_eq_nat @ X_42 @ Y_30 )
<=> ( X_42 = Y_30 ) ) ) ).
thf(fact_924_not__leE,axiom,
! [Y_29: nat,X_41: nat] :
( ~ ( ord_less_eq_nat @ Y_29 @ X_41 )
=> ( ord_less_nat @ X_41 @ Y_29 ) ) ).
thf(fact_925_leI,axiom,
! [X_40: nat,Y_28: nat] :
( ~ ( ord_less_nat @ X_40 @ Y_28 )
=> ( ord_less_eq_nat @ Y_28 @ X_40 ) ) ).
thf(fact_926_order__le__less,axiom,
! [X_39: nat,Y_27: nat] :
( ( ord_less_eq_nat @ X_39 @ Y_27 )
<=> ( ( ord_less_nat @ X_39 @ Y_27 )
| ( X_39 = Y_27 ) ) ) ).
thf(fact_927_less__le__not__le,axiom,
! [X_38: nat,Y_26: nat] :
( ( ord_less_nat @ X_38 @ Y_26 )
<=> ( ( ord_less_eq_nat @ X_38 @ Y_26 )
& ~ ( ord_less_eq_nat @ Y_26 @ X_38 ) ) ) ).
thf(fact_928_order__less__le,axiom,
! [X_37: nat,Y_25: nat] :
( ( ord_less_nat @ X_37 @ Y_25 )
<=> ( ( ord_less_eq_nat @ X_37 @ Y_25 )
& ( X_37 != Y_25 ) ) ) ).
thf(fact_929_linorder__le__less__linear,axiom,
! [X_36: nat,Y_24: nat] :
( ( ord_less_eq_nat @ X_36 @ Y_24 )
| ( ord_less_nat @ Y_24 @ X_36 ) ) ).
thf(fact_930_linorder__not__le,axiom,
! [X_35: nat,Y_23: nat] :
( ~ ( ord_less_eq_nat @ X_35 @ Y_23 )
<=> ( ord_less_nat @ Y_23 @ X_35 ) ) ).
thf(fact_931_linorder__not__less,axiom,
! [X_34: nat,Y_22: nat] :
( ~ ( ord_less_nat @ X_34 @ Y_22 )
<=> ( ord_less_eq_nat @ Y_22 @ X_34 ) ) ).
thf(fact_932_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_6: nat] :
( ( ord_less_nat @ X @ Z_6 )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_6 ) ) ) ).
thf(fact_933_termination__basic__simps_I1_J,axiom,
! [Z_6: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_6 ) ) ) ).
thf(fact_934_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_935_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_936_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_937_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_938_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_939_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_940_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_941_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_942_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_943_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_944_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_945_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_946_less__trans__Suc,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
thf(fact_947_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_948_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_949_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_950_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_951_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_952_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_953_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_954_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_955_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_956_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_957_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_958_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
thf(fact_959_less__max__iff__disj,axiom,
! [Z_5: nat,X_33: nat,Y_21: nat] :
( ( ord_less_nat @ Z_5 @ ( ord_max_nat @ X_33 @ Y_21 ) )
<=> ( ( ord_less_nat @ Z_5 @ X_33 )
| ( ord_less_nat @ Z_5 @ Y_21 ) ) ) ).
thf(fact_960_max__less__iff__conj,axiom,
! [X_32: nat,Y_20: nat,Z_4: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X_32 @ Y_20 ) @ Z_4 )
<=> ( ( ord_less_nat @ X_32 @ Z_4 )
& ( ord_less_nat @ Y_20 @ Z_4 ) ) ) ).
thf(fact_961_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_962_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_963_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_964_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_965_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_966_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P @ N @ M ) )
=> ( ( ( M = N )
=> ( P @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P @ N @ M ) )
=> ( P @ N @ M ) ) ) ) ).
thf(fact_967_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_968_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_969_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_970_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_971_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_972_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_973_add__less__cancel__right,axiom,
! [A_30: nat,C_18: nat,B_31: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_30 @ C_18 ) @ ( plus_plus_nat @ B_31 @ C_18 ) )
<=> ( ord_less_nat @ A_30 @ B_31 ) ) ).
thf(fact_974_add__less__cancel__left,axiom,
! [C_17: nat,A_29: nat,B_30: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_17 @ A_29 ) @ ( plus_plus_nat @ C_17 @ B_30 ) )
<=> ( ord_less_nat @ A_29 @ B_30 ) ) ).
thf(fact_975_add__strict__right__mono,axiom,
! [C_16: nat,A_28: nat,B_29: nat] :
( ( ord_less_nat @ A_28 @ B_29 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_28 @ C_16 ) @ ( plus_plus_nat @ B_29 @ C_16 ) ) ) ).
thf(fact_976_add__strict__left__mono,axiom,
! [C_15: nat,A_27: nat,B_28: nat] :
( ( ord_less_nat @ A_27 @ B_28 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_15 @ A_27 ) @ ( plus_plus_nat @ C_15 @ B_28 ) ) ) ).
thf(fact_977_add__strict__mono,axiom,
! [C_14: nat,D_3: nat,A_26: nat,B_27: nat] :
( ( ord_less_nat @ A_26 @ B_27 )
=> ( ( ord_less_nat @ C_14 @ D_3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_26 @ C_14 ) @ ( plus_plus_nat @ B_27 @ D_3 ) ) ) ) ).
thf(fact_978_add__less__imp__less__right,axiom,
! [A_25: nat,C_13: nat,B_26: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_25 @ C_13 ) @ ( plus_plus_nat @ B_26 @ C_13 ) )
=> ( ord_less_nat @ A_25 @ B_26 ) ) ).
thf(fact_979_add__less__imp__less__left,axiom,
! [C_12: nat,A_24: nat,B_25: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_12 @ A_24 ) @ ( plus_plus_nat @ C_12 @ B_25 ) )
=> ( ord_less_nat @ A_24 @ B_25 ) ) ).
thf(fact_980_linorder__cases,axiom,
! [X_31: nat,Y_19: nat] :
( ~ ( ord_less_nat @ X_31 @ Y_19 )
=> ( ( X_31 != Y_19 )
=> ( ord_less_nat @ Y_19 @ X_31 ) ) ) ).
thf(fact_981_order__less__asym,axiom,
! [X_30: nat,Y_18: nat] :
( ( ord_less_nat @ X_30 @ Y_18 )
=> ~ ( ord_less_nat @ Y_18 @ X_30 ) ) ).
thf(fact_982_xt1_I10_J,axiom,
! [Z_3: nat,Y_17: nat,X_29: nat] :
( ( ord_less_nat @ Y_17 @ X_29 )
=> ( ( ord_less_nat @ Z_3 @ Y_17 )
=> ( ord_less_nat @ Z_3 @ X_29 ) ) ) ).
thf(fact_983_order__less__trans,axiom,
! [Z_2: nat,X_28: nat,Y_16: nat] :
( ( ord_less_nat @ X_28 @ Y_16 )
=> ( ( ord_less_nat @ Y_16 @ Z_2 )
=> ( ord_less_nat @ X_28 @ Z_2 ) ) ) ).
thf(fact_984_xt1_I2_J,axiom,
! [C_11: nat,B_24: nat,A_23: nat] :
( ( ord_less_nat @ B_24 @ A_23 )
=> ( ( B_24 = C_11 )
=> ( ord_less_nat @ C_11 @ A_23 ) ) ) ).
thf(fact_985_ord__less__eq__trans,axiom,
! [C_10: nat,A_22: nat,B_23: nat] :
( ( ord_less_nat @ A_22 @ B_23 )
=> ( ( B_23 = C_10 )
=> ( ord_less_nat @ A_22 @ C_10 ) ) ) ).
thf(fact_986_xt1_I1_J,axiom,
! [C_9: nat,A_21: nat,B_22: nat] :
( ( A_21 = B_22 )
=> ( ( ord_less_nat @ C_9 @ B_22 )
=> ( ord_less_nat @ C_9 @ A_21 ) ) ) ).
thf(fact_987_ord__eq__less__trans,axiom,
! [C_8: nat,A_20: nat,B_21: nat] :
( ( A_20 = B_21 )
=> ( ( ord_less_nat @ B_21 @ C_8 )
=> ( ord_less_nat @ A_20 @ C_8 ) ) ) ).
thf(fact_988_xt1_I9_J,axiom,
! [B_20: nat,A_19: nat] :
( ( ord_less_nat @ B_20 @ A_19 )
=> ~ ( ord_less_nat @ A_19 @ B_20 ) ) ).
thf(fact_989_order__less__asym_H,axiom,
! [A_18: nat,B_19: nat] :
( ( ord_less_nat @ A_18 @ B_19 )
=> ~ ( ord_less_nat @ B_19 @ A_18 ) ) ).
thf(fact_990_order__less__imp__triv,axiom,
! [P_11: $o,X_27: nat,Y_15: nat] :
( ( ord_less_nat @ X_27 @ Y_15 )
=> ( ( ord_less_nat @ Y_15 @ X_27 )
=> P_11 ) ) ).
thf(fact_991_order__less__imp__not__eq2,axiom,
! [X_26: nat,Y_14: nat] :
( ( ord_less_nat @ X_26 @ Y_14 )
=> ( Y_14 != X_26 ) ) ).
thf(fact_992_order__less__imp__not__eq,axiom,
! [X_25: nat,Y_13: nat] :
( ( ord_less_nat @ X_25 @ Y_13 )
=> ( X_25 != Y_13 ) ) ).
thf(fact_993_order__less__imp__not__less,axiom,
! [X_24: nat,Y_12: nat] :
( ( ord_less_nat @ X_24 @ Y_12 )
=> ~ ( ord_less_nat @ Y_12 @ X_24 ) ) ).
thf(fact_994_order__less__not__sym,axiom,
! [X_23: nat,Y_11: nat] :
( ( ord_less_nat @ X_23 @ Y_11 )
=> ~ ( ord_less_nat @ Y_11 @ X_23 ) ) ).
