TPTP Problem File: SCT169^2.p
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%------------------------------------------------------------------------------
% File : SCT169^2 : TPTP v8.2.0. Released v5.3.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 29, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : arrow_500_thf_l29 [Bla11]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.92 v8.1.0, 0.91 v7.5.0, 1.00 v5.3.0
% Syntax : Number of formulae : 813 ( 283 unt; 109 typ; 0 def)
% Number of atoms : 1376 ( 806 equ; 2 cnn)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 4739 ( 229 ~; 31 |; 88 &;3800 @)
% ( 161 <=>; 430 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 358 ( 358 >; 0 *; 0 +; 0 <<)
% Number of symbols : 105 ( 101 usr; 14 con; 0-3 aty)
% Number of variables : 1547 ( 2 ^;1492 !; 53 ?;1547 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:13:13
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
thf(ty_ty_tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
arrow_475358991le_alt: $tType ).
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__List__Olist_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_J,type,
list_A2115238852le_alt: $tType ).
thf(ty_ty_tc__List__Olist_Itc__Int__Oint_J,type,
list_int: $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_Itc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_J,type,
option514678660le_alt: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Nat__Onat_J,type,
option_nat: $tType ).
%----Explicit typings (101)
thf(sy_c_Ex1,type,
ex1: ( nat > $o ) > $o ).
thf(sy_c_Ex2,type,
ex2: ( arrow_475358991le_alt > $o ) > $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__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > int > int ).
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__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Oequal__class_Oequal_000tc__List__Olist_Itc__Arrow____Order____Mirabell,type,
equal_484611810le_alt: list_A2115238852le_alt > list_A2115238852le_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_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
thf(sy_c_Int_Oring__1__class_OInts_000tc__Int__Oint,type,
ring_1_Ints_int: int > $o ).
thf(sy_c_List_OListMem_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
listMe78047147le_alt: arrow_475358991le_alt > list_A2115238852le_alt > $o ).
thf(sy_c_List_Oappend_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
append179082452le_alt: list_A2115238852le_alt > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Oappend_000tc__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obutlast_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
butlas274947851le_alt: list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Obutlast_000tc__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Odistinct_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
distin236324274le_alt: list_A2115238852le_alt > $o ).
thf(sy_c_List_Odistinct_000tc__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_OdropWhile_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
dropWh1316781920le_alt: ( arrow_475358991le_alt > $o ) > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_OdropWhile_000tc__Nat__Onat,type,
dropWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Odrop_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
drop_A1346709759le_alt: nat > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Odrop_000tc__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_000tc__Int__Oint_000tc__Int__Oint,type,
foldr_int_int: ( int > int > int ) > list_int > int > int ).
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_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
hd_Arr1965683346le_alt: list_A2115238852le_alt > arrow_475358991le_alt ).
thf(sy_c_List_Ohd_000tc__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Oinsert_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
insert2120566741le_alt: arrow_475358991le_alt > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Oinsert_000tc__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olast_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
last_A1217315288le_alt: list_A2115238852le_alt > arrow_475358991le_alt ).
thf(sy_c_List_Olast_000tc__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted_000tc__Int__Oint,type,
linorder_sorted_int: list_int > $o ).
thf(sy_c_List_Olinorder__class_Osorted_000tc__Nat__Onat,type,
linorder_sorted_nat: list_nat > $o ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_000tc__Nat__Onat,type,
linord810813188et_nat: ( nat > $o ) > list_nat ).
thf(sy_c_List_Olist_OCons_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
cons_A228743023le_alt: arrow_475358991le_alt > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Olist_OCons_000tc__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
nil_Ar1286194111le_alt: list_A2115238852le_alt ).
thf(sy_c_List_Olist_ONil_000tc__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_000tc__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Olist__size_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oa,type,
list_s250295873le_alt: ( arrow_475358991le_alt > nat ) > list_A2115238852le_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__ex1_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
list_e239244930le_alt: ( arrow_475358991le_alt > $o ) > list_A2115238852le_alt > $o ).
thf(sy_c_List_Olist__ex1_000tc__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
list_e1252802557le_alt: ( arrow_475358991le_alt > $o ) > list_A2115238852le_alt > $o ).
thf(sy_c_List_Olist__ex_000tc__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__update_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
list_u62477863le_alt: list_A2115238852le_alt > nat > arrow_475358991le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Olist__update_000tc__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olistrelp_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
listre1213162009le_alt: ( arrow_475358991le_alt > arrow_475358991le_alt > $o ) > list_A2115238852le_alt > list_A2115238852le_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__filter_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_00,type,
map_fi312151301lt_nat: ( arrow_475358991le_alt > option_nat ) > list_A2115238852le_alt > list_nat ).
thf(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____lc,type,
map_fi1462697295le_alt: ( nat > option514678660le_alt ) > list_nat > list_A2115238852le_alt ).
thf(sy_c_List_Omaps_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_000tc__Ar,type,
maps_A409156225le_alt: ( arrow_475358991le_alt > list_A2115238852le_alt ) > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Omaps_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt_000tc__Na,type,
maps_A2124894076lt_nat: ( arrow_475358991le_alt > list_nat ) > list_A2115238852le_alt > list_nat ).
thf(sy_c_List_Omaps_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____lcilvlkkz,type,
maps_n1127956422le_alt: ( nat > list_A2115238852le_alt ) > list_nat > list_A2115238852le_alt ).
thf(sy_c_List_Omember_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
member1504968148le_alt: list_A2115238852le_alt > arrow_475358991le_alt > $o ).
thf(sy_c_List_Omember_000tc__Nat__Onat,type,
member_nat_1: list_nat > nat > $o ).
thf(sy_c_List_Onth_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
nth_Ar125878976le_alt: list_A2115238852le_alt > nat > arrow_475358991le_alt ).
thf(sy_c_List_Onth_000tc__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_000tc__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onull_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
null_A1520965063le_alt: list_A2115238852le_alt > $o ).
thf(sy_c_List_Onull_000tc__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Oreplicate_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
replic1511538809le_alt: nat > arrow_475358991le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Oreplicate_000tc__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Orev_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
rev_Ar1106406943le_alt: list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Orev_000tc__Int__Oint,type,
rev_int: list_int > list_int ).
thf(sy_c_List_Orev_000tc__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
rotate335349260le_alt: list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Orotate1_000tc__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Osplice_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
splice1520898450le_alt: list_A2115238852le_alt > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Osplice_000tc__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Osublist_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
sublis2047953604le_alt: list_A2115238852le_alt > ( nat > $o ) > list_A2115238852le_alt ).
thf(sy_c_List_Osublist_000tc__Nat__Onat,type,
sublist_nat: list_nat > ( nat > $o ) > list_nat ).
thf(sy_c_List_OtakeWhile_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
takeWh1696291512le_alt: ( arrow_475358991le_alt > $o ) > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_OtakeWhile_000tc__Nat__Onat,type,
takeWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Otake_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
take_A614047655le_alt: nat > list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Otl_000tc__Arrow____Order____Mirabelle____lcilvlkkzv__Oalt,type,
tl_Arr465451158le_alt: list_A2115238852le_alt > list_A2115238852le_alt ).
thf(sy_c_List_Otl_000tc__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Arrow____Order____Mirabelle_,type,
size_s1858781230le_alt: list_A2115238852le_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_Orderings_Obot__class_Obot_000_062_Itc__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_000tc__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $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_000_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_000tc__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
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____lcilvlkkzv__O,type,
fequal781288069le_alt: list_A2115238852le_alt > list_A2115238852le_alt > $o ).
thf(sy_c_member_000tc__Int__Oint,type,
member_int: int > ( int > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_v_a,type,
a: arrow_475358991le_alt ).
thf(sy_v_b,type,
b: arrow_475358991le_alt ).
%----Relevant facts (700)
thf(fact_0_alt3,axiom,
? [A_90: arrow_475358991le_alt,B_48: arrow_475358991le_alt,C: arrow_475358991le_alt] : ( distin236324274le_alt @ ( cons_A228743023le_alt @ A_90 @ ( cons_A228743023le_alt @ B_48 @ ( cons_A228743023le_alt @ C @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_1_distinct_Osimps_I1_J,axiom,
distin236324274le_alt @ nil_Ar1286194111le_alt ).
thf(fact_2_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
thf(fact_3_list_Osimps_I2_J,axiom,
! [A_92: arrow_475358991le_alt,List_6: list_A2115238852le_alt] :
( nil_Ar1286194111le_alt
!= ( cons_A228743023le_alt @ A_92 @ List_6 ) ) ).
thf(fact_4_list_Osimps_I2_J,axiom,
! [A_92: nat,List_6: list_nat] :
( nil_nat
!= ( cons_nat @ A_92 @ List_6 ) ) ).
thf(fact_5_list_Osimps_I3_J,axiom,
! [A_91: arrow_475358991le_alt,List_5: list_A2115238852le_alt] :
( ( cons_A228743023le_alt @ A_91 @ List_5 )
!= nil_Ar1286194111le_alt ) ).
thf(fact_6_list_Osimps_I3_J,axiom,
! [A_91: nat,List_5: list_nat] :
( ( cons_nat @ A_91 @ List_5 )
!= nil_nat ) ).
thf(fact_7_neq__Nil__conv,axiom,
! [Xs_183: list_A2115238852le_alt] :
( ( Xs_183 != nil_Ar1286194111le_alt )
<=> ? [Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] :
( Xs_183
= ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) ) ) ).
thf(fact_8_neq__Nil__conv,axiom,
! [Xs_183: list_nat] :
( ( Xs_183 != nil_nat )
<=> ? [Y_3: nat,Ys_5: list_nat] :
( Xs_183
= ( cons_nat @ Y_3 @ Ys_5 ) ) ) ).
thf(fact_9_list_Oexhaust,axiom,
! [Y_44: list_A2115238852le_alt] :
( ( Y_44 != nil_Ar1286194111le_alt )
=> ~ ! [A_90: arrow_475358991le_alt,List_4: list_A2115238852le_alt] :
( Y_44
!= ( cons_A228743023le_alt @ A_90 @ List_4 ) ) ) ).
thf(fact_10_list_Oexhaust,axiom,
! [Y_44: list_nat] :
( ( Y_44 != nil_nat )
=> ~ ! [A_90: nat,List_4: list_nat] :
( Y_44
!= ( cons_nat @ A_90 @ List_4 ) ) ) ).
thf(fact_11_not__Cons__self,axiom,
! [Xs_182: list_A2115238852le_alt,X_115: arrow_475358991le_alt] :
( Xs_182
!= ( cons_A228743023le_alt @ X_115 @ Xs_182 ) ) ).
thf(fact_12_not__Cons__self2,axiom,
! [X_114: arrow_475358991le_alt,Xs_181: list_A2115238852le_alt] :
( ( cons_A228743023le_alt @ X_114 @ Xs_181 )
!= Xs_181 ) ).
thf(fact_13_list_Oinject,axiom,
! [A_89: arrow_475358991le_alt,List_3: list_A2115238852le_alt,A_88: arrow_475358991le_alt,List_2: list_A2115238852le_alt] :
( ( ( cons_A228743023le_alt @ A_89 @ List_3 )
= ( cons_A228743023le_alt @ A_88 @ List_2 ) )
<=> ( ( A_89 = A_88 )
& ( List_3 = List_2 ) ) ) ).
thf(fact_14_splice_Osimps_I2_J,axiom,
! [V_4: arrow_475358991le_alt,Va: list_A2115238852le_alt] :
( ( splice1520898450le_alt @ ( cons_A228743023le_alt @ V_4 @ Va ) @ nil_Ar1286194111le_alt )
= ( cons_A228743023le_alt @ V_4 @ Va ) ) ).
thf(fact_15_splice_Osimps_I2_J,axiom,
! [V_4: nat,Va: list_nat] :
( ( splice_nat @ ( cons_nat @ V_4 @ Va ) @ nil_nat )
= ( cons_nat @ V_4 @ Va ) ) ).
thf(fact_16_insert__Nil,axiom,
! [X_113: arrow_475358991le_alt] :
( ( insert2120566741le_alt @ X_113 @ nil_Ar1286194111le_alt )
= ( cons_A228743023le_alt @ X_113 @ nil_Ar1286194111le_alt ) ) ).
thf(fact_17_insert__Nil,axiom,
! [X_113: nat] :
( ( insert_nat @ X_113 @ nil_nat )
= ( cons_nat @ X_113 @ nil_nat ) ) ).
thf(fact_18_list__nonempty__induct,axiom,
! [P_33: list_A2115238852le_alt > $o,Xs_180: list_A2115238852le_alt] :
( ( Xs_180 != nil_Ar1286194111le_alt )
=> ( ! [X_1: arrow_475358991le_alt] : ( P_33 @ ( cons_A228743023le_alt @ X_1 @ nil_Ar1286194111le_alt ) )
=> ( ! [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt] :
( ( Xs_54 != nil_Ar1286194111le_alt )
=> ( ( P_33 @ Xs_54 )
=> ( P_33 @ ( cons_A228743023le_alt @ X_1 @ Xs_54 ) ) ) )
=> ( P_33 @ Xs_180 ) ) ) ) ).