thf(fact_995_less__imp__neq,axiom,
! [X_22: nat,Y_10: nat] :
( ( ord_less_nat @ X_22 @ Y_10 )
=> ( X_22 != Y_10 ) ) ).
thf(fact_996_linorder__neqE,axiom,
! [X_21: nat,Y_9: nat] :
( ( X_21 != Y_9 )
=> ( ~ ( ord_less_nat @ X_21 @ Y_9 )
=> ( ord_less_nat @ Y_9 @ X_21 ) ) ) ).
thf(fact_997_linorder__antisym__conv3,axiom,
! [Y_8: nat,X_20: nat] :
( ~ ( ord_less_nat @ Y_8 @ X_20 )
=> ( ~ ( ord_less_nat @ X_20 @ Y_8 )
<=> ( X_20 = Y_8 ) ) ) ).
thf(fact_998_linorder__less__linear,axiom,
! [X_19: nat,Y_7: nat] :
( ( ord_less_nat @ X_19 @ Y_7 )
| ( X_19 = Y_7 )
| ( ord_less_nat @ Y_7 @ X_19 ) ) ).
thf(fact_999_not__less__iff__gr__or__eq,axiom,
! [X_18: nat,Y_6: nat] :
( ~ ( ord_less_nat @ X_18 @ Y_6 )
<=> ( ( ord_less_nat @ Y_6 @ X_18 )
| ( X_18 = Y_6 ) ) ) ).
thf(fact_1000_linorder__neq__iff,axiom,
! [X_17: nat,Y_5: nat] :
( ( X_17 != Y_5 )
<=> ( ( ord_less_nat @ X_17 @ Y_5 )
| ( ord_less_nat @ Y_5 @ X_17 ) ) ) ).
thf(fact_1001_order__less__irrefl,axiom,
! [X_16: nat] :
~ ( ord_less_nat @ X_16 @ X_16 ) ).
thf(fact_1002_min__max_Oless__supI2,axiom,
! [A_17: nat,X_15: nat,B_18: nat] :
( ( ord_less_nat @ X_15 @ B_18 )
=> ( ord_less_nat @ X_15 @ ( ord_max_nat @ A_17 @ B_18 ) ) ) ).
thf(fact_1003_min__max_Oless__supI1,axiom,
! [B_17: nat,X_14: nat,A_16: nat] :
( ( ord_less_nat @ X_14 @ A_16 )
=> ( ord_less_nat @ X_14 @ ( ord_max_nat @ A_16 @ B_17 ) ) ) ).
thf(fact_1004_min__less__iff__disj,axiom,
! [X_13: nat,Y_4: nat,Z_1: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X_13 @ Y_4 ) @ Z_1 )
<=> ( ( ord_less_nat @ X_13 @ Z_1 )
| ( ord_less_nat @ Y_4 @ Z_1 ) ) ) ).
thf(fact_1005_min__less__iff__conj,axiom,
! [Z: nat,X_12: nat,Y_3: nat] :
( ( ord_less_nat @ Z @ ( ord_min_nat @ X_12 @ Y_3 ) )
<=> ( ( ord_less_nat @ Z @ X_12 )
& ( ord_less_nat @ Z @ Y_3 ) ) ) ).
thf(fact_1006_min__max_Oless__infI2,axiom,
! [A_15: nat,B_16: nat,X_11: nat] :
( ( ord_less_nat @ B_16 @ X_11 )
=> ( ord_less_nat @ ( ord_min_nat @ A_15 @ B_16 ) @ X_11 ) ) ).
thf(fact_1007_min__max_Oless__infI1,axiom,
! [B_15: nat,A_14: nat,X_10: nat] :
( ( ord_less_nat @ A_14 @ X_10 )
=> ( ord_less_nat @ ( ord_min_nat @ A_14 @ B_15 ) @ X_10 ) ) ).
thf(fact_1008_add__le__less__mono,axiom,
! [C_7: nat,D_2: nat,A_13: nat,B_14: nat] :
( ( ord_less_eq_nat @ A_13 @ B_14 )
=> ( ( ord_less_nat @ C_7 @ D_2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_13 @ C_7 ) @ ( plus_plus_nat @ B_14 @ D_2 ) ) ) ) ).
thf(fact_1009_add__less__le__mono,axiom,
! [C_6: nat,D_1: nat,A_12: nat,B_13: nat] :
( ( ord_less_nat @ A_12 @ B_13 )
=> ( ( ord_less_eq_nat @ C_6 @ D_1 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_12 @ C_6 ) @ ( plus_plus_nat @ B_13 @ D_1 ) ) ) ) ).
thf(fact_1010_add__pos__pos,axiom,
! [B_12: nat,A_11: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_11 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_12 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_11 @ B_12 ) ) ) ) ).
thf(fact_1011_add__neg__neg,axiom,
! [B_11: nat,A_10: nat] :
( ( ord_less_nat @ A_10 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_11 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_10 @ B_11 ) @ zero_zero_nat ) ) ) ).
thf(fact_1012_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M_6: nat] :
( N
= ( suc @ M_6 ) ) ) ).
thf(fact_1013_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1014_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( M = zero_zero_nat )
| ? [J_1: nat] :
( ( M
= ( suc @ J_1 ) )
& ( ord_less_nat @ J_1 @ N ) ) ) ) ).
thf(fact_1015_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_1016_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_1017_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_1018_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ? [K_2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_2 ) ) ) ) ).
thf(fact_1019_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_1020_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1021_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1022_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_1023_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_1024_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_1025_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1026_diff__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
thf(fact_1027_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1028_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_1029_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_1030_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
thf(fact_1031_diff__less__mono,axiom,
! [C_5: nat,A_3: nat,B_4: nat] :
( ( ord_less_nat @ A_3 @ B_4 )
=> ( ( ord_less_eq_nat @ C_5 @ A_3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A_3 @ C_5 ) @ ( minus_minus_nat @ B_4 @ C_5 ) ) ) ) ).
thf(fact_1032_less__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_1033_list__eq__iff__nth__eq,axiom,
! [Xs_50: list_o,Ys_6: list_o] :
( ( Xs_50 = Ys_6 )
<=> ( ( ( size_size_list_o @ Xs_50 )
= ( size_size_list_o @ Ys_6 ) )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_50 ) )
=> ( ( nth_o @ Xs_50 @ I_3 )
<=> ( nth_o @ Ys_6 @ I_3 ) ) ) ) ) ).
thf(fact_1034_list__eq__iff__nth__eq,axiom,
! [Xs_50: list_A1402527603le_alt,Ys_6: list_A1402527603le_alt] :
( ( Xs_50 = Ys_6 )
<=> ( ( ( size_s1146069981le_alt @ Xs_50 )
= ( size_s1146069981le_alt @ Ys_6 ) )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_50 ) )
=> ( ( nth_Ar1039551983le_alt @ Xs_50 @ I_3 )
= ( nth_Ar1039551983le_alt @ Ys_6 @ I_3 ) ) ) ) ) ).
thf(fact_1035_list__eq__iff__nth__eq,axiom,
! [Xs_50: list_nat,Ys_6: list_nat] :
( ( Xs_50 = Ys_6 )
<=> ( ( ( size_size_list_nat @ Xs_50 )
= ( size_size_list_nat @ Ys_6 ) )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_50 ) )
=> ( ( nth_nat @ Xs_50 @ I_3 )
= ( nth_nat @ Ys_6 @ I_3 ) ) ) ) ) ).
thf(fact_1036_nth__take,axiom,
! [Xs_49: list_o,I_7: nat,N_33: nat] :
( ( ord_less_nat @ I_7 @ N_33 )
=> ( ( nth_o @ ( take_o @ N_33 @ Xs_49 ) @ I_7 )
<=> ( nth_o @ Xs_49 @ I_7 ) ) ) ).
thf(fact_1037_nth__take,axiom,
! [Xs_49: list_A1402527603le_alt,I_7: nat,N_33: nat] :
( ( ord_less_nat @ I_7 @ N_33 )
=> ( ( nth_Ar1039551983le_alt @ ( take_A1527720662le_alt @ N_33 @ Xs_49 ) @ I_7 )
= ( nth_Ar1039551983le_alt @ Xs_49 @ I_7 ) ) ) ).
thf(fact_1038_nth__take,axiom,
! [Xs_49: list_nat,I_7: nat,N_33: nat] :
( ( ord_less_nat @ I_7 @ N_33 )
=> ( ( nth_nat @ ( take_nat @ N_33 @ Xs_49 ) @ I_7 )
= ( nth_nat @ Xs_49 @ I_7 ) ) ) ).
thf(fact_1039_nth__replicate,axiom,
! [X_9: $o,I_6: nat,N_32: nat] :
( ( ord_less_nat @ I_6 @ N_32 )
=> ( ( nth_o @ ( replicate_o @ N_32 @ X_9 ) @ I_6 )
<=> X_9 ) ) ).
thf(fact_1040_nth__replicate,axiom,
! [X_9: arrow_1389031998le_alt,I_6: nat,N_32: nat] :
( ( ord_less_nat @ I_6 @ N_32 )
=> ( ( nth_Ar1039551983le_alt @ ( replic277728168le_alt @ N_32 @ X_9 ) @ I_6 )
= X_9 ) ) ).
thf(fact_1041_nth__replicate,axiom,
! [X_9: nat,I_6: nat,N_32: nat] :
( ( ord_less_nat @ I_6 @ N_32 )
=> ( ( nth_nat @ ( replicate_nat @ N_32 @ X_9 ) @ I_6 )
= X_9 ) ) ).
thf(fact_1042_list__size__estimation,axiom,
! [Y_2: nat,F_4: arrow_1389031998le_alt > nat,X_8: arrow_1389031998le_alt,Xs_48: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_8 @ ( set_Ar1491127311le_alt @ Xs_48 ) )
=> ( ( ord_less_nat @ Y_2 @ ( F_4 @ X_8 ) )
=> ( ord_less_nat @ Y_2 @ ( list_s1163968880le_alt @ F_4 @ Xs_48 ) ) ) ) ).
thf(fact_1043_list__size__estimation,axiom,
! [Y_2: nat,F_4: nat > nat,X_8: nat,Xs_48: list_nat] :
( ( member_nat @ X_8 @ ( set_nat @ Xs_48 ) )
=> ( ( ord_less_nat @ Y_2 @ ( F_4 @ X_8 ) )
=> ( ord_less_nat @ Y_2 @ ( list_size_nat @ F_4 @ Xs_48 ) ) ) ) ).
thf(fact_1044_less__eq__Suc__le__raw,axiom,
! [X_4: nat] :
( ( ord_less_nat @ X_4 )
= ( ord_less_eq_nat @ ( suc @ X_4 ) ) ) ).
thf(fact_1045_add__nonpos__neg,axiom,
! [B_10: nat,A_9: nat] :
( ( ord_less_eq_nat @ A_9 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_10 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_9 @ B_10 ) @ zero_zero_nat ) ) ) ).
thf(fact_1046_add__neg__nonpos,axiom,
! [B_9: nat,A_8: nat] :
( ( ord_less_nat @ A_8 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_9 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_8 @ B_9 ) @ zero_zero_nat ) ) ) ).
thf(fact_1047_add__strict__increasing2,axiom,
! [B_8: nat,C_4: nat,A_7: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_7 )
=> ( ( ord_less_nat @ B_8 @ C_4 )
=> ( ord_less_nat @ B_8 @ ( plus_plus_nat @ A_7 @ C_4 ) ) ) ) ).
thf(fact_1048_add__strict__increasing,axiom,
! [B_7: nat,C_3: nat,A_6: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_6 )
=> ( ( ord_less_eq_nat @ B_7 @ C_3 )
=> ( ord_less_nat @ B_7 @ ( plus_plus_nat @ A_6 @ C_3 ) ) ) ) ).
thf(fact_1049_add__nonneg__pos,axiom,
! [B_6: nat,A_5: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_5 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_6 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_5 @ B_6 ) ) ) ) ).
thf(fact_1050_add__pos__nonneg,axiom,
! [B_5: nat,A_4: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_5 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_4 @ B_5 ) ) ) ) ).
thf(fact_1051_length__greater__0__conv,axiom,
! [Xs_47: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs_47 ) )
<=> ( Xs_47 != nil_nat ) ) ).
thf(fact_1052_length__greater__0__conv,axiom,
! [Xs_47: list_A1402527603le_alt] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1146069981le_alt @ Xs_47 ) )
<=> ( Xs_47 != nil_Ar52383470le_alt ) ) ).
thf(fact_1053_length__pos__if__in__set,axiom,
! [X_7: arrow_1389031998le_alt,Xs_46: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_7 @ ( set_Ar1491127311le_alt @ Xs_46 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1146069981le_alt @ Xs_46 ) ) ) ).
thf(fact_1054_length__pos__if__in__set,axiom,
! [X_7: nat,Xs_46: list_nat] :
( ( member_nat @ X_7 @ ( set_nat @ Xs_46 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs_46 ) ) ) ).
thf(fact_1055_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
thf(fact_1056_diff__Suc__less,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
thf(fact_1057_nat__diff__split__asm,axiom,
! [P: nat > $o,A_3: nat,B_4: nat] :
( ( P @ ( minus_minus_nat @ A_3 @ B_4 ) )
<=> ~ ( ( ( ord_less_nat @ A_3 @ B_4 )
& ~ ( P @ zero_zero_nat ) )
| ? [D: nat] :
( ( A_3
= ( plus_plus_nat @ B_4 @ D ) )
& ~ ( P @ D ) ) ) ) ).
thf(fact_1058_nat__diff__split,axiom,
! [P: nat > $o,A_3: nat,B_4: nat] :
( ( P @ ( minus_minus_nat @ A_3 @ B_4 ) )
<=> ( ( ( ord_less_nat @ A_3 @ B_4 )
=> ( P @ zero_zero_nat ) )
& ! [D: nat] :
( ( A_3
= ( plus_plus_nat @ B_4 @ D ) )
=> ( P @ D ) ) ) ) ).
thf(fact_1059_in__set__conv__nth,axiom,
! [X_6: $o,Xs_45: list_o] :
( ( member_o @ X_6 @ ( set_o @ Xs_45 ) )
<=> ? [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_45 ) )
& ( ( nth_o @ Xs_45 @ I_3 )
<=> X_6 ) ) ) ).
thf(fact_1060_in__set__conv__nth,axiom,
! [X_6: arrow_1389031998le_alt,Xs_45: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_6 @ ( set_Ar1491127311le_alt @ Xs_45 ) )
<=> ? [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_45 ) )
& ( ( nth_Ar1039551983le_alt @ Xs_45 @ I_3 )
= X_6 ) ) ) ).
thf(fact_1061_in__set__conv__nth,axiom,
! [X_6: nat,Xs_45: list_nat] :
( ( member_nat @ X_6 @ ( set_nat @ Xs_45 ) )
<=> ? [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_45 ) )
& ( ( nth_nat @ Xs_45 @ I_3 )
= X_6 ) ) ) ).
thf(fact_1062_all__set__conv__all__nth,axiom,
! [P_10: $o > $o,Xs_44: list_o] :
( ! [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_44 ) )
=> ( P_10 @ X_4 ) )
<=> ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_44 ) )
=> ( P_10 @ ( nth_o @ Xs_44 @ I_3 ) ) ) ) ).
thf(fact_1063_all__set__conv__all__nth,axiom,
! [P_10: arrow_1389031998le_alt > $o,Xs_44: list_A1402527603le_alt] :
( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_44 ) )
=> ( P_10 @ X_4 ) )
<=> ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_44 ) )
=> ( P_10 @ ( nth_Ar1039551983le_alt @ Xs_44 @ I_3 ) ) ) ) ).
thf(fact_1064_all__set__conv__all__nth,axiom,
! [P_10: nat > $o,Xs_44: list_nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_44 ) )
=> ( P_10 @ X_4 ) )
<=> ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_44 ) )
=> ( P_10 @ ( nth_nat @ Xs_44 @ I_3 ) ) ) ) ).
thf(fact_1065_nth__mem,axiom,
! [N_31: nat,Xs_43: list_o] :
( ( ord_less_nat @ N_31 @ ( size_size_list_o @ Xs_43 ) )
=> ( member_o @ ( nth_o @ Xs_43 @ N_31 ) @ ( set_o @ Xs_43 ) ) ) ).
thf(fact_1066_nth__mem,axiom,
! [N_31: nat,Xs_43: list_A1402527603le_alt] :
( ( ord_less_nat @ N_31 @ ( size_s1146069981le_alt @ Xs_43 ) )
=> ( member998036369le_alt @ ( nth_Ar1039551983le_alt @ Xs_43 @ N_31 ) @ ( set_Ar1491127311le_alt @ Xs_43 ) ) ) ).
thf(fact_1067_nth__mem,axiom,
! [N_31: nat,Xs_43: list_nat] :
( ( ord_less_nat @ N_31 @ ( size_size_list_nat @ Xs_43 ) )
=> ( member_nat @ ( nth_nat @ Xs_43 @ N_31 ) @ ( set_nat @ Xs_43 ) ) ) ).
thf(fact_1068_distinct__conv__nth,axiom,
! [Xs_42: list_o] :
( ( distinct_o @ Xs_42 )
<=> ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_42 ) )
=> ! [J_1: nat] :
( ( ord_less_nat @ J_1 @ ( size_size_list_o @ Xs_42 ) )
=> ( ( I_3 != J_1 )
=> ~ ( ( nth_o @ Xs_42 @ I_3 )
<=> ( nth_o @ Xs_42 @ J_1 ) ) ) ) ) ) ).
thf(fact_1069_distinct__conv__nth,axiom,
! [Xs_42: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_42 )
<=> ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_42 ) )
=> ! [J_1: nat] :
( ( ord_less_nat @ J_1 @ ( size_s1146069981le_alt @ Xs_42 ) )
=> ( ( I_3 != J_1 )
=> ( ( nth_Ar1039551983le_alt @ Xs_42 @ I_3 )
!= ( nth_Ar1039551983le_alt @ Xs_42 @ J_1 ) ) ) ) ) ) ).
thf(fact_1070_distinct__conv__nth,axiom,
! [Xs_42: list_nat] :
( ( distinct_nat @ Xs_42 )
<=> ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_42 ) )
=> ! [J_1: nat] :
( ( ord_less_nat @ J_1 @ ( size_size_list_nat @ Xs_42 ) )
=> ( ( I_3 != J_1 )
=> ( ( nth_nat @ Xs_42 @ I_3 )
!= ( nth_nat @ Xs_42 @ J_1 ) ) ) ) ) ) ).
thf(fact_1071_nth__eq__iff__index__eq,axiom,
! [J_6: nat,I_5: nat,Xs_41: list_o] :
( ( distinct_o @ Xs_41 )
=> ( ( ord_less_nat @ I_5 @ ( size_size_list_o @ Xs_41 ) )
=> ( ( ord_less_nat @ J_6 @ ( size_size_list_o @ Xs_41 ) )
=> ( ( ( nth_o @ Xs_41 @ I_5 )
<=> ( nth_o @ Xs_41 @ J_6 ) )
<=> ( I_5 = J_6 ) ) ) ) ) ).