thf(fact_19_list__nonempty__induct,axiom,
! [P_33: list_nat > $o,Xs_180: list_nat] :
( ( Xs_180 != nil_nat )
=> ( ! [X_1: nat] : ( P_33 @ ( cons_nat @ X_1 @ nil_nat ) )
=> ( ! [X_1: nat,Xs_54: list_nat] :
( ( Xs_54 != nil_nat )
=> ( ( P_33 @ Xs_54 )
=> ( P_33 @ ( cons_nat @ X_1 @ Xs_54 ) ) ) )
=> ( P_33 @ Xs_180 ) ) ) ) ).
thf(fact_20_distinct__butlast,axiom,
! [Xs_179: list_A2115238852le_alt] :
( ( Xs_179 != nil_Ar1286194111le_alt )
=> ( ( distin236324274le_alt @ Xs_179 )
=> ( distin236324274le_alt @ ( butlas274947851le_alt @ Xs_179 ) ) ) ) ).
thf(fact_21_distinct__butlast,axiom,
! [Xs_179: list_nat] :
( ( Xs_179 != nil_nat )
=> ( ( distinct_nat @ Xs_179 )
=> ( distinct_nat @ ( butlast_nat @ Xs_179 ) ) ) ) ).
thf(fact_22_butlast_Osimps_I1_J,axiom,
( ( butlas274947851le_alt @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_23_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
thf(fact_24_distinct__insert,axiom,
! [X_112: arrow_475358991le_alt,Xs_178: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_178 )
=> ( distin236324274le_alt @ ( insert2120566741le_alt @ X_112 @ Xs_178 ) ) ) ).
thf(fact_25_splice_Osimps_I3_J,axiom,
! [X_111: arrow_475358991le_alt,Xs_177: list_A2115238852le_alt,Y_43: arrow_475358991le_alt,Ys_54: list_A2115238852le_alt] :
( ( splice1520898450le_alt @ ( cons_A228743023le_alt @ X_111 @ Xs_177 ) @ ( cons_A228743023le_alt @ Y_43 @ Ys_54 ) )
= ( cons_A228743023le_alt @ X_111 @ ( cons_A228743023le_alt @ Y_43 @ ( splice1520898450le_alt @ Xs_177 @ Ys_54 ) ) ) ) ).
thf(fact_26_splice__Nil2,axiom,
! [Xs_176: list_A2115238852le_alt] :
( ( splice1520898450le_alt @ Xs_176 @ nil_Ar1286194111le_alt )
= Xs_176 ) ).
thf(fact_27_splice__Nil2,axiom,
! [Xs_176: list_nat] :
( ( splice_nat @ Xs_176 @ nil_nat )
= Xs_176 ) ).
thf(fact_28_splice_Osimps_I1_J,axiom,
! [Ys_53: list_A2115238852le_alt] :
( ( splice1520898450le_alt @ nil_Ar1286194111le_alt @ Ys_53 )
= Ys_53 ) ).
thf(fact_29_splice_Osimps_I1_J,axiom,
! [Ys_53: list_nat] :
( ( splice_nat @ nil_nat @ Ys_53 )
= Ys_53 ) ).
thf(fact_30_butlast_Osimps_I2_J,axiom,
! [X_110: arrow_475358991le_alt,Xs_175: list_A2115238852le_alt] :
( ( ( Xs_175 = nil_Ar1286194111le_alt )
=> ( ( butlas274947851le_alt @ ( cons_A228743023le_alt @ X_110 @ Xs_175 ) )
= nil_Ar1286194111le_alt ) )
& ( ( Xs_175 != nil_Ar1286194111le_alt )
=> ( ( butlas274947851le_alt @ ( cons_A228743023le_alt @ X_110 @ Xs_175 ) )
= ( cons_A228743023le_alt @ X_110 @ ( butlas274947851le_alt @ Xs_175 ) ) ) ) ) ).
thf(fact_31_butlast_Osimps_I2_J,axiom,
! [X_110: nat,Xs_175: list_nat] :
( ( ( Xs_175 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X_110 @ Xs_175 ) )
= nil_nat ) )
& ( ( Xs_175 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X_110 @ Xs_175 ) )
= ( cons_nat @ X_110 @ ( butlast_nat @ Xs_175 ) ) ) ) ) ).
thf(fact_32_eq__Nil__null,axiom,
! [Xs_174: list_A2115238852le_alt] :
( ( Xs_174 = nil_Ar1286194111le_alt )
<=> ( null_A1520965063le_alt @ Xs_174 ) ) ).
thf(fact_33_eq__Nil__null,axiom,
! [Xs_174: list_nat] :
( ( Xs_174 = nil_nat )
<=> ( null_nat @ Xs_174 ) ) ).
thf(fact_34_List_Onull__def,axiom,
! [Xs_173: list_A2115238852le_alt] :
( ( null_A1520965063le_alt @ Xs_173 )
<=> ( Xs_173 = nil_Ar1286194111le_alt ) ) ).
thf(fact_35_List_Onull__def,axiom,
! [Xs_173: list_nat] :
( ( null_nat @ Xs_173 )
<=> ( Xs_173 = nil_nat ) ) ).
thf(fact_36_null__rec_I2_J,axiom,
null_A1520965063le_alt @ nil_Ar1286194111le_alt ).
thf(fact_37_null__rec_I2_J,axiom,
null_nat @ nil_nat ).
thf(fact_38_null__rec_I1_J,axiom,
! [X_109: arrow_475358991le_alt,Xs_172: list_A2115238852le_alt] :
~ ( null_A1520965063le_alt @ ( cons_A228743023le_alt @ X_109 @ Xs_172 ) ) ).
thf(fact_39_butlast__snoc,axiom,
! [Xs_171: list_A2115238852le_alt,X_108: arrow_475358991le_alt] :
( ( butlas274947851le_alt @ ( append179082452le_alt @ Xs_171 @ ( cons_A228743023le_alt @ X_108 @ nil_Ar1286194111le_alt ) ) )
= Xs_171 ) ).
thf(fact_40_butlast__snoc,axiom,
! [Xs_171: list_nat,X_108: nat] :
( ( butlast_nat @ ( append_nat @ Xs_171 @ ( cons_nat @ X_108 @ nil_nat ) ) )
= Xs_171 ) ).
thf(fact_41_maps__simps_I2_J,axiom,
! [F_7: nat > list_A2115238852le_alt] :
( ( maps_n1127956422le_alt @ F_7 @ nil_nat )
= nil_Ar1286194111le_alt ) ).
thf(fact_42_maps__simps_I2_J,axiom,
! [F_7: arrow_475358991le_alt > list_nat] :
( ( maps_A2124894076lt_nat @ F_7 @ nil_Ar1286194111le_alt )
= nil_nat ) ).
thf(fact_43_maps__simps_I2_J,axiom,
! [F_7: arrow_475358991le_alt > list_A2115238852le_alt] :
( ( maps_A409156225le_alt @ F_7 @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_44_sorted__single,axiom,
! [X_107: nat] : ( linorder_sorted_nat @ ( cons_nat @ X_107 @ nil_nat ) ) ).
thf(fact_45_sorted_ONil,axiom,
linorder_sorted_nat @ nil_nat ).
thf(fact_46_append__assoc,axiom,
! [Xs_170: list_A2115238852le_alt,Ys_52: list_A2115238852le_alt,Zs_9: list_A2115238852le_alt] :
( ( append179082452le_alt @ ( append179082452le_alt @ Xs_170 @ Ys_52 ) @ Zs_9 )
= ( append179082452le_alt @ Xs_170 @ ( append179082452le_alt @ Ys_52 @ Zs_9 ) ) ) ).
thf(fact_47_append__eq__append__conv2,axiom,
! [Xs_169: list_A2115238852le_alt,Ys_51: list_A2115238852le_alt,Zs_8: list_A2115238852le_alt,Ts: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_169 @ Ys_51 )
= ( append179082452le_alt @ Zs_8 @ Ts ) )
<=> ? [Us_1: list_A2115238852le_alt] :
( ( ( Xs_169
= ( append179082452le_alt @ Zs_8 @ Us_1 ) )
& ( ( append179082452le_alt @ Us_1 @ Ys_51 )
= Ts ) )
| ( ( ( append179082452le_alt @ Xs_169 @ Us_1 )
= Zs_8 )
& ( Ys_51
= ( append179082452le_alt @ Us_1 @ Ts ) ) ) ) ) ).
thf(fact_48_same__append__eq,axiom,
! [Xs_168: list_A2115238852le_alt,Ys_50: list_A2115238852le_alt,Zs_7: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_168 @ Ys_50 )
= ( append179082452le_alt @ Xs_168 @ Zs_7 ) )
<=> ( Ys_50 = Zs_7 ) ) ).
thf(fact_49_append__same__eq,axiom,
! [Ys_49: list_A2115238852le_alt,Xs_167: list_A2115238852le_alt,Zs_6: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Ys_49 @ Xs_167 )
= ( append179082452le_alt @ Zs_6 @ Xs_167 ) )
<=> ( Ys_49 = Zs_6 ) ) ).
thf(fact_50_append__eq__appendI,axiom,
! [Ys_48: list_A2115238852le_alt,Us: list_A2115238852le_alt,Xs_166: list_A2115238852le_alt,Xs1_1: list_A2115238852le_alt,Zs_5: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_166 @ Xs1_1 )
= Zs_5 )
=> ( ( Ys_48
= ( append179082452le_alt @ Xs1_1 @ Us ) )
=> ( ( append179082452le_alt @ Xs_166 @ Ys_48 )
= ( append179082452le_alt @ Zs_5 @ Us ) ) ) ) ).
thf(fact_51_maps__simps_I1_J,axiom,
! [F_6: arrow_475358991le_alt > list_A2115238852le_alt,X_106: arrow_475358991le_alt,Xs_165: list_A2115238852le_alt] :
( ( maps_A409156225le_alt @ F_6 @ ( cons_A228743023le_alt @ X_106 @ Xs_165 ) )
= ( append179082452le_alt @ ( F_6 @ X_106 ) @ ( maps_A409156225le_alt @ F_6 @ Xs_165 ) ) ) ).
thf(fact_52_append__Cons,axiom,
! [X_105: arrow_475358991le_alt,Xs_164: list_A2115238852le_alt,Ys_47: list_A2115238852le_alt] :
( ( append179082452le_alt @ ( cons_A228743023le_alt @ X_105 @ Xs_164 ) @ Ys_47 )
= ( cons_A228743023le_alt @ X_105 @ ( append179082452le_alt @ Xs_164 @ Ys_47 ) ) ) ).
thf(fact_53_Cons__eq__appendI,axiom,
! [Xs_163: list_A2115238852le_alt,Zs_4: list_A2115238852le_alt,X_104: arrow_475358991le_alt,Xs1: list_A2115238852le_alt,Ys_46: list_A2115238852le_alt] :
( ( ( cons_A228743023le_alt @ X_104 @ Xs1 )
= Ys_46 )
=> ( ( Xs_163
= ( append179082452le_alt @ Xs1 @ Zs_4 ) )
=> ( ( cons_A228743023le_alt @ X_104 @ Xs_163 )
= ( append179082452le_alt @ Ys_46 @ Zs_4 ) ) ) ) ).
thf(fact_54_append__Nil,axiom,
! [Ys_45: list_A2115238852le_alt] :
( ( append179082452le_alt @ nil_Ar1286194111le_alt @ Ys_45 )
= Ys_45 ) ).
thf(fact_55_append__Nil,axiom,
! [Ys_45: list_nat] :
( ( append_nat @ nil_nat @ Ys_45 )
= Ys_45 ) ).
thf(fact_56_Nil__is__append__conv,axiom,
! [Xs_162: list_A2115238852le_alt,Ys_44: list_A2115238852le_alt] :
( ( nil_Ar1286194111le_alt
= ( append179082452le_alt @ Xs_162 @ Ys_44 ) )
<=> ( ( Xs_162 = nil_Ar1286194111le_alt )
& ( Ys_44 = nil_Ar1286194111le_alt ) ) ) ).
thf(fact_57_Nil__is__append__conv,axiom,
! [Xs_162: list_nat,Ys_44: list_nat] :
( ( nil_nat
= ( append_nat @ Xs_162 @ Ys_44 ) )
<=> ( ( Xs_162 = nil_nat )
& ( Ys_44 = nil_nat ) ) ) ).
thf(fact_58_append__Nil2,axiom,
! [Xs_161: list_A2115238852le_alt] :
( ( append179082452le_alt @ Xs_161 @ nil_Ar1286194111le_alt )
= Xs_161 ) ).
thf(fact_59_append__Nil2,axiom,
! [Xs_161: list_nat] :
( ( append_nat @ Xs_161 @ nil_nat )
= Xs_161 ) ).