thf(fact_1072_nth__eq__iff__index__eq,axiom,
! [J_6: nat,I_5: nat,Xs_41: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_41 )
=> ( ( ord_less_nat @ I_5 @ ( size_s1146069981le_alt @ Xs_41 ) )
=> ( ( ord_less_nat @ J_6 @ ( size_s1146069981le_alt @ Xs_41 ) )
=> ( ( ( nth_Ar1039551983le_alt @ Xs_41 @ I_5 )
= ( nth_Ar1039551983le_alt @ Xs_41 @ J_6 ) )
<=> ( I_5 = J_6 ) ) ) ) ) ).
thf(fact_1073_nth__eq__iff__index__eq,axiom,
! [J_6: nat,I_5: nat,Xs_41: list_nat] :
( ( distinct_nat @ Xs_41 )
=> ( ( ord_less_nat @ I_5 @ ( size_size_list_nat @ Xs_41 ) )
=> ( ( ord_less_nat @ J_6 @ ( size_size_list_nat @ Xs_41 ) )
=> ( ( ( nth_nat @ Xs_41 @ I_5 )
= ( nth_nat @ Xs_41 @ J_6 ) )
<=> ( I_5 = J_6 ) ) ) ) ) ).
thf(fact_1074_nth__length__takeWhile,axiom,
! [P_9: $o > $o,Xs_40: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ ( takeWhile_o @ P_9 @ Xs_40 ) ) @ ( size_size_list_o @ Xs_40 ) )
=> ~ ( P_9 @ ( nth_o @ Xs_40 @ ( size_size_list_o @ ( takeWhile_o @ P_9 @ Xs_40 ) ) ) ) ) ).
thf(fact_1075_nth__length__takeWhile,axiom,
! [P_9: arrow_1389031998le_alt > $o,Xs_40: list_A1402527603le_alt] :
( ( ord_less_nat @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_9 @ Xs_40 ) ) @ ( size_s1146069981le_alt @ Xs_40 ) )
=> ~ ( P_9 @ ( nth_Ar1039551983le_alt @ Xs_40 @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_9 @ Xs_40 ) ) ) ) ) ).
thf(fact_1076_nth__length__takeWhile,axiom,
! [P_9: nat > $o,Xs_40: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P_9 @ Xs_40 ) ) @ ( size_size_list_nat @ Xs_40 ) )
=> ~ ( P_9 @ ( nth_nat @ Xs_40 @ ( size_size_list_nat @ ( takeWhile_nat @ P_9 @ Xs_40 ) ) ) ) ) ).
thf(fact_1077_takeWhile__nth,axiom,
! [J_5: nat,P_8: $o > $o,Xs_39: list_o] :
( ( ord_less_nat @ J_5 @ ( size_size_list_o @ ( takeWhile_o @ P_8 @ Xs_39 ) ) )
=> ( ( nth_o @ ( takeWhile_o @ P_8 @ Xs_39 ) @ J_5 )
<=> ( nth_o @ Xs_39 @ J_5 ) ) ) ).
thf(fact_1078_takeWhile__nth,axiom,
! [J_5: nat,P_8: arrow_1389031998le_alt > $o,Xs_39: list_A1402527603le_alt] :
( ( ord_less_nat @ J_5 @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_8 @ Xs_39 ) ) )
=> ( ( nth_Ar1039551983le_alt @ ( takeWh462480871le_alt @ P_8 @ Xs_39 ) @ J_5 )
= ( nth_Ar1039551983le_alt @ Xs_39 @ J_5 ) ) ) ).
thf(fact_1079_takeWhile__nth,axiom,
! [J_5: nat,P_8: nat > $o,Xs_39: list_nat] :
( ( ord_less_nat @ J_5 @ ( size_size_list_nat @ ( takeWhile_nat @ P_8 @ Xs_39 ) ) )
=> ( ( nth_nat @ ( takeWhile_nat @ P_8 @ Xs_39 ) @ J_5 )
= ( nth_nat @ Xs_39 @ J_5 ) ) ) ).
thf(fact_1080_take__butlast,axiom,
! [N_30: nat,Xs_38: list_A1402527603le_alt] :
( ( ord_less_nat @ N_30 @ ( size_s1146069981le_alt @ Xs_38 ) )
=> ( ( take_A1527720662le_alt @ N_30 @ ( butlas1188620858le_alt @ Xs_38 ) )
= ( take_A1527720662le_alt @ N_30 @ Xs_38 ) ) ) ).
thf(fact_1081_take__butlast,axiom,
! [N_30: nat,Xs_38: list_nat] :
( ( ord_less_nat @ N_30 @ ( size_size_list_nat @ Xs_38 ) )
=> ( ( take_nat @ N_30 @ ( butlast_nat @ Xs_38 ) )
= ( take_nat @ N_30 @ Xs_38 ) ) ) ).
thf(fact_1082_list__all__length,axiom,
! [P_7: $o > $o,Xs_37: list_o] :
( ( list_all_o @ P_7 @ Xs_37 )
<=> ! [N_23: nat] :
( ( ord_less_nat @ N_23 @ ( size_size_list_o @ Xs_37 ) )
=> ( P_7 @ ( nth_o @ Xs_37 @ N_23 ) ) ) ) ).
thf(fact_1083_list__all__length,axiom,
! [P_7: arrow_1389031998le_alt > $o,Xs_37: list_A1402527603le_alt] :
( ( list_a1483787182le_alt @ P_7 @ Xs_37 )
<=> ! [N_23: nat] :
( ( ord_less_nat @ N_23 @ ( size_s1146069981le_alt @ Xs_37 ) )
=> ( P_7 @ ( nth_Ar1039551983le_alt @ Xs_37 @ N_23 ) ) ) ) ).
thf(fact_1084_list__all__length,axiom,
! [P_7: nat > $o,Xs_37: list_nat] :
( ( list_all_nat @ P_7 @ Xs_37 )
<=> ! [N_23: nat] :
( ( ord_less_nat @ N_23 @ ( size_size_list_nat @ Xs_37 ) )
=> ( P_7 @ ( nth_nat @ Xs_37 @ N_23 ) ) ) ) ).
thf(fact_1085_nth__append,axiom,
! [Xs_36: list_o,Ys_5: list_o,N_29: nat] :
( ( nth_o @ ( append_o @ Xs_36 @ Ys_5 ) @ N_29 )
<=> ( ( ( ord_less_nat @ N_29 @ ( size_size_list_o @ Xs_36 ) )
=> ( nth_o @ Xs_36 @ N_29 ) )
& ( ~ ( ord_less_nat @ N_29 @ ( size_size_list_o @ Xs_36 ) )
=> ( nth_o @ Ys_5 @ ( minus_minus_nat @ N_29 @ ( size_size_list_o @ Xs_36 ) ) ) ) ) ) ).
thf(fact_1086_nth__append,axiom,
! [Ys_5: list_A1402527603le_alt,N_29: nat,Xs_36: list_A1402527603le_alt] :
( ( ( ord_less_nat @ N_29 @ ( size_s1146069981le_alt @ Xs_36 ) )
=> ( ( nth_Ar1039551983le_alt @ ( append1092755459le_alt @ Xs_36 @ Ys_5 ) @ N_29 )
= ( nth_Ar1039551983le_alt @ Xs_36 @ N_29 ) ) )
& ( ~ ( ord_less_nat @ N_29 @ ( size_s1146069981le_alt @ Xs_36 ) )
=> ( ( nth_Ar1039551983le_alt @ ( append1092755459le_alt @ Xs_36 @ Ys_5 ) @ N_29 )
= ( nth_Ar1039551983le_alt @ Ys_5 @ ( minus_minus_nat @ N_29 @ ( size_s1146069981le_alt @ Xs_36 ) ) ) ) ) ) ).
thf(fact_1087_nth__append,axiom,
! [Ys_5: list_nat,N_29: nat,Xs_36: list_nat] :
( ( ( ord_less_nat @ N_29 @ ( size_size_list_nat @ Xs_36 ) )
=> ( ( nth_nat @ ( append_nat @ Xs_36 @ Ys_5 ) @ N_29 )
= ( nth_nat @ Xs_36 @ N_29 ) ) )
& ( ~ ( ord_less_nat @ N_29 @ ( size_size_list_nat @ Xs_36 ) )
=> ( ( nth_nat @ ( append_nat @ Xs_36 @ Ys_5 ) @ N_29 )
= ( nth_nat @ Ys_5 @ ( minus_minus_nat @ N_29 @ ( size_size_list_nat @ Xs_36 ) ) ) ) ) ) ).
thf(fact_1088_sorted__nth__mono,axiom,
! [I_4: nat,J_4: nat,Xs_35: list_nat] :
( ( linorder_sorted_nat @ Xs_35 )
=> ( ( ord_less_eq_nat @ I_4 @ J_4 )
=> ( ( ord_less_nat @ J_4 @ ( size_size_list_nat @ Xs_35 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs_35 @ I_4 ) @ ( nth_nat @ Xs_35 @ J_4 ) ) ) ) ) ).
thf(fact_1089_sorted__equals__nth__mono,axiom,
! [Xs_34: list_nat] :
( ( linorder_sorted_nat @ Xs_34 )
<=> ! [J_1: nat] :
( ( ord_less_nat @ J_1 @ ( size_size_list_nat @ Xs_34 ) )
=> ! [I_3: nat] :
( ( ord_less_eq_nat @ I_3 @ J_1 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs_34 @ I_3 ) @ ( nth_nat @ Xs_34 @ J_1 ) ) ) ) ) ).
thf(fact_1090_rev__nth,axiom,
! [N_28: nat,Xs_33: list_o] :
( ( ord_less_nat @ N_28 @ ( size_size_list_o @ Xs_33 ) )
=> ( ( nth_o @ ( rev_o @ Xs_33 ) @ N_28 )
<=> ( nth_o @ Xs_33 @ ( minus_minus_nat @ ( size_size_list_o @ Xs_33 ) @ ( suc @ N_28 ) ) ) ) ) ).
thf(fact_1091_rev__nth,axiom,
! [N_28: nat,Xs_33: list_A1402527603le_alt] :
( ( ord_less_nat @ N_28 @ ( size_s1146069981le_alt @ Xs_33 ) )
=> ( ( nth_Ar1039551983le_alt @ ( rev_Ar2020079950le_alt @ Xs_33 ) @ N_28 )
= ( nth_Ar1039551983le_alt @ Xs_33 @ ( minus_minus_nat @ ( size_s1146069981le_alt @ Xs_33 ) @ ( suc @ N_28 ) ) ) ) ) ).
thf(fact_1092_rev__nth,axiom,
! [N_28: nat,Xs_33: list_nat] :
( ( ord_less_nat @ N_28 @ ( size_size_list_nat @ Xs_33 ) )
=> ( ( nth_nat @ ( rev_nat @ Xs_33 ) @ N_28 )
= ( nth_nat @ Xs_33 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs_33 ) @ ( suc @ N_28 ) ) ) ) ) ).
thf(fact_1093_dropWhile__nth,axiom,
! [J_3: nat,P_6: $o > $o,Xs_32: list_o] :
( ( ord_less_nat @ J_3 @ ( size_size_list_o @ ( dropWhile_o @ P_6 @ Xs_32 ) ) )
=> ( ( nth_o @ ( dropWhile_o @ P_6 @ Xs_32 ) @ J_3 )
<=> ( nth_o @ Xs_32 @ ( plus_plus_nat @ J_3 @ ( size_size_list_o @ ( takeWhile_o @ P_6 @ Xs_32 ) ) ) ) ) ) ).
thf(fact_1094_dropWhile__nth,axiom,
! [J_3: nat,P_6: arrow_1389031998le_alt > $o,Xs_32: list_A1402527603le_alt] :
( ( ord_less_nat @ J_3 @ ( size_s1146069981le_alt @ ( dropWh82971279le_alt @ P_6 @ Xs_32 ) ) )
=> ( ( nth_Ar1039551983le_alt @ ( dropWh82971279le_alt @ P_6 @ Xs_32 ) @ J_3 )
= ( nth_Ar1039551983le_alt @ Xs_32 @ ( plus_plus_nat @ J_3 @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_6 @ Xs_32 ) ) ) ) ) ) ).
thf(fact_1095_dropWhile__nth,axiom,
! [J_3: nat,P_6: nat > $o,Xs_32: list_nat] :
( ( ord_less_nat @ J_3 @ ( size_size_list_nat @ ( dropWhile_nat @ P_6 @ Xs_32 ) ) )
=> ( ( nth_nat @ ( dropWhile_nat @ P_6 @ Xs_32 ) @ J_3 )
= ( nth_nat @ Xs_32 @ ( plus_plus_nat @ J_3 @ ( size_size_list_nat @ ( takeWhile_nat @ P_6 @ Xs_32 ) ) ) ) ) ) ).
thf(fact_1096_sorted__nth__monoI,axiom,
! [Xs_31: list_nat] :
( ! [I_3: nat,J_1: nat] :
( ( ord_less_eq_nat @ I_3 @ J_1 )
=> ( ( ord_less_nat @ J_1 @ ( size_size_list_nat @ Xs_31 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs_31 @ I_3 ) @ ( nth_nat @ Xs_31 @ J_1 ) ) ) )
=> ( linorder_sorted_nat @ Xs_31 ) ) ).
thf(fact_1097_takeWhile__eq__take__P__nth,axiom,
! [P_5: $o > $o,Xs_30: list_o,N_27: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ N_27 )
=> ( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_30 ) )
=> ( P_5 @ ( nth_o @ Xs_30 @ I_3 ) ) ) )
=> ( ( ( ord_less_nat @ N_27 @ ( size_size_list_o @ Xs_30 ) )
=> ~ ( P_5 @ ( nth_o @ Xs_30 @ N_27 ) ) )
=> ( ( takeWhile_o @ P_5 @ Xs_30 )
= ( take_o @ N_27 @ Xs_30 ) ) ) ) ).
thf(fact_1098_takeWhile__eq__take__P__nth,axiom,
! [P_5: arrow_1389031998le_alt > $o,Xs_30: list_A1402527603le_alt,N_27: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ N_27 )
=> ( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_30 ) )
=> ( P_5 @ ( nth_Ar1039551983le_alt @ Xs_30 @ I_3 ) ) ) )
=> ( ( ( ord_less_nat @ N_27 @ ( size_s1146069981le_alt @ Xs_30 ) )
=> ~ ( P_5 @ ( nth_Ar1039551983le_alt @ Xs_30 @ N_27 ) ) )
=> ( ( takeWh462480871le_alt @ P_5 @ Xs_30 )
= ( take_A1527720662le_alt @ N_27 @ Xs_30 ) ) ) ) ).
thf(fact_1099_takeWhile__eq__take__P__nth,axiom,
! [P_5: nat > $o,Xs_30: list_nat,N_27: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ N_27 )
=> ( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_30 ) )
=> ( P_5 @ ( nth_nat @ Xs_30 @ I_3 ) ) ) )
=> ( ( ( ord_less_nat @ N_27 @ ( size_size_list_nat @ Xs_30 ) )
=> ~ ( P_5 @ ( nth_nat @ Xs_30 @ N_27 ) ) )
=> ( ( takeWhile_nat @ P_5 @ Xs_30 )
= ( take_nat @ N_27 @ Xs_30 ) ) ) ) ).
thf(fact_1100_length__takeWhile__less__P__nth,axiom,
! [P_4: $o > $o,Xs_29: list_o,J_2: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ J_2 )
=> ( P_4 @ ( nth_o @ Xs_29 @ I_3 ) ) )
=> ( ( ord_less_eq_nat @ J_2 @ ( size_size_list_o @ Xs_29 ) )
=> ( ord_less_eq_nat @ J_2 @ ( size_size_list_o @ ( takeWhile_o @ P_4 @ Xs_29 ) ) ) ) ) ).
thf(fact_1101_length__takeWhile__less__P__nth,axiom,
! [P_4: arrow_1389031998le_alt > $o,Xs_29: list_A1402527603le_alt,J_2: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ J_2 )
=> ( P_4 @ ( nth_Ar1039551983le_alt @ Xs_29 @ I_3 ) ) )
=> ( ( ord_less_eq_nat @ J_2 @ ( size_s1146069981le_alt @ Xs_29 ) )
=> ( ord_less_eq_nat @ J_2 @ ( size_s1146069981le_alt @ ( takeWh462480871le_alt @ P_4 @ Xs_29 ) ) ) ) ) ).
thf(fact_1102_length__takeWhile__less__P__nth,axiom,
! [P_4: nat > $o,Xs_29: list_nat,J_2: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ J_2 )
=> ( P_4 @ ( nth_nat @ Xs_29 @ I_3 ) ) )
=> ( ( ord_less_eq_nat @ J_2 @ ( size_size_list_nat @ Xs_29 ) )
=> ( ord_less_eq_nat @ J_2 @ ( size_size_list_nat @ ( takeWhile_nat @ P_4 @ Xs_29 ) ) ) ) ) ).
thf(fact_1103_nth__take__lemma,axiom,
! [Ys_4: list_o,K_4: nat,Xs_28: list_o] :
( ( ord_less_eq_nat @ K_4 @ ( size_size_list_o @ Xs_28 ) )
=> ( ( ord_less_eq_nat @ K_4 @ ( size_size_list_o @ Ys_4 ) )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_4 )
=> ( ( nth_o @ Xs_28 @ I_3 )
<=> ( nth_o @ Ys_4 @ I_3 ) ) )
=> ( ( take_o @ K_4 @ Xs_28 )
= ( take_o @ K_4 @ Ys_4 ) ) ) ) ) ).
thf(fact_1104_nth__take__lemma,axiom,
! [Ys_4: list_A1402527603le_alt,K_4: nat,Xs_28: list_A1402527603le_alt] :
( ( ord_less_eq_nat @ K_4 @ ( size_s1146069981le_alt @ Xs_28 ) )
=> ( ( ord_less_eq_nat @ K_4 @ ( size_s1146069981le_alt @ Ys_4 ) )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_4 )
=> ( ( nth_Ar1039551983le_alt @ Xs_28 @ I_3 )
= ( nth_Ar1039551983le_alt @ Ys_4 @ I_3 ) ) )
=> ( ( take_A1527720662le_alt @ K_4 @ Xs_28 )
= ( take_A1527720662le_alt @ K_4 @ Ys_4 ) ) ) ) ) ).
thf(fact_1105_nth__take__lemma,axiom,
! [Ys_4: list_nat,K_4: nat,Xs_28: list_nat] :
( ( ord_less_eq_nat @ K_4 @ ( size_size_list_nat @ Xs_28 ) )
=> ( ( ord_less_eq_nat @ K_4 @ ( size_size_list_nat @ Ys_4 ) )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_4 )
=> ( ( nth_nat @ Xs_28 @ I_3 )
= ( nth_nat @ Ys_4 @ I_3 ) ) )
=> ( ( take_nat @ K_4 @ Xs_28 )
= ( take_nat @ K_4 @ Ys_4 ) ) ) ) ) ).
thf(fact_1106_list__size__pointwise,axiom,
! [F_3: arrow_1389031998le_alt > nat,G: arrow_1389031998le_alt > nat,Xs_27: list_A1402527603le_alt] :
( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_27 ) )
=> ( ord_less_nat @ ( F_3 @ X_4 ) @ ( G @ X_4 ) ) )
=> ( ord_less_eq_nat @ ( list_s1163968880le_alt @ F_3 @ Xs_27 ) @ ( list_s1163968880le_alt @ G @ Xs_27 ) ) ) ).
thf(fact_1107_list__size__pointwise,axiom,
! [F_3: nat > nat,G: nat > nat,Xs_27: list_nat] :
( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_27 ) )
=> ( ord_less_nat @ ( F_3 @ X_4 ) @ ( G @ X_4 ) ) )
=> ( ord_less_eq_nat @ ( list_size_nat @ F_3 @ Xs_27 ) @ ( list_size_nat @ G @ Xs_27 ) ) ) ).