thf(fact_60_self__append__conv,axiom,
! [Xs_160: list_A2115238852le_alt,Ys_43: list_A2115238852le_alt] :
( ( Xs_160
= ( append179082452le_alt @ Xs_160 @ Ys_43 ) )
<=> ( Ys_43 = nil_Ar1286194111le_alt ) ) ).
thf(fact_61_self__append__conv,axiom,
! [Xs_160: list_nat,Ys_43: list_nat] :
( ( Xs_160
= ( append_nat @ Xs_160 @ Ys_43 ) )
<=> ( Ys_43 = nil_nat ) ) ).
thf(fact_62_self__append__conv2,axiom,
! [Ys_42: list_A2115238852le_alt,Xs_159: list_A2115238852le_alt] :
( ( Ys_42
= ( append179082452le_alt @ Xs_159 @ Ys_42 ) )
<=> ( Xs_159 = nil_Ar1286194111le_alt ) ) ).
thf(fact_63_self__append__conv2,axiom,
! [Ys_42: list_nat,Xs_159: list_nat] :
( ( Ys_42
= ( append_nat @ Xs_159 @ Ys_42 ) )
<=> ( Xs_159 = nil_nat ) ) ).
thf(fact_64_append__is__Nil__conv,axiom,
! [Xs_158: list_A2115238852le_alt,Ys_41: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_158 @ Ys_41 )
= nil_Ar1286194111le_alt )
<=> ( ( Xs_158 = nil_Ar1286194111le_alt )
& ( Ys_41 = nil_Ar1286194111le_alt ) ) ) ).
thf(fact_65_append__is__Nil__conv,axiom,
! [Xs_158: list_nat,Ys_41: list_nat] :
( ( ( append_nat @ Xs_158 @ Ys_41 )
= nil_nat )
<=> ( ( Xs_158 = nil_nat )
& ( Ys_41 = nil_nat ) ) ) ).
thf(fact_66_append__self__conv,axiom,
! [Xs_157: list_A2115238852le_alt,Ys_40: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_157 @ Ys_40 )
= Xs_157 )
<=> ( Ys_40 = nil_Ar1286194111le_alt ) ) ).
thf(fact_67_append__self__conv,axiom,
! [Xs_157: list_nat,Ys_40: list_nat] :
( ( ( append_nat @ Xs_157 @ Ys_40 )
= Xs_157 )
<=> ( Ys_40 = nil_nat ) ) ).
thf(fact_68_append__self__conv2,axiom,
! [Xs_156: list_A2115238852le_alt,Ys_39: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_156 @ Ys_39 )
= Ys_39 )
<=> ( Xs_156 = nil_Ar1286194111le_alt ) ) ).
thf(fact_69_append__self__conv2,axiom,
! [Xs_156: list_nat,Ys_39: list_nat] :
( ( ( append_nat @ Xs_156 @ Ys_39 )
= Ys_39 )
<=> ( Xs_156 = nil_nat ) ) ).
thf(fact_70_eq__Nil__appendI,axiom,
! [Xs_155: list_A2115238852le_alt,Ys_38: list_A2115238852le_alt] :
( ( Xs_155 = Ys_38 )
=> ( Xs_155
= ( append179082452le_alt @ nil_Ar1286194111le_alt @ Ys_38 ) ) ) ).
thf(fact_71_eq__Nil__appendI,axiom,
! [Xs_155: list_nat,Ys_38: list_nat] :
( ( Xs_155 = Ys_38 )
=> ( Xs_155
= ( append_nat @ nil_nat @ Ys_38 ) ) ) ).
thf(fact_72_append__eq__Cons__conv,axiom,
! [Ys_37: list_A2115238852le_alt,Zs_3: list_A2115238852le_alt,X_103: arrow_475358991le_alt,Xs_154: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Ys_37 @ Zs_3 )
= ( cons_A228743023le_alt @ X_103 @ Xs_154 ) )
<=> ( ( ( Ys_37 = nil_Ar1286194111le_alt )
& ( Zs_3
= ( cons_A228743023le_alt @ X_103 @ Xs_154 ) ) )
| ? [Ys_36: list_A2115238852le_alt] :
( ( Ys_37
= ( cons_A228743023le_alt @ X_103 @ Ys_36 ) )
& ( ( append179082452le_alt @ Ys_36 @ Zs_3 )
= Xs_154 ) ) ) ) ).
thf(fact_73_append__eq__Cons__conv,axiom,
! [Ys_37: list_nat,Zs_3: list_nat,X_103: nat,Xs_154: list_nat] :
( ( ( append_nat @ Ys_37 @ Zs_3 )
= ( cons_nat @ X_103 @ Xs_154 ) )
<=> ( ( ( Ys_37 = nil_nat )
& ( Zs_3
= ( cons_nat @ X_103 @ Xs_154 ) ) )
| ? [Ys_36: list_nat] :
( ( Ys_37
= ( cons_nat @ X_103 @ Ys_36 ) )
& ( ( append_nat @ Ys_36 @ Zs_3 )
= Xs_154 ) ) ) ) ).
thf(fact_74_Cons__eq__append__conv,axiom,
! [X_102: arrow_475358991le_alt,Xs_153: list_A2115238852le_alt,Ys_35: list_A2115238852le_alt,Zs_2: list_A2115238852le_alt] :
( ( ( cons_A228743023le_alt @ X_102 @ Xs_153 )
= ( append179082452le_alt @ Ys_35 @ Zs_2 ) )
<=> ( ( ( Ys_35 = nil_Ar1286194111le_alt )
& ( ( cons_A228743023le_alt @ X_102 @ Xs_153 )
= Zs_2 ) )
| ? [Ys_36: list_A2115238852le_alt] :
( ( ( cons_A228743023le_alt @ X_102 @ Ys_36 )
= Ys_35 )
& ( Xs_153
= ( append179082452le_alt @ Ys_36 @ Zs_2 ) ) ) ) ) ).
thf(fact_75_Cons__eq__append__conv,axiom,
! [X_102: nat,Xs_153: list_nat,Ys_35: list_nat,Zs_2: list_nat] :
( ( ( cons_nat @ X_102 @ Xs_153 )
= ( append_nat @ Ys_35 @ Zs_2 ) )
<=> ( ( ( Ys_35 = nil_nat )
& ( ( cons_nat @ X_102 @ Xs_153 )
= Zs_2 ) )
| ? [Ys_36: list_nat] :
( ( ( cons_nat @ X_102 @ Ys_36 )
= Ys_35 )
& ( Xs_153
= ( append_nat @ Ys_36 @ Zs_2 ) ) ) ) ) ).
thf(fact_76_append1__eq__conv,axiom,
! [Xs_152: list_A2115238852le_alt,X_101: arrow_475358991le_alt,Ys_34: list_A2115238852le_alt,Y_42: arrow_475358991le_alt] :
( ( ( append179082452le_alt @ Xs_152 @ ( cons_A228743023le_alt @ X_101 @ nil_Ar1286194111le_alt ) )
= ( append179082452le_alt @ Ys_34 @ ( cons_A228743023le_alt @ Y_42 @ nil_Ar1286194111le_alt ) ) )
<=> ( ( Xs_152 = Ys_34 )
& ( X_101 = Y_42 ) ) ) ).
thf(fact_77_append1__eq__conv,axiom,
! [Xs_152: list_nat,X_101: nat,Ys_34: list_nat,Y_42: nat] :
( ( ( append_nat @ Xs_152 @ ( cons_nat @ X_101 @ nil_nat ) )
= ( append_nat @ Ys_34 @ ( cons_nat @ Y_42 @ nil_nat ) ) )
<=> ( ( Xs_152 = Ys_34 )
& ( X_101 = Y_42 ) ) ) ).
thf(fact_78_butlast__append,axiom,
! [Xs_151: list_A2115238852le_alt,Ys_33: list_A2115238852le_alt] :
( ( ( Ys_33 = nil_Ar1286194111le_alt )
=> ( ( butlas274947851le_alt @ ( append179082452le_alt @ Xs_151 @ Ys_33 ) )
= ( butlas274947851le_alt @ Xs_151 ) ) )
& ( ( Ys_33 != nil_Ar1286194111le_alt )
=> ( ( butlas274947851le_alt @ ( append179082452le_alt @ Xs_151 @ Ys_33 ) )
= ( append179082452le_alt @ Xs_151 @ ( butlas274947851le_alt @ Ys_33 ) ) ) ) ) ).
thf(fact_79_butlast__append,axiom,
! [Xs_151: list_nat,Ys_33: list_nat] :
( ( ( Ys_33 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs_151 @ Ys_33 ) )
= ( butlast_nat @ Xs_151 ) ) )
& ( ( Ys_33 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs_151 @ Ys_33 ) )
= ( append_nat @ Xs_151 @ ( butlast_nat @ Ys_33 ) ) ) ) ) ).
thf(fact_80_sorted__butlast,axiom,
! [Xs_150: list_nat] :
( ( Xs_150 != nil_nat )
=> ( ( linorder_sorted_nat @ Xs_150 )
=> ( linorder_sorted_nat @ ( butlast_nat @ Xs_150 ) ) ) ) ).
thf(fact_81_rev__induct,axiom,
! [Xs_149: list_A2115238852le_alt,P_32: list_A2115238852le_alt > $o] :
( ( P_32 @ nil_Ar1286194111le_alt )
=> ( ! [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt] :
( ( P_32 @ Xs_54 )
=> ( P_32 @ ( append179082452le_alt @ Xs_54 @ ( cons_A228743023le_alt @ X_1 @ nil_Ar1286194111le_alt ) ) ) )
=> ( P_32 @ Xs_149 ) ) ) ).
thf(fact_82_rev__induct,axiom,
! [Xs_149: list_nat,P_32: list_nat > $o] :
( ( P_32 @ nil_nat )
=> ( ! [X_1: nat,Xs_54: list_nat] :
( ( P_32 @ Xs_54 )
=> ( P_32 @ ( append_nat @ Xs_54 @ ( cons_nat @ X_1 @ nil_nat ) ) ) )
=> ( P_32 @ Xs_149 ) ) ) ).
thf(fact_83_rev__cases,axiom,
! [Xs_148: list_A2115238852le_alt] :
( ( Xs_148 != nil_Ar1286194111le_alt )
=> ~ ! [Ys_5: list_A2115238852le_alt,Y_3: arrow_475358991le_alt] :
( Xs_148
!= ( append179082452le_alt @ Ys_5 @ ( cons_A228743023le_alt @ Y_3 @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_84_rev__cases,axiom,
! [Xs_148: list_nat] :
( ( Xs_148 != nil_nat )
=> ~ ! [Ys_5: list_nat,Y_3: nat] :
( Xs_148
!= ( append_nat @ Ys_5 @ ( cons_nat @ Y_3 @ nil_nat ) ) ) ) ).
thf(fact_85_equal__Nil__null,axiom,
! [Xs_147: list_A2115238852le_alt] :
( ( equal_484611810le_alt @ Xs_147 @ nil_Ar1286194111le_alt )
<=> ( null_A1520965063le_alt @ Xs_147 ) ) ).
thf(fact_86_equal__Nil__null,axiom,
! [Xs_147: list_nat] :
( ( equal_equal_list_nat @ Xs_147 @ nil_nat )
<=> ( null_nat @ Xs_147 ) ) ).
thf(fact_87_snoc__eq__iff__butlast,axiom,
! [Xs_146: list_A2115238852le_alt,X_100: arrow_475358991le_alt,Ys_32: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_146 @ ( cons_A228743023le_alt @ X_100 @ nil_Ar1286194111le_alt ) )
= Ys_32 )
<=> ( ( Ys_32 != nil_Ar1286194111le_alt )
& ( ( butlas274947851le_alt @ Ys_32 )
= Xs_146 )
& ( ( last_A1217315288le_alt @ Ys_32 )
= X_100 ) ) ) ).
thf(fact_88_snoc__eq__iff__butlast,axiom,
! [Xs_146: list_nat,X_100: nat,Ys_32: list_nat] :
( ( ( append_nat @ Xs_146 @ ( cons_nat @ X_100 @ nil_nat ) )
= Ys_32 )
<=> ( ( Ys_32 != nil_nat )
& ( ( butlast_nat @ Ys_32 )
= Xs_146 )
& ( ( last_nat @ Ys_32 )
= X_100 ) ) ) ).
thf(fact_89_append__butlast__last__id,axiom,
! [Xs_145: list_A2115238852le_alt] :
( ( Xs_145 != nil_Ar1286194111le_alt )
=> ( ( append179082452le_alt @ ( butlas274947851le_alt @ Xs_145 ) @ ( cons_A228743023le_alt @ ( last_A1217315288le_alt @ Xs_145 ) @ nil_Ar1286194111le_alt ) )
= Xs_145 ) ) ).