thf(fact_1108_psubsetD,axiom,
! [C_2: arrow_1389031998le_alt,A_2: arrow_1389031998le_alt > $o,B_3: arrow_1389031998le_alt > $o] :
( ( ord_le1573998569_alt_o @ A_2 @ B_3 )
=> ( ( member998036369le_alt @ C_2 @ A_2 )
=> ( member998036369le_alt @ C_2 @ B_3 ) ) ) ).
thf(fact_1109_psubsetD,axiom,
! [C_2: nat,A_2: nat > $o,B_3: nat > $o] :
( ( ord_less_nat_o @ A_2 @ B_3 )
=> ( ( member_nat @ C_2 @ A_2 )
=> ( member_nat @ C_2 @ B_3 ) ) ) ).
thf(fact_1110_all__nth__imp__all__set,axiom,
! [X_5: $o,P_3: $o > $o,Xs_26: list_o] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_26 ) )
=> ( P_3 @ ( nth_o @ Xs_26 @ I_3 ) ) )
=> ( ( member_o @ X_5 @ ( set_o @ Xs_26 ) )
=> ( P_3 @ X_5 ) ) ) ).
thf(fact_1111_all__nth__imp__all__set,axiom,
! [X_5: arrow_1389031998le_alt,P_3: arrow_1389031998le_alt > $o,Xs_26: list_A1402527603le_alt] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_26 ) )
=> ( P_3 @ ( nth_Ar1039551983le_alt @ Xs_26 @ I_3 ) ) )
=> ( ( member998036369le_alt @ X_5 @ ( set_Ar1491127311le_alt @ Xs_26 ) )
=> ( P_3 @ X_5 ) ) ) ).
thf(fact_1112_all__nth__imp__all__set,axiom,
! [X_5: nat,P_3: nat > $o,Xs_26: list_nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_26 ) )
=> ( P_3 @ ( nth_nat @ Xs_26 @ I_3 ) ) )
=> ( ( member_nat @ X_5 @ ( set_nat @ Xs_26 ) )
=> ( P_3 @ X_5 ) ) ) ).
thf(fact_1113_list__ball__nth,axiom,
! [P_2: $o > $o,N_26: nat,Xs_25: list_o] :
( ( ord_less_nat @ N_26 @ ( size_size_list_o @ Xs_25 ) )
=> ( ! [X_4: $o] :
( ( member_o @ X_4 @ ( set_o @ Xs_25 ) )
=> ( P_2 @ X_4 ) )
=> ( P_2 @ ( nth_o @ Xs_25 @ N_26 ) ) ) ) ).
thf(fact_1114_list__ball__nth,axiom,
! [P_2: arrow_1389031998le_alt > $o,N_26: nat,Xs_25: list_A1402527603le_alt] :
( ( ord_less_nat @ N_26 @ ( size_s1146069981le_alt @ Xs_25 ) )
=> ( ! [X_4: arrow_1389031998le_alt] :
( ( member998036369le_alt @ X_4 @ ( set_Ar1491127311le_alt @ Xs_25 ) )
=> ( P_2 @ X_4 ) )
=> ( P_2 @ ( nth_Ar1039551983le_alt @ Xs_25 @ N_26 ) ) ) ) ).
thf(fact_1115_list__ball__nth,axiom,
! [P_2: nat > $o,N_26: nat,Xs_25: list_nat] :
( ( ord_less_nat @ N_26 @ ( size_size_list_nat @ Xs_25 ) )
=> ( ! [X_4: nat] :
( ( member_nat @ X_4 @ ( set_nat @ Xs_25 ) )
=> ( P_2 @ X_4 ) )
=> ( P_2 @ ( nth_nat @ Xs_25 @ N_26 ) ) ) ) ).
thf(fact_1116_nth__equalityI,axiom,
! [Xs_24: list_o,Ys_3: list_o] :
( ( ( size_size_list_o @ Xs_24 )
= ( size_size_list_o @ Ys_3 ) )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_o @ Xs_24 ) )
=> ( ( nth_o @ Xs_24 @ I_3 )
<=> ( nth_o @ Ys_3 @ I_3 ) ) )
=> ( Xs_24 = Ys_3 ) ) ) ).
thf(fact_1117_nth__equalityI,axiom,
! [Xs_24: list_A1402527603le_alt,Ys_3: list_A1402527603le_alt] :
( ( ( size_s1146069981le_alt @ Xs_24 )
= ( size_s1146069981le_alt @ Ys_3 ) )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_s1146069981le_alt @ Xs_24 ) )
=> ( ( nth_Ar1039551983le_alt @ Xs_24 @ I_3 )
= ( nth_Ar1039551983le_alt @ Ys_3 @ I_3 ) ) )
=> ( Xs_24 = Ys_3 ) ) ) ).
thf(fact_1118_nth__equalityI,axiom,
! [Xs_24: list_nat,Ys_3: list_nat] :
( ( ( size_size_list_nat @ Xs_24 )
= ( size_size_list_nat @ Ys_3 ) )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ ( size_size_list_nat @ Xs_24 ) )
=> ( ( nth_nat @ Xs_24 @ I_3 )
= ( nth_nat @ Ys_3 @ I_3 ) ) )
=> ( Xs_24 = Ys_3 ) ) ) ).
thf(fact_1119_Skolem__list__nth,axiom,
! [P_1: nat > $o > $o,K_3: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_3 )
=> ( ?? @ $o @ ( P_1 @ I_3 ) ) )
<=> ? [Xs_23: list_o] :
( ( ( size_size_list_o @ Xs_23 )
= K_3 )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_3 )
=> ( P_1 @ I_3 @ ( nth_o @ Xs_23 @ I_3 ) ) ) ) ) ).
thf(fact_1120_Skolem__list__nth,axiom,
! [P_1: nat > arrow_1389031998le_alt > $o,K_3: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_3 )
=> ( ?? @ arrow_1389031998le_alt @ ( P_1 @ I_3 ) ) )
<=> ? [Xs_23: list_A1402527603le_alt] :
( ( ( size_s1146069981le_alt @ Xs_23 )
= K_3 )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_3 )
=> ( P_1 @ I_3 @ ( nth_Ar1039551983le_alt @ Xs_23 @ I_3 ) ) ) ) ) ).
thf(fact_1121_Skolem__list__nth,axiom,
! [P_1: nat > nat > $o,K_3: nat] :
( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_3 )
=> ( ?? @ nat @ ( P_1 @ I_3 ) ) )
<=> ? [Xs_23: list_nat] :
( ( ( size_size_list_nat @ Xs_23 )
= K_3 )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_3 )
=> ( P_1 @ I_3 @ ( nth_nat @ Xs_23 @ I_3 ) ) ) ) ) ).
thf(fact_1122_mono__nat__linear__lb,axiom,
! [M: nat,K: nat,F: nat > nat] :
( ! [M_6: nat,N_23: nat] :
( ( ord_less_nat @ M_6 @ N_23 )
=> ( ord_less_nat @ ( F @ M_6 ) @ ( F @ N_23 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_1123_inc__induct,axiom,
! [P: nat > $o,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ J )
=> ( ( P @ ( suc @ I_3 ) )
=> ( P @ I_3 ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1124_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K_2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_2 ) ) ) ) ).
thf(fact_1125_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_2: nat] :
( ( ord_less_eq_nat @ K_2 @ N )
& ! [I_3: nat] :
( ( ord_less_nat @ I_3 @ K_2 )
=> ~ ( P @ I_3 ) )
& ( P @ K_2 ) ) ) ) ).
thf(fact_1126_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K_2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_2 )
& ( ( plus_plus_nat @ I @ K_2 )
= J ) ) ) ).
thf(fact_1127_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M_6: nat] :
( N
= ( suc @ M_6 ) ) ) ).
thf(fact_1128_lift__Suc__mono__less,axiom,
! [N_25: nat,N_24: nat,F_2: nat > nat] :
( ! [N_23: nat] : ( ord_less_nat @ ( F_2 @ N_23 ) @ ( F_2 @ ( suc @ N_23 ) ) )
=> ( ( ord_less_nat @ N_25 @ N_24 )
=> ( ord_less_nat @ ( F_2 @ N_25 ) @ ( F_2 @ N_24 ) ) ) ) ).
thf(fact_1129_lift__Suc__mono__less__iff,axiom,
! [N_22: nat,M_5: nat,F_1: nat > nat] :
( ! [N_23: nat] : ( ord_less_nat @ ( F_1 @ N_23 ) @ ( F_1 @ ( suc @ N_23 ) ) )
=> ( ( ord_less_nat @ ( F_1 @ N_22 ) @ ( F_1 @ M_5 ) )
<=> ( ord_less_nat @ N_22 @ M_5 ) ) ) ).
thf(fact_1130_pos__add__strict,axiom,
! [B_2: nat,C_1: nat,A_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_1 )
=> ( ( ord_less_nat @ B_2 @ C_1 )
=> ( ord_less_nat @ B_2 @ ( plus_plus_nat @ A_1 @ C_1 ) ) ) ) ).
thf(fact_1131_psubset__imp__ex__mem,axiom,
! [A: arrow_1389031998le_alt > $o,B: arrow_1389031998le_alt > $o] :
( ( ord_le1573998569_alt_o @ A @ B )
=> ? [B_1: arrow_1389031998le_alt] : ( member998036369le_alt @ B_1 @ ( minus_1594223470_alt_o @ B @ A ) ) ) ).
thf(fact_1132_psubset__imp__ex__mem,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_nat_o @ A @ B )
=> ? [B_1: nat] : ( member_nat @ B_1 @ ( minus_minus_nat_o @ B @ A ) ) ) ).