thf(fact_90_append__butlast__last__id,axiom,
! [Xs_145: list_nat] :
( ( Xs_145 != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs_145 ) @ ( cons_nat @ ( last_nat @ Xs_145 ) @ nil_nat ) )
= Xs_145 ) ) ).
thf(fact_91_rotate__simps,axiom,
! [X_99: arrow_475358991le_alt,Xs_144: list_A2115238852le_alt] :
( ( ( rotate1_nat @ nil_nat )
= nil_nat )
& ( ( rotate335349260le_alt @ ( cons_A228743023le_alt @ X_99 @ Xs_144 ) )
= ( append179082452le_alt @ Xs_144 @ ( cons_A228743023le_alt @ X_99 @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_92_rotate__simps,axiom,
! [X_99: arrow_475358991le_alt,Xs_144: list_A2115238852le_alt] :
( ( ( rotate335349260le_alt @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt )
& ( ( rotate335349260le_alt @ ( cons_A228743023le_alt @ X_99 @ Xs_144 ) )
= ( append179082452le_alt @ Xs_144 @ ( cons_A228743023le_alt @ X_99 @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_93_rotate__simps,axiom,
! [X_99: nat,Xs_144: list_nat] :
( ( ( rotate335349260le_alt @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt )
& ( ( rotate1_nat @ ( cons_nat @ X_99 @ Xs_144 ) )
= ( append_nat @ Xs_144 @ ( cons_nat @ X_99 @ nil_nat ) ) ) ) ).
thf(fact_94_last__snoc,axiom,
! [Xs_143: list_A2115238852le_alt,X_98: arrow_475358991le_alt] :
( ( last_A1217315288le_alt @ ( append179082452le_alt @ Xs_143 @ ( cons_A228743023le_alt @ X_98 @ nil_Ar1286194111le_alt ) ) )
= X_98 ) ).
thf(fact_95_last__snoc,axiom,
! [Xs_143: list_nat,X_98: nat] :
( ( last_nat @ ( append_nat @ Xs_143 @ ( cons_nat @ X_98 @ nil_nat ) ) )
= X_98 ) ).
thf(fact_96_not__distinct__decomp,axiom,
! [Ws: list_A2115238852le_alt] :
( ~ ( distin236324274le_alt @ Ws )
=> ? [Xs_54: list_A2115238852le_alt,Ys_5: list_A2115238852le_alt,Zs_1: list_A2115238852le_alt,Y_3: arrow_475358991le_alt] :
( Ws
= ( append179082452le_alt @ Xs_54 @ ( append179082452le_alt @ ( cons_A228743023le_alt @ Y_3 @ nil_Ar1286194111le_alt ) @ ( append179082452le_alt @ Ys_5 @ ( append179082452le_alt @ ( cons_A228743023le_alt @ Y_3 @ nil_Ar1286194111le_alt ) @ Zs_1 ) ) ) ) ) ) ).
thf(fact_97_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs_54: list_nat,Ys_5: list_nat,Zs_1: list_nat,Y_3: nat] :
( Ws
= ( append_nat @ Xs_54 @ ( append_nat @ ( cons_nat @ Y_3 @ nil_nat ) @ ( append_nat @ Ys_5 @ ( append_nat @ ( cons_nat @ Y_3 @ nil_nat ) @ Zs_1 ) ) ) ) ) ) ).
thf(fact_98_replicate__append__same,axiom,
! [I_28: nat,X_97: arrow_475358991le_alt] :
( ( append179082452le_alt @ ( replic1511538809le_alt @ I_28 @ X_97 ) @ ( cons_A228743023le_alt @ X_97 @ nil_Ar1286194111le_alt ) )
= ( cons_A228743023le_alt @ X_97 @ ( replic1511538809le_alt @ I_28 @ X_97 ) ) ) ).
thf(fact_99_replicate__append__same,axiom,
! [I_28: nat,X_97: nat] :
( ( append_nat @ ( replicate_nat @ I_28 @ X_97 ) @ ( cons_nat @ X_97 @ nil_nat ) )
= ( cons_nat @ X_97 @ ( replicate_nat @ I_28 @ X_97 ) ) ) ).
thf(fact_100_rev_Osimps_I2_J,axiom,
! [X_96: arrow_475358991le_alt,Xs_142: list_A2115238852le_alt] :
( ( rev_Ar1106406943le_alt @ ( cons_A228743023le_alt @ X_96 @ Xs_142 ) )
= ( append179082452le_alt @ ( rev_Ar1106406943le_alt @ Xs_142 ) @ ( cons_A228743023le_alt @ X_96 @ nil_Ar1286194111le_alt ) ) ) ).
thf(fact_101_rev_Osimps_I2_J,axiom,
! [X_96: nat,Xs_142: list_nat] :
( ( rev_nat @ ( cons_nat @ X_96 @ Xs_142 ) )
= ( append_nat @ ( rev_nat @ Xs_142 ) @ ( cons_nat @ X_96 @ nil_nat ) ) ) ).
thf(fact_102_rev__is__rev__conv,axiom,
! [Xs_141: list_nat,Ys_31: list_nat] :
( ( ( rev_nat @ Xs_141 )
= ( rev_nat @ Ys_31 ) )
<=> ( Xs_141 = Ys_31 ) ) ).
thf(fact_103_rev__is__rev__conv,axiom,
! [Xs_141: list_A2115238852le_alt,Ys_31: list_A2115238852le_alt] :
( ( ( rev_Ar1106406943le_alt @ Xs_141 )
= ( rev_Ar1106406943le_alt @ Ys_31 ) )
<=> ( Xs_141 = Ys_31 ) ) ).
thf(fact_104_rev__swap,axiom,
! [Xs_140: list_nat,Ys_30: list_nat] :
( ( ( rev_nat @ Xs_140 )
= Ys_30 )
<=> ( Xs_140
= ( rev_nat @ Ys_30 ) ) ) ).
thf(fact_105_rev__swap,axiom,
! [Xs_140: list_A2115238852le_alt,Ys_30: list_A2115238852le_alt] :
( ( ( rev_Ar1106406943le_alt @ Xs_140 )
= Ys_30 )
<=> ( Xs_140
= ( rev_Ar1106406943le_alt @ Ys_30 ) ) ) ).
thf(fact_106_rev__replicate,axiom,
! [N_53: nat,X_95: nat] :
( ( rev_nat @ ( replicate_nat @ N_53 @ X_95 ) )
= ( replicate_nat @ N_53 @ X_95 ) ) ).
thf(fact_107_rev__replicate,axiom,
! [N_53: nat,X_95: arrow_475358991le_alt] :
( ( rev_Ar1106406943le_alt @ ( replic1511538809le_alt @ N_53 @ X_95 ) )
= ( replic1511538809le_alt @ N_53 @ X_95 ) ) ).
thf(fact_108_rev__rev__ident,axiom,
! [Xs_139: list_nat] :
( ( rev_nat @ ( rev_nat @ Xs_139 ) )
= Xs_139 ) ).
thf(fact_109_rev__rev__ident,axiom,
! [Xs_139: list_A2115238852le_alt] :
( ( rev_Ar1106406943le_alt @ ( rev_Ar1106406943le_alt @ Xs_139 ) )
= Xs_139 ) ).
thf(fact_110_equal__list__def,axiom,
! [X_94: list_A2115238852le_alt,Y_41: list_A2115238852le_alt] :
( ( equal_484611810le_alt @ X_94 @ Y_41 )
<=> ( X_94 = Y_41 ) ) ).
thf(fact_111_rev__is__Nil__conv,axiom,
! [Xs_138: list_A2115238852le_alt] :
( ( ( rev_Ar1106406943le_alt @ Xs_138 )
= nil_Ar1286194111le_alt )
<=> ( Xs_138 = nil_Ar1286194111le_alt ) ) ).
thf(fact_112_rev__is__Nil__conv,axiom,
! [Xs_138: list_nat] :
( ( ( rev_nat @ Xs_138 )
= nil_nat )
<=> ( Xs_138 = nil_nat ) ) ).
thf(fact_113_Nil__is__rev__conv,axiom,
! [Xs_137: list_A2115238852le_alt] :
( ( nil_Ar1286194111le_alt
= ( rev_Ar1106406943le_alt @ Xs_137 ) )
<=> ( Xs_137 = nil_Ar1286194111le_alt ) ) ).
thf(fact_114_Nil__is__rev__conv,axiom,
! [Xs_137: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs_137 ) )
<=> ( Xs_137 = nil_nat ) ) ).
thf(fact_115_rev_Osimps_I1_J,axiom,
( ( rev_Ar1106406943le_alt @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_116_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
thf(fact_117_rev__append,axiom,
! [Xs_136: list_nat,Ys_29: list_nat] :
( ( rev_nat @ ( append_nat @ Xs_136 @ Ys_29 ) )
= ( append_nat @ ( rev_nat @ Ys_29 ) @ ( rev_nat @ Xs_136 ) ) ) ).
thf(fact_118_rev__append,axiom,
! [Xs_136: list_A2115238852le_alt,Ys_29: list_A2115238852le_alt] :
( ( rev_Ar1106406943le_alt @ ( append179082452le_alt @ Xs_136 @ Ys_29 ) )
= ( append179082452le_alt @ ( rev_Ar1106406943le_alt @ Ys_29 ) @ ( rev_Ar1106406943le_alt @ Xs_136 ) ) ) ).
thf(fact_119_distinct__rev,axiom,
! [Xs_135: list_A2115238852le_alt] :
( ( distin236324274le_alt @ ( rev_Ar1106406943le_alt @ Xs_135 ) )
<=> ( distin236324274le_alt @ Xs_135 ) ) ).
thf(fact_120_distinct__rev,axiom,
! [Xs_135: list_nat] :
( ( distinct_nat @ ( rev_nat @ Xs_135 ) )
<=> ( distinct_nat @ Xs_135 ) ) ).
thf(fact_121_append__replicate__commute,axiom,
! [N_52: nat,X_93: arrow_475358991le_alt,K_8: nat] :
( ( append179082452le_alt @ ( replic1511538809le_alt @ N_52 @ X_93 ) @ ( replic1511538809le_alt @ K_8 @ X_93 ) )
= ( append179082452le_alt @ ( replic1511538809le_alt @ K_8 @ X_93 ) @ ( replic1511538809le_alt @ N_52 @ X_93 ) ) ) ).
thf(fact_122_rotate1__is__Nil__conv,axiom,
! [Xs_134: list_A2115238852le_alt] :
( ( ( rotate335349260le_alt @ Xs_134 )
= nil_Ar1286194111le_alt )
<=> ( Xs_134 = nil_Ar1286194111le_alt ) ) ).
thf(fact_123_rotate1__is__Nil__conv,axiom,
! [Xs_134: list_nat] :
( ( ( rotate1_nat @ Xs_134 )
= nil_nat )
<=> ( Xs_134 = nil_nat ) ) ).
thf(fact_124_mem__def,axiom,
! [X_92: nat,A_87: nat > $o] :
( ( member_nat @ X_92 @ A_87 )
<=> ( A_87 @ X_92 ) ) ).
thf(fact_125_mem__def,axiom,
! [X_92: int,A_87: int > $o] :
( ( member_int @ X_92 @ A_87 )
<=> ( A_87 @ X_92 ) ) ).
thf(fact_126_Collect__def,axiom,
! [P_31: nat > $o] :
( ( collect_nat @ P_31 )
= P_31 ) ).
thf(fact_127_distinct1__rotate,axiom,
! [Xs_133: list_A2115238852le_alt] :
( ( distin236324274le_alt @ ( rotate335349260le_alt @ Xs_133 ) )
<=> ( distin236324274le_alt @ Xs_133 ) ) ).
thf(fact_128_singleton__rev__conv,axiom,
! [X_91: arrow_475358991le_alt,Xs_132: list_A2115238852le_alt] :
( ( ( cons_A228743023le_alt @ X_91 @ nil_Ar1286194111le_alt )
= ( rev_Ar1106406943le_alt @ Xs_132 ) )
<=> ( Xs_132
= ( cons_A228743023le_alt @ X_91 @ nil_Ar1286194111le_alt ) ) ) ).
thf(fact_129_singleton__rev__conv,axiom,
! [X_91: nat,Xs_132: list_nat] :
( ( ( cons_nat @ X_91 @ nil_nat )
= ( rev_nat @ Xs_132 ) )
<=> ( Xs_132
= ( cons_nat @ X_91 @ nil_nat ) ) ) ).
thf(fact_130_rev__singleton__conv,axiom,
! [Xs_131: list_A2115238852le_alt,X_90: arrow_475358991le_alt] :
( ( ( rev_Ar1106406943le_alt @ Xs_131 )
= ( cons_A228743023le_alt @ X_90 @ nil_Ar1286194111le_alt ) )
<=> ( Xs_131
= ( cons_A228743023le_alt @ X_90 @ nil_Ar1286194111le_alt ) ) ) ).
thf(fact_131_rev__singleton__conv,axiom,
! [Xs_131: list_nat,X_90: nat] :
( ( ( rev_nat @ Xs_131 )
= ( cons_nat @ X_90 @ nil_nat ) )
<=> ( Xs_131
= ( cons_nat @ X_90 @ nil_nat ) ) ) ).
thf(fact_132_replicate__app__Cons__same,axiom,
! [N_51: nat,X_89: arrow_475358991le_alt,Xs_130: list_A2115238852le_alt] :
( ( append179082452le_alt @ ( replic1511538809le_alt @ N_51 @ X_89 ) @ ( cons_A228743023le_alt @ X_89 @ Xs_130 ) )
= ( cons_A228743023le_alt @ X_89 @ ( append179082452le_alt @ ( replic1511538809le_alt @ N_51 @ X_89 ) @ Xs_130 ) ) ) ).
thf(fact_133_last__ConsL,axiom,
! [X_88: arrow_475358991le_alt,Xs_129: list_A2115238852le_alt] :
( ( Xs_129 = nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( cons_A228743023le_alt @ X_88 @ Xs_129 ) )
= X_88 ) ) ).
thf(fact_134_last__ConsL,axiom,
! [X_88: nat,Xs_129: list_nat] :
( ( Xs_129 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_88 @ Xs_129 ) )
= X_88 ) ) ).
thf(fact_135_last__ConsR,axiom,
! [X_87: arrow_475358991le_alt,Xs_128: list_A2115238852le_alt] :
( ( Xs_128 != nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( cons_A228743023le_alt @ X_87 @ Xs_128 ) )
= ( last_A1217315288le_alt @ Xs_128 ) ) ) ).
thf(fact_136_last__ConsR,axiom,
! [X_87: nat,Xs_128: list_nat] :
( ( Xs_128 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_87 @ Xs_128 ) )
= ( last_nat @ Xs_128 ) ) ) ).
thf(fact_137_last_Osimps,axiom,
! [X_86: arrow_475358991le_alt,Xs_127: list_A2115238852le_alt] :
( ( ( Xs_127 = nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( cons_A228743023le_alt @ X_86 @ Xs_127 ) )
= X_86 ) )
& ( ( Xs_127 != nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( cons_A228743023le_alt @ X_86 @ Xs_127 ) )
= ( last_A1217315288le_alt @ Xs_127 ) ) ) ) ).
thf(fact_138_last_Osimps,axiom,
! [X_86: nat,Xs_127: list_nat] :
( ( ( Xs_127 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_86 @ Xs_127 ) )
= X_86 ) )
& ( ( Xs_127 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X_86 @ Xs_127 ) )
= ( last_nat @ Xs_127 ) ) ) ) ).
thf(fact_139_last__append,axiom,
! [Xs_126: list_A2115238852le_alt,Ys_28: list_A2115238852le_alt] :
( ( ( Ys_28 = nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( append179082452le_alt @ Xs_126 @ Ys_28 ) )
= ( last_A1217315288le_alt @ Xs_126 ) ) )
& ( ( Ys_28 != nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( append179082452le_alt @ Xs_126 @ Ys_28 ) )
= ( last_A1217315288le_alt @ Ys_28 ) ) ) ) ).
thf(fact_140_last__append,axiom,
! [Xs_126: list_nat,Ys_28: list_nat] :
( ( ( Ys_28 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_126 @ Ys_28 ) )
= ( last_nat @ Xs_126 ) ) )
& ( ( Ys_28 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_126 @ Ys_28 ) )
= ( last_nat @ Ys_28 ) ) ) ) ).
thf(fact_141_last__appendR,axiom,
! [Xs_125: list_A2115238852le_alt,Ys_27: list_A2115238852le_alt] :
( ( Ys_27 != nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( append179082452le_alt @ Xs_125 @ Ys_27 ) )
= ( last_A1217315288le_alt @ Ys_27 ) ) ) ).
thf(fact_142_last__appendR,axiom,
! [Xs_125: list_nat,Ys_27: list_nat] :
( ( Ys_27 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_125 @ Ys_27 ) )
= ( last_nat @ Ys_27 ) ) ) ).
thf(fact_143_last__appendL,axiom,
! [Xs_124: list_A2115238852le_alt,Ys_26: list_A2115238852le_alt] :
( ( Ys_26 = nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( append179082452le_alt @ Xs_124 @ Ys_26 ) )
= ( last_A1217315288le_alt @ Xs_124 ) ) ) ).
thf(fact_144_last__appendL,axiom,
! [Xs_124: list_nat,Ys_26: list_nat] :
( ( Ys_26 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs_124 @ Ys_26 ) )
= ( last_nat @ Xs_124 ) ) ) ).
thf(fact_145_rev__eq__Cons__iff,axiom,
! [Xs_123: list_A2115238852le_alt,Y_40: arrow_475358991le_alt,Ys_25: list_A2115238852le_alt] :
( ( ( rev_Ar1106406943le_alt @ Xs_123 )
= ( cons_A228743023le_alt @ Y_40 @ Ys_25 ) )
<=> ( Xs_123
= ( append179082452le_alt @ ( rev_Ar1106406943le_alt @ Ys_25 ) @ ( cons_A228743023le_alt @ Y_40 @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_146_rev__eq__Cons__iff,axiom,
! [Xs_123: list_nat,Y_40: nat,Ys_25: list_nat] :
( ( ( rev_nat @ Xs_123 )
= ( cons_nat @ Y_40 @ Ys_25 ) )
<=> ( Xs_123
= ( append_nat @ ( rev_nat @ Ys_25 ) @ ( cons_nat @ Y_40 @ nil_nat ) ) ) ) ).
thf(fact_147_last__rev,axiom,
! [Xs_122: list_A2115238852le_alt] :
( ( Xs_122 != nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ ( rev_Ar1106406943le_alt @ Xs_122 ) )
= ( hd_Arr1965683346le_alt @ Xs_122 ) ) ) ).
thf(fact_148_last__rev,axiom,
! [Xs_122: list_nat] :
( ( Xs_122 != nil_nat )
=> ( ( last_nat @ ( rev_nat @ Xs_122 ) )
= ( hd_nat @ Xs_122 ) ) ) ).
thf(fact_149_hd__rev,axiom,
! [Xs_121: list_A2115238852le_alt] :
( ( Xs_121 != nil_Ar1286194111le_alt )
=> ( ( hd_Arr1965683346le_alt @ ( rev_Ar1106406943le_alt @ Xs_121 ) )
= ( last_A1217315288le_alt @ Xs_121 ) ) ) ).
thf(fact_150_hd__rev,axiom,
! [Xs_121: list_nat] :
( ( Xs_121 != nil_nat )
=> ( ( hd_nat @ ( rev_nat @ Xs_121 ) )
= ( last_nat @ Xs_121 ) ) ) ).
thf(fact_151_equal,axiom,
equal_484611810le_alt = fequal781288069le_alt ).
thf(fact_152_equal__refl,axiom,
! [X_85: list_A2115238852le_alt] : ( equal_484611810le_alt @ X_85 @ X_85 ) ).
thf(fact_153_equal__eq,axiom,
! [X_84: list_A2115238852le_alt,Y_39: list_A2115238852le_alt] :
( ( equal_484611810le_alt @ X_84 @ Y_39 )
<=> ( X_84 = Y_39 ) ) ).
thf(fact_154_eq__equal,axiom,
fequal781288069le_alt = equal_484611810le_alt ).
thf(fact_155_hd__append,axiom,
! [Ys_24: list_A2115238852le_alt,Xs_120: list_A2115238852le_alt] :
( ( ( Xs_120 = nil_Ar1286194111le_alt )
=> ( ( hd_Arr1965683346le_alt @ ( append179082452le_alt @ Xs_120 @ Ys_24 ) )
= ( hd_Arr1965683346le_alt @ Ys_24 ) ) )
& ( ( Xs_120 != nil_Ar1286194111le_alt )
=> ( ( hd_Arr1965683346le_alt @ ( append179082452le_alt @ Xs_120 @ Ys_24 ) )
= ( hd_Arr1965683346le_alt @ Xs_120 ) ) ) ) ).
thf(fact_156_hd__append,axiom,
! [Ys_24: list_nat,Xs_120: list_nat] :
( ( ( Xs_120 = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs_120 @ Ys_24 ) )
= ( hd_nat @ Ys_24 ) ) )
& ( ( Xs_120 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs_120 @ Ys_24 ) )
= ( hd_nat @ Xs_120 ) ) ) ) ).
thf(fact_157_hd__append2,axiom,
! [Ys_23: list_A2115238852le_alt,Xs_119: list_A2115238852le_alt] :
( ( Xs_119 != nil_Ar1286194111le_alt )
=> ( ( hd_Arr1965683346le_alt @ ( append179082452le_alt @ Xs_119 @ Ys_23 ) )
= ( hd_Arr1965683346le_alt @ Xs_119 ) ) ) ).
thf(fact_158_hd__append2,axiom,
! [Ys_23: list_nat,Xs_119: list_nat] :
( ( Xs_119 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs_119 @ Ys_23 ) )
= ( hd_nat @ Xs_119 ) ) ) ).
thf(fact_159_rotate1__hd__tl,axiom,
! [Xs_118: list_A2115238852le_alt] :
( ( Xs_118 != nil_Ar1286194111le_alt )
=> ( ( rotate335349260le_alt @ Xs_118 )
= ( append179082452le_alt @ ( tl_Arr465451158le_alt @ Xs_118 ) @ ( cons_A228743023le_alt @ ( hd_Arr1965683346le_alt @ Xs_118 ) @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_160_rotate1__hd__tl,axiom,
! [Xs_118: list_nat] :
( ( Xs_118 != nil_nat )
=> ( ( rotate1_nat @ Xs_118 )
= ( append_nat @ ( tl_nat @ Xs_118 ) @ ( cons_nat @ ( hd_nat @ Xs_118 ) @ nil_nat ) ) ) ) ).
thf(fact_161_tl_Osimps_I2_J,axiom,
! [X_83: arrow_475358991le_alt,Xs_117: list_A2115238852le_alt] :
( ( tl_Arr465451158le_alt @ ( cons_A228743023le_alt @ X_83 @ Xs_117 ) )
= Xs_117 ) ).
thf(fact_162_tl_Osimps_I1_J,axiom,
( ( tl_Arr465451158le_alt @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_163_tl_Osimps_I1_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
thf(fact_164_hd_Osimps,axiom,
! [X_82: arrow_475358991le_alt,Xs_116: list_A2115238852le_alt] :
( ( hd_Arr1965683346le_alt @ ( cons_A228743023le_alt @ X_82 @ Xs_116 ) )
= X_82 ) ).
thf(fact_165_distinct__tl,axiom,
! [Xs_115: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_115 )
=> ( distin236324274le_alt @ ( tl_Arr465451158le_alt @ Xs_115 ) ) ) ).
thf(fact_166_sorted__tl,axiom,
! [Xs_114: list_nat] :
( ( linorder_sorted_nat @ Xs_114 )
=> ( linorder_sorted_nat @ ( tl_nat @ Xs_114 ) ) ) ).
thf(fact_167_tl__append2,axiom,
! [Ys_22: list_A2115238852le_alt,Xs_113: list_A2115238852le_alt] :
( ( Xs_113 != nil_Ar1286194111le_alt )
=> ( ( tl_Arr465451158le_alt @ ( append179082452le_alt @ Xs_113 @ Ys_22 ) )
= ( append179082452le_alt @ ( tl_Arr465451158le_alt @ Xs_113 ) @ Ys_22 ) ) ) ).
thf(fact_168_tl__append2,axiom,
! [Ys_22: list_nat,Xs_113: list_nat] :
( ( Xs_113 != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs_113 @ Ys_22 ) )
= ( append_nat @ ( tl_nat @ Xs_113 ) @ Ys_22 ) ) ) ).
thf(fact_169_hd__dropWhile,axiom,
! [P_30: arrow_475358991le_alt > $o,Xs_112: list_A2115238852le_alt] :
( ( ( dropWh1316781920le_alt @ P_30 @ Xs_112 )
!= nil_Ar1286194111le_alt )
=> ~ ( P_30 @ ( hd_Arr1965683346le_alt @ ( dropWh1316781920le_alt @ P_30 @ Xs_112 ) ) ) ) ).
thf(fact_170_hd__dropWhile,axiom,
! [P_30: nat > $o,Xs_112: list_nat] :
( ( ( dropWhile_nat @ P_30 @ Xs_112 )
!= nil_nat )
=> ~ ( P_30 @ ( hd_nat @ ( dropWhile_nat @ P_30 @ Xs_112 ) ) ) ) ).
thf(fact_171_member__rec_I2_J,axiom,
! [Y_38: arrow_475358991le_alt] :
~ ( member1504968148le_alt @ nil_Ar1286194111le_alt @ Y_38 ) ).
thf(fact_172_member__rec_I2_J,axiom,
! [Y_38: nat] :
~ ( member_nat_1 @ nil_nat @ Y_38 ) ).