thf(fact_1133_less__mono__imp__le__mono,axiom,
! [I: nat,J: nat,F: nat > nat] :
( ! [I_3: nat,J_1: nat] :
( ( ord_less_nat @ I_3 @ J_1 )
=> ( ord_less_nat @ ( F @ I_3 ) @ ( F @ J_1 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
thf(fact_1134_take__hd__drop,axiom,
! [N_21: nat,Xs_22: list_A1402527603le_alt] :
( ( ord_less_nat @ N_21 @ ( size_s1146069981le_alt @ Xs_22 ) )
=> ( ( append1092755459le_alt @ ( take_A1527720662le_alt @ N_21 @ Xs_22 ) @ ( cons_A1142416030le_alt @ ( hd_Arr731872705le_alt @ ( drop_A112899118le_alt @ N_21 @ Xs_22 ) ) @ nil_Ar52383470le_alt ) )
= ( take_A1527720662le_alt @ ( suc @ N_21 ) @ Xs_22 ) ) ) ).
thf(fact_1135_take__hd__drop,axiom,
! [N_21: nat,Xs_22: list_nat] :
( ( ord_less_nat @ N_21 @ ( size_size_list_nat @ Xs_22 ) )
=> ( ( append_nat @ ( take_nat @ N_21 @ Xs_22 ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N_21 @ Xs_22 ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N_21 ) @ Xs_22 ) ) ) ).
thf(fact_1136_take__hd__drop,axiom,
! [N_21: nat,Xs_22: list_o] :
( ( ord_less_nat @ N_21 @ ( size_size_list_o @ Xs_22 ) )
=> ( ( append_o @ ( take_o @ N_21 @ Xs_22 ) @ ( cons_o @ ( hd_o @ ( drop_o @ N_21 @ Xs_22 ) ) @ nil_o ) )
= ( take_o @ ( suc @ N_21 ) @ Xs_22 ) ) ) ).
thf(fact_1137_drop__Suc__Cons,axiom,
! [N_20: nat,X_3: nat,Xs_21: list_nat] :
( ( drop_nat @ ( suc @ N_20 ) @ ( cons_nat @ X_3 @ Xs_21 ) )
= ( drop_nat @ N_20 @ Xs_21 ) ) ).
thf(fact_1138_drop__Suc__Cons,axiom,
! [N_20: nat,X_3: $o,Xs_21: list_o] :
( ( drop_o @ ( suc @ N_20 ) @ ( cons_o @ X_3 @ Xs_21 ) )
= ( drop_o @ N_20 @ Xs_21 ) ) ).
thf(fact_1139_drop__Suc__Cons,axiom,
! [N_20: nat,X_3: arrow_1389031998le_alt,Xs_21: list_A1402527603le_alt] :
( ( drop_A112899118le_alt @ ( suc @ N_20 ) @ ( cons_A1142416030le_alt @ X_3 @ Xs_21 ) )
= ( drop_A112899118le_alt @ N_20 @ Xs_21 ) ) ).
thf(fact_1140_set__drop__subset,axiom,
! [N_19: nat,Xs_20: list_A1402527603le_alt] : ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ ( drop_A112899118le_alt @ N_19 @ Xs_20 ) ) @ ( set_Ar1491127311le_alt @ Xs_20 ) ) ).
thf(fact_1141_set__drop__subset,axiom,
! [N_19: nat,Xs_20: list_nat] : ( ord_less_eq_nat_o @ ( set_nat @ ( drop_nat @ N_19 @ Xs_20 ) ) @ ( set_nat @ Xs_20 ) ) ).
thf(fact_1142_length__drop,axiom,
! [N_18: nat,Xs_19: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N_18 @ Xs_19 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs_19 ) @ N_18 ) ) ).
thf(fact_1143_nth__via__drop,axiom,
! [N_17: nat,Xs_18: list_o,Y_1: $o,Ys_2: list_o] :
( ( ( drop_o @ N_17 @ Xs_18 )
= ( cons_o @ Y_1 @ Ys_2 ) )
=> ( ( nth_o @ Xs_18 @ N_17 )
<=> Y_1 ) ) ).
thf(fact_1144_nth__via__drop,axiom,
! [N_17: nat,Xs_18: list_A1402527603le_alt,Y_1: arrow_1389031998le_alt,Ys_2: list_A1402527603le_alt] :
( ( ( drop_A112899118le_alt @ N_17 @ Xs_18 )
= ( cons_A1142416030le_alt @ Y_1 @ Ys_2 ) )
=> ( ( nth_Ar1039551983le_alt @ Xs_18 @ N_17 )
= Y_1 ) ) ).
thf(fact_1145_nth__via__drop,axiom,
! [N_17: nat,Xs_18: list_nat,Y_1: nat,Ys_2: list_nat] :
( ( ( drop_nat @ N_17 @ Xs_18 )
= ( cons_nat @ Y_1 @ Ys_2 ) )
=> ( ( nth_nat @ Xs_18 @ N_17 )
= Y_1 ) ) ).
thf(fact_1146_take__drop,axiom,
! [N_16: nat,M_4: nat,Xs_17: list_nat] :
( ( take_nat @ N_16 @ ( drop_nat @ M_4 @ Xs_17 ) )
= ( drop_nat @ M_4 @ ( take_nat @ ( plus_plus_nat @ N_16 @ M_4 ) @ Xs_17 ) ) ) ).
thf(fact_1147_take__drop,axiom,
! [N_16: nat,M_4: nat,Xs_17: list_A1402527603le_alt] :
( ( take_A1527720662le_alt @ N_16 @ ( drop_A112899118le_alt @ M_4 @ Xs_17 ) )
= ( drop_A112899118le_alt @ M_4 @ ( take_A1527720662le_alt @ ( plus_plus_nat @ N_16 @ M_4 ) @ Xs_17 ) ) ) ).
thf(fact_1148_drop__take,axiom,
! [N_15: nat,M_3: nat,Xs_16: list_nat] :
( ( drop_nat @ N_15 @ ( take_nat @ M_3 @ Xs_16 ) )
= ( take_nat @ ( minus_minus_nat @ M_3 @ N_15 ) @ ( drop_nat @ N_15 @ Xs_16 ) ) ) ).
thf(fact_1149_drop__take,axiom,
! [N_15: nat,M_3: nat,Xs_16: list_A1402527603le_alt] :
( ( drop_A112899118le_alt @ N_15 @ ( take_A1527720662le_alt @ M_3 @ Xs_16 ) )
= ( take_A1527720662le_alt @ ( minus_minus_nat @ M_3 @ N_15 ) @ ( drop_A112899118le_alt @ N_15 @ Xs_16 ) ) ) ).
thf(fact_1150_append__take__drop__id,axiom,
! [N_14: nat,Xs_15: list_nat] :
( ( append_nat @ ( take_nat @ N_14 @ Xs_15 ) @ ( drop_nat @ N_14 @ Xs_15 ) )
= Xs_15 ) ).
thf(fact_1151_append__take__drop__id,axiom,
! [N_14: nat,Xs_15: list_A1402527603le_alt] :
( ( append1092755459le_alt @ ( take_A1527720662le_alt @ N_14 @ Xs_15 ) @ ( drop_A112899118le_alt @ N_14 @ Xs_15 ) )
= Xs_15 ) ).
thf(fact_1152_drop__replicate,axiom,
! [I_2: nat,K_1: nat,X_2: nat] :
( ( drop_nat @ I_2 @ ( replicate_nat @ K_1 @ X_2 ) )
= ( replicate_nat @ ( minus_minus_nat @ K_1 @ I_2 ) @ X_2 ) ) ).
thf(fact_1153_drop__replicate,axiom,
! [I_2: nat,K_1: nat,X_2: arrow_1389031998le_alt] :
( ( drop_A112899118le_alt @ I_2 @ ( replic277728168le_alt @ K_1 @ X_2 ) )
= ( replic277728168le_alt @ ( minus_minus_nat @ K_1 @ I_2 ) @ X_2 ) ) ).
thf(fact_1154_drop__Suc,axiom,
! [N_13: nat,Xs_14: list_nat] :
( ( drop_nat @ ( suc @ N_13 ) @ Xs_14 )
= ( drop_nat @ N_13 @ ( tl_nat @ Xs_14 ) ) ) ).
thf(fact_1155_drop__Suc,axiom,
! [N_13: nat,Xs_14: list_A1402527603le_alt] :
( ( drop_A112899118le_alt @ ( suc @ N_13 ) @ Xs_14 )
= ( drop_A112899118le_alt @ N_13 @ ( tl_Arr1379124165le_alt @ Xs_14 ) ) ) ).
thf(fact_1156_drop__butlast,axiom,
! [N_12: nat,Xs_13: list_nat] :
( ( drop_nat @ N_12 @ ( butlast_nat @ Xs_13 ) )
= ( butlast_nat @ ( drop_nat @ N_12 @ Xs_13 ) ) ) ).
thf(fact_1157_drop__butlast,axiom,
! [N_12: nat,Xs_13: list_A1402527603le_alt] :
( ( drop_A112899118le_alt @ N_12 @ ( butlas1188620858le_alt @ Xs_13 ) )
= ( butlas1188620858le_alt @ ( drop_A112899118le_alt @ N_12 @ Xs_13 ) ) ) ).
thf(fact_1158_butlast__drop,axiom,
! [N_11: nat,Xs_12: list_nat] :
( ( butlast_nat @ ( drop_nat @ N_11 @ Xs_12 ) )
= ( drop_nat @ N_11 @ ( butlast_nat @ Xs_12 ) ) ) ).
thf(fact_1159_butlast__drop,axiom,
! [N_11: nat,Xs_12: list_A1402527603le_alt] :
( ( butlas1188620858le_alt @ ( drop_A112899118le_alt @ N_11 @ Xs_12 ) )
= ( drop_A112899118le_alt @ N_11 @ ( butlas1188620858le_alt @ Xs_12 ) ) ) ).