thf(fact_173_list__ex1__simps_I1_J,axiom,
! [P_29: arrow_475358991le_alt > $o] :
~ ( list_e239244930le_alt @ P_29 @ nil_Ar1286194111le_alt ) ).
thf(fact_174_list__ex1__simps_I1_J,axiom,
! [P_29: nat > $o] :
~ ( list_ex1_nat @ P_29 @ nil_nat ) ).
thf(fact_175_elem,axiom,
! [X_81: arrow_475358991le_alt,Xs_111: list_A2115238852le_alt] : ( listMe78047147le_alt @ X_81 @ ( cons_A228743023le_alt @ X_81 @ Xs_111 ) ) ).
thf(fact_176_member__rec_I1_J,axiom,
! [X_80: arrow_475358991le_alt,Xs_110: list_A2115238852le_alt,Y_37: arrow_475358991le_alt] :
( ( member1504968148le_alt @ ( cons_A228743023le_alt @ X_80 @ Xs_110 ) @ Y_37 )
<=> ( ( X_80 = Y_37 )
| ( member1504968148le_alt @ Xs_110 @ Y_37 ) ) ) ).
thf(fact_177_insert,axiom,
! [Y_36: arrow_475358991le_alt,X_79: arrow_475358991le_alt,Xs_109: list_A2115238852le_alt] :
( ( listMe78047147le_alt @ X_79 @ Xs_109 )
=> ( listMe78047147le_alt @ X_79 @ ( cons_A228743023le_alt @ Y_36 @ Xs_109 ) ) ) ).
thf(fact_178_hd__replicate,axiom,
! [X_78: arrow_475358991le_alt,N_50: nat] :
( ( N_50 != zero_zero_nat )
=> ( ( hd_Arr1965683346le_alt @ ( replic1511538809le_alt @ N_50 @ X_78 ) )
= X_78 ) ) ).
thf(fact_179_last__replicate,axiom,
! [X_77: arrow_475358991le_alt,N_49: nat] :
( ( N_49 != zero_zero_nat )
=> ( ( last_A1217315288le_alt @ ( replic1511538809le_alt @ N_49 @ X_77 ) )
= X_77 ) ) ).
thf(fact_180_replicate__eq__replicate,axiom,
! [M_5: nat,X_76: arrow_475358991le_alt,N_48: nat,Y_35: arrow_475358991le_alt] :
( ( ( replic1511538809le_alt @ M_5 @ X_76 )
= ( replic1511538809le_alt @ N_48 @ Y_35 ) )
<=> ( ( M_5 = N_48 )
& ( ( M_5 != zero_zero_nat )
=> ( X_76 = Y_35 ) ) ) ) ).
thf(fact_181_dropWhile_Osimps_I2_J,axiom,
! [Xs_108: list_A2115238852le_alt,P_28: arrow_475358991le_alt > $o,X_75: arrow_475358991le_alt] :
( ( ( P_28 @ X_75 )
=> ( ( dropWh1316781920le_alt @ P_28 @ ( cons_A228743023le_alt @ X_75 @ Xs_108 ) )
= ( dropWh1316781920le_alt @ P_28 @ Xs_108 ) ) )
& ( ~ ( P_28 @ X_75 )
=> ( ( dropWh1316781920le_alt @ P_28 @ ( cons_A228743023le_alt @ X_75 @ Xs_108 ) )
= ( cons_A228743023le_alt @ X_75 @ Xs_108 ) ) ) ) ).
thf(fact_182_dropWhile_Osimps_I1_J,axiom,
! [P_27: arrow_475358991le_alt > $o] :
( ( dropWh1316781920le_alt @ P_27 @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_183_dropWhile_Osimps_I1_J,axiom,
! [P_27: nat > $o] :
( ( dropWhile_nat @ P_27 @ nil_nat )
= nil_nat ) ).
thf(fact_184_distinct__dropWhile,axiom,
! [P_26: arrow_475358991le_alt > $o,Xs_107: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_107 )
=> ( distin236324274le_alt @ ( dropWh1316781920le_alt @ P_26 @ Xs_107 ) ) ) ).
thf(fact_185_sorted__dropWhile,axiom,
! [P_25: nat > $o,Xs_106: list_nat] :
( ( linorder_sorted_nat @ Xs_106 )
=> ( linorder_sorted_nat @ ( dropWhile_nat @ P_25 @ Xs_106 ) ) ) ).
thf(fact_186_replicate__0,axiom,
! [X_74: arrow_475358991le_alt] :
( ( replic1511538809le_alt @ zero_zero_nat @ X_74 )
= nil_Ar1286194111le_alt ) ).
thf(fact_187_replicate__0,axiom,
! [X_74: nat] :
( ( replicate_nat @ zero_zero_nat @ X_74 )
= nil_nat ) ).
thf(fact_188_empty__replicate,axiom,
! [N_47: nat,X_73: arrow_475358991le_alt] :
( ( nil_Ar1286194111le_alt
= ( replic1511538809le_alt @ N_47 @ X_73 ) )
<=> ( N_47 = zero_zero_nat ) ) ).
thf(fact_189_empty__replicate,axiom,
! [N_47: nat,X_73: nat] :
( ( nil_nat
= ( replicate_nat @ N_47 @ X_73 ) )
<=> ( N_47 = zero_zero_nat ) ) ).
thf(fact_190_replicate__empty,axiom,
! [N_46: nat,X_72: arrow_475358991le_alt] :
( ( ( replic1511538809le_alt @ N_46 @ X_72 )
= nil_Ar1286194111le_alt )
<=> ( N_46 = zero_zero_nat ) ) ).
thf(fact_191_replicate__empty,axiom,
! [N_46: nat,X_72: nat] :
( ( ( replicate_nat @ N_46 @ X_72 )
= nil_nat )
<=> ( N_46 = zero_zero_nat ) ) ).
thf(fact_192_sublist__singleton,axiom,
! [X_71: arrow_475358991le_alt,A_86: nat > $o] :
( ( ( member_nat @ zero_zero_nat @ A_86 )
=> ( ( sublis2047953604le_alt @ ( cons_A228743023le_alt @ X_71 @ nil_Ar1286194111le_alt ) @ A_86 )
= ( cons_A228743023le_alt @ X_71 @ nil_Ar1286194111le_alt ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A_86 )
=> ( ( sublis2047953604le_alt @ ( cons_A228743023le_alt @ X_71 @ nil_Ar1286194111le_alt ) @ A_86 )
= nil_Ar1286194111le_alt ) ) ) ).
thf(fact_193_sublist__singleton,axiom,
! [X_71: nat,A_86: nat > $o] :
( ( ( member_nat @ zero_zero_nat @ A_86 )
=> ( ( sublist_nat @ ( cons_nat @ X_71 @ nil_nat ) @ A_86 )
= ( cons_nat @ X_71 @ nil_nat ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A_86 )
=> ( ( sublist_nat @ ( cons_nat @ X_71 @ nil_nat ) @ A_86 )
= nil_nat ) ) ) ).
thf(fact_194_dropWhile__eq__Cons__conv,axiom,
! [P_24: arrow_475358991le_alt > $o,Xs_105: list_A2115238852le_alt,Y_34: arrow_475358991le_alt,Ys_21: list_A2115238852le_alt] :
( ( ( dropWh1316781920le_alt @ P_24 @ Xs_105 )
= ( cons_A228743023le_alt @ Y_34 @ Ys_21 ) )
<=> ( ( Xs_105
= ( append179082452le_alt @ ( takeWh1696291512le_alt @ P_24 @ Xs_105 ) @ ( cons_A228743023le_alt @ Y_34 @ Ys_21 ) ) )
& ~ ( P_24 @ Y_34 ) ) ) ).
thf(fact_195_list_Osize_I1_J,axiom,
! [Fa_1: arrow_475358991le_alt > nat] :
( ( list_s250295873le_alt @ Fa_1 @ nil_Ar1286194111le_alt )
= zero_zero_nat ) ).
thf(fact_196_list_Osize_I1_J,axiom,
! [Fa_1: nat > nat] :
( ( list_size_nat @ Fa_1 @ nil_nat )
= zero_zero_nat ) ).
thf(fact_197_ListMem_Osimps,axiom,
! [A1_1: arrow_475358991le_alt,A2_1: list_A2115238852le_alt] :
( ( listMe78047147le_alt @ A1_1 @ A2_1 )
<=> ( ? [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt] :
( ( A1_1 = X_1 )
& ( A2_1
= ( cons_A228743023le_alt @ X_1 @ Xs_54 ) ) )
| ? [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt,Y_3: arrow_475358991le_alt] :
( ( A1_1 = X_1 )
& ( A2_1
= ( cons_A228743023le_alt @ Y_3 @ Xs_54 ) )
& ( listMe78047147le_alt @ X_1 @ Xs_54 ) ) ) ) ).
thf(fact_198_zero__reorient,axiom,
! [X_70: int] :
( ( zero_zero_int = X_70 )
<=> ( X_70 = zero_zero_int ) ) ).
thf(fact_199_zero__reorient,axiom,
! [X_70: nat] :
( ( zero_zero_nat = X_70 )
<=> ( X_70 = zero_zero_nat ) ) ).
thf(fact_200_hd__conv__nth,axiom,
! [Xs_104: list_nat] :
( ( Xs_104 != nil_nat )
=> ( ( hd_nat @ Xs_104 )
= ( nth_nat @ Xs_104 @ zero_zero_nat ) ) ) ).
thf(fact_201_hd__conv__nth,axiom,
! [Xs_104: list_A2115238852le_alt] :
( ( Xs_104 != nil_Ar1286194111le_alt )
=> ( ( hd_Arr1965683346le_alt @ Xs_104 )
= ( nth_Ar125878976le_alt @ Xs_104 @ zero_zero_nat ) ) ) ).
thf(fact_202_takeWhile_Osimps_I1_J,axiom,
! [P_23: nat > $o] :
( ( takeWhile_nat @ P_23 @ nil_nat )
= nil_nat ) ).
thf(fact_203_takeWhile_Osimps_I1_J,axiom,
! [P_23: arrow_475358991le_alt > $o] :
( ( takeWh1696291512le_alt @ P_23 @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_204_distinct__takeWhile,axiom,
! [P_22: arrow_475358991le_alt > $o,Xs_103: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_103 )
=> ( distin236324274le_alt @ ( takeWh1696291512le_alt @ P_22 @ Xs_103 ) ) ) ).
thf(fact_205_sorted__takeWhile,axiom,
! [P_21: nat > $o,Xs_102: list_nat] :
( ( linorder_sorted_nat @ Xs_102 )
=> ( linorder_sorted_nat @ ( takeWhile_nat @ P_21 @ Xs_102 ) ) ) ).
thf(fact_206_sublist__nil,axiom,
! [A_85: nat > $o] :
( ( sublist_nat @ nil_nat @ A_85 )
= nil_nat ) ).
thf(fact_207_sublist__nil,axiom,
! [A_85: nat > $o] :
( ( sublis2047953604le_alt @ nil_Ar1286194111le_alt @ A_85 )
= nil_Ar1286194111le_alt ) ).
thf(fact_208_distinct__sublistI,axiom,
! [I_27: nat > $o,Xs_101: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_101 )
=> ( distin236324274le_alt @ ( sublis2047953604le_alt @ Xs_101 @ I_27 ) ) ) ).
thf(fact_209_nth__Cons__0,axiom,
! [X_69: nat,Xs_100: list_nat] :
( ( nth_nat @ ( cons_nat @ X_69 @ Xs_100 ) @ zero_zero_nat )
= X_69 ) ).
thf(fact_210_nth__Cons__0,axiom,
! [X_69: arrow_475358991le_alt,Xs_100: list_A2115238852le_alt] :
( ( nth_Ar125878976le_alt @ ( cons_A228743023le_alt @ X_69 @ Xs_100 ) @ zero_zero_nat )
= X_69 ) ).
thf(fact_211_takeWhile_Osimps_I2_J,axiom,
! [Xs_99: list_nat,P_20: nat > $o,X_68: nat] :
( ( ( P_20 @ X_68 )
=> ( ( takeWhile_nat @ P_20 @ ( cons_nat @ X_68 @ Xs_99 ) )
= ( cons_nat @ X_68 @ ( takeWhile_nat @ P_20 @ Xs_99 ) ) ) )
& ( ~ ( P_20 @ X_68 )
=> ( ( takeWhile_nat @ P_20 @ ( cons_nat @ X_68 @ Xs_99 ) )
= nil_nat ) ) ) ).
thf(fact_212_takeWhile_Osimps_I2_J,axiom,
! [Xs_99: list_A2115238852le_alt,P_20: arrow_475358991le_alt > $o,X_68: arrow_475358991le_alt] :
( ( ( P_20 @ X_68 )
=> ( ( takeWh1696291512le_alt @ P_20 @ ( cons_A228743023le_alt @ X_68 @ Xs_99 ) )
= ( cons_A228743023le_alt @ X_68 @ ( takeWh1696291512le_alt @ P_20 @ Xs_99 ) ) ) )
& ( ~ ( P_20 @ X_68 )
=> ( ( takeWh1696291512le_alt @ P_20 @ ( cons_A228743023le_alt @ X_68 @ Xs_99 ) )
= nil_Ar1286194111le_alt ) ) ) ).
thf(fact_213_takeWhile__tail,axiom,
! [Xs_98: list_A2115238852le_alt,L_1: list_A2115238852le_alt,P_19: arrow_475358991le_alt > $o,X_67: arrow_475358991le_alt] :
( ~ ( P_19 @ X_67 )
=> ( ( takeWh1696291512le_alt @ P_19 @ ( append179082452le_alt @ Xs_98 @ ( cons_A228743023le_alt @ X_67 @ L_1 ) ) )
= ( takeWh1696291512le_alt @ P_19 @ Xs_98 ) ) ) ).
thf(fact_214_takeWhile__dropWhile__id,axiom,
! [P_18: arrow_475358991le_alt > $o,Xs_97: list_A2115238852le_alt] :
( ( append179082452le_alt @ ( takeWh1696291512le_alt @ P_18 @ Xs_97 ) @ ( dropWh1316781920le_alt @ P_18 @ Xs_97 ) )
= Xs_97 ) ).
thf(fact_215_nth__append__length,axiom,
! [Xs_96: list_nat,X_66: nat,Ys_20: list_nat] :
( ( nth_nat @ ( append_nat @ Xs_96 @ ( cons_nat @ X_66 @ Ys_20 ) ) @ ( size_size_list_nat @ Xs_96 ) )
= X_66 ) ).
thf(fact_216_nth__append__length,axiom,
! [Xs_96: list_A2115238852le_alt,X_66: arrow_475358991le_alt,Ys_20: list_A2115238852le_alt] :
( ( nth_Ar125878976le_alt @ ( append179082452le_alt @ Xs_96 @ ( cons_A228743023le_alt @ X_66 @ Ys_20 ) ) @ ( size_s1858781230le_alt @ Xs_96 ) )
= X_66 ) ).
thf(fact_217_foldr__max__sorted,axiom,
! [Y_33: int,Xs_95: list_int] :
( ( linorder_sorted_int @ ( rev_int @ Xs_95 ) )
=> ( ( ( Xs_95 = nil_int )
=> ( ( foldr_int_int @ ord_max_int @ Xs_95 @ Y_33 )
= Y_33 ) )
& ( ( Xs_95 != nil_int )
=> ( ( foldr_int_int @ ord_max_int @ Xs_95 @ Y_33 )
= ( ord_max_int @ ( nth_int @ Xs_95 @ zero_zero_nat ) @ Y_33 ) ) ) ) ) ).
thf(fact_218_foldr__max__sorted,axiom,
! [Y_33: nat,Xs_95: list_nat] :
( ( linorder_sorted_nat @ ( rev_nat @ Xs_95 ) )
=> ( ( ( Xs_95 = nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs_95 @ Y_33 )
= Y_33 ) )
& ( ( Xs_95 != nil_nat )
=> ( ( foldr_nat_nat @ ord_max_nat @ Xs_95 @ Y_33 )
= ( ord_max_nat @ ( nth_nat @ Xs_95 @ zero_zero_nat ) @ Y_33 ) ) ) ) ) ).
thf(fact_219_list__induct2_H,axiom,
! [Xs_94: list_A2115238852le_alt,Ys_19: list_nat,P_17: list_A2115238852le_alt > list_nat > $o] :
( ( P_17 @ nil_Ar1286194111le_alt @ nil_nat )
=> ( ! [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt] : ( P_17 @ ( cons_A228743023le_alt @ X_1 @ Xs_54 ) @ nil_nat )
=> ( ! [Y_3: nat,Ys_5: list_nat] : ( P_17 @ nil_Ar1286194111le_alt @ ( cons_nat @ Y_3 @ Ys_5 ) )
=> ( ! [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt,Y_3: nat,Ys_5: list_nat] :
( ( P_17 @ Xs_54 @ Ys_5 )
=> ( P_17 @ ( cons_A228743023le_alt @ X_1 @ Xs_54 ) @ ( cons_nat @ Y_3 @ Ys_5 ) ) )
=> ( P_17 @ Xs_94 @ Ys_19 ) ) ) ) ) ).
thf(fact_220_list__induct2_H,axiom,
! [Xs_94: list_nat,Ys_19: list_A2115238852le_alt,P_17: list_nat > list_A2115238852le_alt > $o] :
( ( P_17 @ nil_nat @ nil_Ar1286194111le_alt )
=> ( ! [X_1: nat,Xs_54: list_nat] : ( P_17 @ ( cons_nat @ X_1 @ Xs_54 ) @ nil_Ar1286194111le_alt )
=> ( ! [Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] : ( P_17 @ nil_nat @ ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) )
=> ( ! [X_1: nat,Xs_54: list_nat,Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] :
( ( P_17 @ Xs_54 @ Ys_5 )
=> ( P_17 @ ( cons_nat @ X_1 @ Xs_54 ) @ ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) ) )
=> ( P_17 @ Xs_94 @ Ys_19 ) ) ) ) ) ).
thf(fact_221_map__filter__simps_I2_J,axiom,
! [F_5: arrow_475358991le_alt > option_nat] :
( ( map_fi312151301lt_nat @ F_5 @ nil_Ar1286194111le_alt )
= nil_nat ) ).
thf(fact_222_map__filter__simps_I2_J,axiom,
! [F_5: nat > option514678660le_alt] :
( ( map_fi1462697295le_alt @ F_5 @ nil_nat )
= nil_Ar1286194111le_alt ) ).
thf(fact_223_listrelp_ONil,axiom,
! [R_2: nat > nat > $o] : ( listrelp_nat @ R_2 @ nil_nat @ nil_nat ) ).
thf(fact_224_listrelp_ONil,axiom,
! [R_2: arrow_475358991le_alt > arrow_475358991le_alt > $o] : ( listre1213162009le_alt @ R_2 @ nil_Ar1286194111le_alt @ nil_Ar1286194111le_alt ) ).
thf(fact_225_listrelp_OCons,axiom,
! [Xs_93: list_A2115238852le_alt,Ys_18: list_A2115238852le_alt,R_1: arrow_475358991le_alt > arrow_475358991le_alt > $o,X_65: arrow_475358991le_alt,Y_32: arrow_475358991le_alt] :
( ( R_1 @ X_65 @ Y_32 )
=> ( ( listre1213162009le_alt @ R_1 @ Xs_93 @ Ys_18 )
=> ( listre1213162009le_alt @ R_1 @ ( cons_A228743023le_alt @ X_65 @ Xs_93 ) @ ( cons_A228743023le_alt @ Y_32 @ Ys_18 ) ) ) ) ).
thf(fact_226_sublist__empty,axiom,
! [Xs_92: list_nat] :
( ( sublist_nat @ Xs_92 @ bot_bot_nat_o )
= nil_nat ) ).
thf(fact_227_sublist__empty,axiom,
! [Xs_92: list_A2115238852le_alt] :
( ( sublis2047953604le_alt @ Xs_92 @ bot_bot_nat_o )
= nil_Ar1286194111le_alt ) ).
thf(fact_228_neq__if__length__neq,axiom,
! [Xs_91: list_A2115238852le_alt,Ys_17: list_A2115238852le_alt] :
( ( ( size_s1858781230le_alt @ Xs_91 )
!= ( size_s1858781230le_alt @ Ys_17 ) )
=> ( Xs_91 != Ys_17 ) ) ).
thf(fact_229_length__rev,axiom,
! [Xs_90: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs_90 ) )
= ( size_size_list_nat @ Xs_90 ) ) ).
thf(fact_230_length__rev,axiom,
! [Xs_90: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( rev_Ar1106406943le_alt @ Xs_90 ) )
= ( size_s1858781230le_alt @ Xs_90 ) ) ).
thf(fact_231_length__replicate,axiom,
! [N_45: nat,X_64: arrow_475358991le_alt] :
( ( size_s1858781230le_alt @ ( replic1511538809le_alt @ N_45 @ X_64 ) )
= N_45 ) ).
thf(fact_232_length__rotate1,axiom,
! [Xs_89: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( rotate335349260le_alt @ Xs_89 ) )
= ( size_s1858781230le_alt @ Xs_89 ) ) ).
thf(fact_233_foldr_Osimps_I2_J,axiom,
! [F_4: nat > nat > nat,X_63: nat,Xs_88: list_nat,A_84: nat] :
( ( foldr_nat_nat @ F_4 @ ( cons_nat @ X_63 @ Xs_88 ) @ A_84 )
= ( F_4 @ X_63 @ ( foldr_nat_nat @ F_4 @ Xs_88 @ A_84 ) ) ) ).
thf(fact_234_foldr_Osimps_I1_J,axiom,
! [F_3: nat > nat > nat,A_83: nat] :
( ( foldr_nat_nat @ F_3 @ nil_nat @ A_83 )
= A_83 ) ).
thf(fact_235_foldr__append,axiom,
! [F_2: nat > nat > nat,Xs_87: list_nat,Ys_16: list_nat,A_82: nat] :
( ( foldr_nat_nat @ F_2 @ ( append_nat @ Xs_87 @ Ys_16 ) @ A_82 )
= ( foldr_nat_nat @ F_2 @ Xs_87 @ ( foldr_nat_nat @ F_2 @ Ys_16 @ A_82 ) ) ) ).
thf(fact_236_length__0__conv,axiom,
! [Xs_86: list_nat] :
( ( ( size_size_list_nat @ Xs_86 )
= zero_zero_nat )
<=> ( Xs_86 = nil_nat ) ) ).
thf(fact_237_length__0__conv,axiom,
! [Xs_86: list_A2115238852le_alt] :
( ( ( size_s1858781230le_alt @ Xs_86 )
= zero_zero_nat )
<=> ( Xs_86 = nil_Ar1286194111le_alt ) ) ).
thf(fact_238_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
thf(fact_239_list_Osize_I3_J,axiom,
( ( size_s1858781230le_alt @ nil_Ar1286194111le_alt )
= zero_zero_nat ) ).
thf(fact_240_list__induct2,axiom,
! [P_16: list_A2115238852le_alt > list_nat > $o,Xs_85: list_A2115238852le_alt,Ys_15: list_nat] :
( ( ( size_s1858781230le_alt @ Xs_85 )
= ( size_size_list_nat @ Ys_15 ) )
=> ( ( P_16 @ nil_Ar1286194111le_alt @ nil_nat )
=> ( ! [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt,Y_3: nat,Ys_5: list_nat] :
( ( ( size_s1858781230le_alt @ Xs_54 )
= ( size_size_list_nat @ Ys_5 ) )
=> ( ( P_16 @ Xs_54 @ Ys_5 )
=> ( P_16 @ ( cons_A228743023le_alt @ X_1 @ Xs_54 ) @ ( cons_nat @ Y_3 @ Ys_5 ) ) ) )
=> ( P_16 @ Xs_85 @ Ys_15 ) ) ) ) ).
thf(fact_241_list__induct2,axiom,
! [P_16: list_nat > list_A2115238852le_alt > $o,Xs_85: list_nat,Ys_15: list_A2115238852le_alt] :
( ( ( size_size_list_nat @ Xs_85 )
= ( size_s1858781230le_alt @ Ys_15 ) )
=> ( ( P_16 @ nil_nat @ nil_Ar1286194111le_alt )
=> ( ! [X_1: nat,Xs_54: list_nat,Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] :
( ( ( size_size_list_nat @ Xs_54 )
= ( size_s1858781230le_alt @ Ys_5 ) )
=> ( ( P_16 @ Xs_54 @ Ys_5 )
=> ( P_16 @ ( cons_nat @ X_1 @ Xs_54 ) @ ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) ) ) )
=> ( P_16 @ Xs_85 @ Ys_15 ) ) ) ) ).
thf(fact_242_list__induct2,axiom,
! [P_16: list_A2115238852le_alt > list_A2115238852le_alt > $o,Xs_85: list_A2115238852le_alt,Ys_15: list_A2115238852le_alt] :
( ( ( size_s1858781230le_alt @ Xs_85 )
= ( size_s1858781230le_alt @ Ys_15 ) )
=> ( ( P_16 @ nil_Ar1286194111le_alt @ nil_Ar1286194111le_alt )
=> ( ! [X_1: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt,Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] :
( ( ( size_s1858781230le_alt @ Xs_54 )
= ( size_s1858781230le_alt @ Ys_5 ) )
=> ( ( P_16 @ Xs_54 @ Ys_5 )
=> ( P_16 @ ( cons_A228743023le_alt @ X_1 @ Xs_54 ) @ ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) ) ) )
=> ( P_16 @ Xs_85 @ Ys_15 ) ) ) ) ).
thf(fact_243_emptyE,axiom,
! [A_81: int] :
~ ( member_int @ A_81 @ bot_bot_int_o ) ).