thf(fact_1160_distinct__drop,axiom,
! [I_1: nat,Xs_11: list_nat] :
( ( distinct_nat @ Xs_11 )
=> ( distinct_nat @ ( drop_nat @ I_1 @ Xs_11 ) ) ) ).
thf(fact_1161_distinct__drop,axiom,
! [I_1: nat,Xs_11: list_A1402527603le_alt] :
( ( distin1149997281le_alt @ Xs_11 )
=> ( distin1149997281le_alt @ ( drop_A112899118le_alt @ I_1 @ Xs_11 ) ) ) ).
thf(fact_1162_sorted__drop,axiom,
! [N_10: nat,Xs_10: list_nat] :
( ( linorder_sorted_nat @ Xs_10 )
=> ( linorder_sorted_nat @ ( drop_nat @ N_10 @ Xs_10 ) ) ) ).
thf(fact_1163_drop__Nil,axiom,
! [N_9: nat] :
( ( drop_nat @ N_9 @ nil_nat )
= nil_nat ) ).
thf(fact_1164_drop__Nil,axiom,
! [N_9: nat] :
( ( drop_A112899118le_alt @ N_9 @ nil_Ar52383470le_alt )
= nil_Ar52383470le_alt ) ).
thf(fact_1165_in__set__dropD,axiom,
! [X_1: arrow_1389031998le_alt,N_8: nat,Xs_9: list_A1402527603le_alt] :
( ( member998036369le_alt @ X_1 @ ( set_Ar1491127311le_alt @ ( drop_A112899118le_alt @ N_8 @ Xs_9 ) ) )
=> ( member998036369le_alt @ X_1 @ ( set_Ar1491127311le_alt @ Xs_9 ) ) ) ).
thf(fact_1166_in__set__dropD,axiom,
! [X_1: nat,N_8: nat,Xs_9: list_nat] :
( ( member_nat @ X_1 @ ( set_nat @ ( drop_nat @ N_8 @ Xs_9 ) ) )
=> ( member_nat @ X_1 @ ( set_nat @ Xs_9 ) ) ) ).
thf(fact_1167_drop__drop,axiom,
! [N_7: nat,M_2: nat,Xs_8: list_nat] :
( ( drop_nat @ N_7 @ ( drop_nat @ M_2 @ Xs_8 ) )
= ( drop_nat @ ( plus_plus_nat @ N_7 @ M_2 ) @ Xs_8 ) ) ).
thf(fact_1168_drop__0,axiom,
! [Xs_7: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs_7 )
= Xs_7 ) ).
thf(fact_1169_tl__drop,axiom,
! [N_6: nat,Xs_6: list_nat] :
( ( tl_nat @ ( drop_nat @ N_6 @ Xs_6 ) )
= ( drop_nat @ N_6 @ ( tl_nat @ Xs_6 ) ) ) ).
thf(fact_1170_tl__drop,axiom,
! [N_6: nat,Xs_6: list_A1402527603le_alt] :
( ( tl_Arr1379124165le_alt @ ( drop_A112899118le_alt @ N_6 @ Xs_6 ) )
= ( drop_A112899118le_alt @ N_6 @ ( tl_Arr1379124165le_alt @ Xs_6 ) ) ) ).
thf(fact_1171_drop__tl,axiom,
! [N_5: nat,Xs_5: list_nat] :
( ( drop_nat @ N_5 @ ( tl_nat @ Xs_5 ) )
= ( tl_nat @ ( drop_nat @ N_5 @ Xs_5 ) ) ) ).
thf(fact_1172_drop__tl,axiom,
! [N_5: nat,Xs_5: list_A1402527603le_alt] :
( ( drop_A112899118le_alt @ N_5 @ ( tl_Arr1379124165le_alt @ Xs_5 ) )
= ( tl_Arr1379124165le_alt @ ( drop_A112899118le_alt @ N_5 @ Xs_5 ) ) ) ).
thf(fact_1173_set__drop__subset__set__drop,axiom,
! [Xs_4: list_A1402527603le_alt,N_4: nat,M_1: nat] :
( ( ord_less_eq_nat @ N_4 @ M_1 )
=> ( ord_le1423491573_alt_o @ ( set_Ar1491127311le_alt @ ( drop_A112899118le_alt @ M_1 @ Xs_4 ) ) @ ( set_Ar1491127311le_alt @ ( drop_A112899118le_alt @ N_4 @ Xs_4 ) ) ) ) ).
thf(fact_1174_set__drop__subset__set__drop,axiom,
! [Xs_4: list_nat,N_4: nat,M_1: nat] :
( ( ord_less_eq_nat @ N_4 @ M_1 )
=> ( ord_less_eq_nat_o @ ( set_nat @ ( drop_nat @ M_1 @ Xs_4 ) ) @ ( set_nat @ ( drop_nat @ N_4 @ Xs_4 ) ) ) ) ).
thf(fact_1175_drop__eq__Nil,axiom,
! [N_3: nat,Xs_3: list_nat] :
( ( ( drop_nat @ N_3 @ Xs_3 )
= nil_nat )
<=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_3 ) @ N_3 ) ) ).
thf(fact_1176_drop__eq__Nil,axiom,
! [N_3: nat,Xs_3: list_A1402527603le_alt] :
( ( ( drop_A112899118le_alt @ N_3 @ Xs_3 )
= nil_Ar52383470le_alt )
<=> ( ord_less_eq_nat @ ( size_s1146069981le_alt @ Xs_3 ) @ N_3 ) ) ).
thf(fact_1177_drop__all,axiom,
! [Xs_2: list_nat,N_2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_2 ) @ N_2 )
=> ( ( drop_nat @ N_2 @ Xs_2 )
= nil_nat ) ) ).
thf(fact_1178_drop__all,axiom,
! [Xs_2: list_A1402527603le_alt,N_2: nat] :
( ( ord_less_eq_nat @ ( size_s1146069981le_alt @ Xs_2 ) @ N_2 )
=> ( ( drop_A112899118le_alt @ N_2 @ Xs_2 )
= nil_Ar52383470le_alt ) ) ).
thf(fact_1179_drop__append,axiom,
! [N_1: nat,Xs_1: list_nat,Ys_1: list_nat] :
( ( drop_nat @ N_1 @ ( append_nat @ Xs_1 @ Ys_1 ) )
= ( append_nat @ ( drop_nat @ N_1 @ Xs_1 ) @ ( drop_nat @ ( minus_minus_nat @ N_1 @ ( size_size_list_nat @ Xs_1 ) ) @ Ys_1 ) ) ) ).
thf(fact_1180_drop__append,axiom,
! [N_1: nat,Xs_1: list_A1402527603le_alt,Ys_1: list_A1402527603le_alt] :
( ( drop_A112899118le_alt @ N_1 @ ( append1092755459le_alt @ Xs_1 @ Ys_1 ) )
= ( append1092755459le_alt @ ( drop_A112899118le_alt @ N_1 @ Xs_1 ) @ ( drop_A112899118le_alt @ ( minus_minus_nat @ N_1 @ ( size_s1146069981le_alt @ Xs_1 ) ) @ Ys_1 ) ) ) ).
thf(fact_1181_append__eq__conv__conj,axiom,
! [Xs: list_A1402527603le_alt,Ys: list_A1402527603le_alt,Zs: list_A1402527603le_alt] :
( ( ( append1092755459le_alt @ Xs @ Ys )
= Zs )
<=> ( ( Xs
= ( take_A1527720662le_alt @ ( size_s1146069981le_alt @ Xs ) @ Zs ) )
& ( Ys
= ( drop_A112899118le_alt @ ( size_s1146069981le_alt @ Xs ) @ Zs ) ) ) ) ).
thf(fact_1182_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ) ).
thf(fact_1183_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ).
thf(fact_1184_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_1185_sorted__upt,axiom,
! [I: nat,J: nat] : ( linorder_sorted_nat @ ( upt @ I @ J ) ) ).
thf(fact_1186_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
thf(fact_1187_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
thf(fact_1188_upt__rec,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) )
& ( ~ ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= nil_nat ) ) ) ).
thf(fact_1189_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
thf(fact_1190_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
thf(fact_1191_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
thf(fact_1192_distinct__upt,axiom,
! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
thf(fact_1193_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
thf(fact_1194_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
thf(fact_1195_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
thf(fact_1196_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
<=> ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
thf(fact_1197_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
thf(fact_1198_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
thf(fact_1199_upt__add__eq__append,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
%----Helper facts (13)
thf(help_If_1_1_If_000tc__List__Olist_I_Eo_J_T,axiom,
! [X: list_o,Y: list_o] :
( ( if_list_o @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__List__Olist_I_Eo_J_T,axiom,
! [X: list_o,Y: list_o] :
( ( if_list_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__List__Olist_I_Eo_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_fequal_1_1_fequal_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ~ ( fequal_list_nat @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( X != Y )
| ( fequal_list_nat @ X @ Y ) ) ).
thf(help_If_1_1_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh,axiom,
! [X: list_A1402527603le_alt,Y: list_A1402527603le_alt] :
( ( if_lis613229059le_alt @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh,axiom,
! [X: list_A1402527603le_alt,Y: list_A1402527603le_alt] :
( ( if_lis613229059le_alt @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____piyeccggrh,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_fequal_1_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____pi,axiom,
! [X: list_A1402527603le_alt,Y: list_A1402527603le_alt] :
( ~ ( fequal68576820le_alt @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____pi,axiom,
! [X: list_A1402527603le_alt,Y: list_A1402527603le_alt] :
( ( X != Y )
| ( fequal68576820le_alt @ X @ Y ) ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
a != b ).
thf(conj_1,conjecture,
? [C: arrow_1389031998le_alt] : ( distin1149997281le_alt @ ( cons_A1142416030le_alt @ a @ ( cons_A1142416030le_alt @ b @ ( cons_A1142416030le_alt @ C @ nil_Ar52383470le_alt ) ) ) ) ).
%------------------------------------------------------------------------------