thf(fact_244_emptyE,axiom,
! [A_81: nat] :
~ ( member_nat @ A_81 @ bot_bot_nat_o ) ).
thf(fact_245_bot__fun__def,axiom,
! [X_1: nat] :
( ( bot_bot_nat_o @ X_1 )
<=> bot_bot_o ) ).
thf(fact_246_bot__apply,axiom,
! [X_62: nat] :
( ( bot_bot_nat_o @ X_62 )
<=> bot_bot_o ) ).
thf(fact_247_Ex__list__of__length,axiom,
! [N_44: nat] :
? [Xs_54: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ Xs_54 )
= N_44 ) ).
thf(fact_248_listrelp_Osimps,axiom,
! [R: nat > nat > $o,A1: list_nat,A2: list_nat] :
( ( listrelp_nat @ R @ A1 @ A2 )
<=> ( ( ( A1 = nil_nat )
& ( A2 = nil_nat ) )
| ? [X_1: nat,Y_3: nat,Xs_54: list_nat,Ys_5: list_nat] :
( ( A1
= ( cons_nat @ X_1 @ Xs_54 ) )
& ( A2
= ( cons_nat @ Y_3 @ Ys_5 ) )
& ( R @ X_1 @ Y_3 )
& ( listrelp_nat @ R @ Xs_54 @ Ys_5 ) ) ) ) ).
thf(fact_249_listrelp_Osimps,axiom,
! [R: arrow_475358991le_alt > arrow_475358991le_alt > $o,A1: list_A2115238852le_alt,A2: list_A2115238852le_alt] :
( ( listre1213162009le_alt @ R @ A1 @ A2 )
<=> ( ( ( A1 = nil_Ar1286194111le_alt )
& ( A2 = nil_Ar1286194111le_alt ) )
| ? [X_1: arrow_475358991le_alt,Y_3: arrow_475358991le_alt,Xs_54: list_A2115238852le_alt,Ys_5: list_A2115238852le_alt] :
( ( A1
= ( cons_A228743023le_alt @ X_1 @ Xs_54 ) )
& ( A2
= ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) )
& ( R @ X_1 @ Y_3 )
& ( listre1213162009le_alt @ R @ Xs_54 @ Ys_5 ) ) ) ) ).
thf(fact_250_equals0D,axiom,
! [A_80: int,A_79: int > $o] :
( ( A_79 = bot_bot_int_o )
=> ~ ( member_int @ A_80 @ A_79 ) ) ).
thf(fact_251_equals0D,axiom,
! [A_80: nat,A_79: nat > $o] :
( ( A_79 = bot_bot_nat_o )
=> ~ ( member_nat @ A_80 @ A_79 ) ) ).
thf(fact_252_Collect__empty__eq,axiom,
! [P_15: nat > $o] :
( ( ( collect_nat @ P_15 )
= bot_bot_nat_o )
<=> ! [X_1: nat] :
~ ( P_15 @ X_1 ) ) ).
thf(fact_253_empty__iff,axiom,
! [C_36: int] :
~ ( member_int @ C_36 @ bot_bot_int_o ) ).
thf(fact_254_empty__iff,axiom,
! [C_36: nat] :
~ ( member_nat @ C_36 @ bot_bot_nat_o ) ).
thf(fact_255_empty__Collect__eq,axiom,
! [P_14: nat > $o] :
( ( bot_bot_nat_o
= ( collect_nat @ P_14 ) )
<=> ! [X_1: nat] :
~ ( P_14 @ X_1 ) ) ).
thf(fact_256_ex__in__conv,axiom,
! [A_78: int > $o] :
( ? [X_1: int] : ( member_int @ X_1 @ A_78 )
<=> ( A_78 != bot_bot_int_o ) ) ).
thf(fact_257_ex__in__conv,axiom,
! [A_78: nat > $o] :
( ? [X_1: nat] : ( member_nat @ X_1 @ A_78 )
<=> ( A_78 != bot_bot_nat_o ) ) ).
thf(fact_258_all__not__in__conv,axiom,
! [A_77: int > $o] :
( ! [X_1: int] :
~ ( member_int @ X_1 @ A_77 )
<=> ( A_77 = bot_bot_int_o ) ) ).
thf(fact_259_all__not__in__conv,axiom,
! [A_77: nat > $o] :
( ! [X_1: nat] :
~ ( member_nat @ X_1 @ A_77 )
<=> ( A_77 = bot_bot_nat_o ) ) ).
thf(fact_260_empty__def,axiom,
( bot_bot_nat_o
= ( collect_nat
@ ^ [X_1: nat] : $false ) ) ).
thf(fact_261_equals0I,axiom,
! [A_76: int > $o] :
( ! [Y_3: int] :
~ ( member_int @ Y_3 @ A_76 )
=> ( A_76 = bot_bot_int_o ) ) ).
thf(fact_262_equals0I,axiom,
! [A_76: nat > $o] :
( ! [Y_3: nat] :
~ ( member_nat @ Y_3 @ A_76 )
=> ( A_76 = bot_bot_nat_o ) ) ).
thf(fact_263_sorted__list__of__set__empty,axiom,
( ( linord810813188et_nat @ bot_bot_nat_o )
= nil_nat ) ).
thf(fact_264_min__max_Osup__assoc,axiom,
! [X_61: int,Y_31: int,Z_8: int] :
( ( ord_max_int @ ( ord_max_int @ X_61 @ Y_31 ) @ Z_8 )
= ( ord_max_int @ X_61 @ ( ord_max_int @ Y_31 @ Z_8 ) ) ) ).
thf(fact_265_min__max_Osup__assoc,axiom,
! [X_61: nat,Y_31: nat,Z_8: nat] :
( ( ord_max_nat @ ( ord_max_nat @ X_61 @ Y_31 ) @ Z_8 )
= ( ord_max_nat @ X_61 @ ( ord_max_nat @ Y_31 @ Z_8 ) ) ) ).
thf(fact_266_min__max_Osup_Oassoc,axiom,
! [A_75: int,B_47: int,C_35: int] :
( ( ord_max_int @ ( ord_max_int @ A_75 @ B_47 ) @ C_35 )
= ( ord_max_int @ A_75 @ ( ord_max_int @ B_47 @ C_35 ) ) ) ).
thf(fact_267_min__max_Osup_Oassoc,axiom,
! [A_75: nat,B_47: nat,C_35: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A_75 @ B_47 ) @ C_35 )
= ( ord_max_nat @ A_75 @ ( ord_max_nat @ B_47 @ C_35 ) ) ) ).
thf(fact_268_min__max_Osup__left__commute,axiom,
! [X_60: int,Y_30: int,Z_7: int] :
( ( ord_max_int @ X_60 @ ( ord_max_int @ Y_30 @ Z_7 ) )
= ( ord_max_int @ Y_30 @ ( ord_max_int @ X_60 @ Z_7 ) ) ) ).
thf(fact_269_min__max_Osup__left__commute,axiom,
! [X_60: nat,Y_30: nat,Z_7: nat] :
( ( ord_max_nat @ X_60 @ ( ord_max_nat @ Y_30 @ Z_7 ) )
= ( ord_max_nat @ Y_30 @ ( ord_max_nat @ X_60 @ Z_7 ) ) ) ).
thf(fact_270_min__max_Osup_Oleft__commute,axiom,
! [B_46: int,A_74: int,C_34: int] :
( ( ord_max_int @ B_46 @ ( ord_max_int @ A_74 @ C_34 ) )
= ( ord_max_int @ A_74 @ ( ord_max_int @ B_46 @ C_34 ) ) ) ).
thf(fact_271_min__max_Osup_Oleft__commute,axiom,
! [B_46: nat,A_74: nat,C_34: nat] :
( ( ord_max_nat @ B_46 @ ( ord_max_nat @ A_74 @ C_34 ) )
= ( ord_max_nat @ A_74 @ ( ord_max_nat @ B_46 @ C_34 ) ) ) ).
thf(fact_272_min__max_Osup_Oidem,axiom,
! [A_73: int] :
( ( ord_max_int @ A_73 @ A_73 )
= A_73 ) ).
thf(fact_273_min__max_Osup_Oidem,axiom,
! [A_73: nat] :
( ( ord_max_nat @ A_73 @ A_73 )
= A_73 ) ).
thf(fact_274_min__max_Osup_Ocommute,axiom,
! [A_72: int,B_45: int] :
( ( ord_max_int @ A_72 @ B_45 )
= ( ord_max_int @ B_45 @ A_72 ) ) ).
thf(fact_275_min__max_Osup_Ocommute,axiom,
! [A_72: nat,B_45: nat] :
( ( ord_max_nat @ A_72 @ B_45 )
= ( ord_max_nat @ B_45 @ A_72 ) ) ).
thf(fact_276_min__max_Osup__commute,axiom,
! [X_59: int,Y_29: int] :
( ( ord_max_int @ X_59 @ Y_29 )
= ( ord_max_int @ Y_29 @ X_59 ) ) ).
thf(fact_277_min__max_Osup__commute,axiom,
! [X_59: nat,Y_29: nat] :
( ( ord_max_nat @ X_59 @ Y_29 )
= ( ord_max_nat @ Y_29 @ X_59 ) ) ).
thf(fact_278_min__max_Osup_Oleft__idem,axiom,
! [A_71: int,B_44: int] :
( ( ord_max_int @ A_71 @ ( ord_max_int @ A_71 @ B_44 ) )
= ( ord_max_int @ A_71 @ B_44 ) ) ).
thf(fact_279_min__max_Osup_Oleft__idem,axiom,
! [A_71: nat,B_44: nat] :
( ( ord_max_nat @ A_71 @ ( ord_max_nat @ A_71 @ B_44 ) )
= ( ord_max_nat @ A_71 @ B_44 ) ) ).
thf(fact_280_min__max_Osup__left__idem,axiom,
! [X_58: int,Y_28: int] :
( ( ord_max_int @ X_58 @ ( ord_max_int @ X_58 @ Y_28 ) )
= ( ord_max_int @ X_58 @ Y_28 ) ) ).
thf(fact_281_min__max_Osup__left__idem,axiom,
! [X_58: nat,Y_28: nat] :
( ( ord_max_nat @ X_58 @ ( ord_max_nat @ X_58 @ Y_28 ) )
= ( ord_max_nat @ X_58 @ Y_28 ) ) ).
thf(fact_282_Max_Oidem,axiom,
! [X_57: int] :
( ( ord_max_int @ X_57 @ X_57 )
= X_57 ) ).
thf(fact_283_Max_Oidem,axiom,
! [X_57: nat] :
( ( ord_max_nat @ X_57 @ X_57 )
= X_57 ) ).
thf(fact_284_dropWhile__eq__drop,axiom,
! [P_13: arrow_475358991le_alt > $o,Xs_84: list_A2115238852le_alt] :
( ( dropWh1316781920le_alt @ P_13 @ Xs_84 )
= ( drop_A1346709759le_alt @ ( size_s1858781230le_alt @ ( takeWh1696291512le_alt @ P_13 @ Xs_84 ) ) @ Xs_84 ) ) ).
thf(fact_285_nth__length__takeWhile,axiom,
! [P_12: nat > $o,Xs_83: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P_12 @ Xs_83 ) ) @ ( size_size_list_nat @ Xs_83 ) )
=> ~ ( P_12 @ ( nth_nat @ Xs_83 @ ( size_size_list_nat @ ( takeWhile_nat @ P_12 @ Xs_83 ) ) ) ) ) ).
thf(fact_286_nth__length__takeWhile,axiom,
! [P_12: arrow_475358991le_alt > $o,Xs_83: list_A2115238852le_alt] :
( ( ord_less_nat @ ( size_s1858781230le_alt @ ( takeWh1696291512le_alt @ P_12 @ Xs_83 ) ) @ ( size_s1858781230le_alt @ Xs_83 ) )
=> ~ ( P_12 @ ( nth_Ar125878976le_alt @ Xs_83 @ ( size_s1858781230le_alt @ ( takeWh1696291512le_alt @ P_12 @ Xs_83 ) ) ) ) ) ).
thf(fact_287_takeWhile__nth,axiom,
! [J_7: nat,P_11: nat > $o,Xs_82: list_nat] :
( ( ord_less_nat @ J_7 @ ( size_size_list_nat @ ( takeWhile_nat @ P_11 @ Xs_82 ) ) )
=> ( ( nth_nat @ ( takeWhile_nat @ P_11 @ Xs_82 ) @ J_7 )
= ( nth_nat @ Xs_82 @ J_7 ) ) ) ).
thf(fact_288_takeWhile__nth,axiom,
! [J_7: nat,P_11: arrow_475358991le_alt > $o,Xs_82: list_A2115238852le_alt] :
( ( ord_less_nat @ J_7 @ ( size_s1858781230le_alt @ ( takeWh1696291512le_alt @ P_11 @ Xs_82 ) ) )
=> ( ( nth_Ar125878976le_alt @ ( takeWh1696291512le_alt @ P_11 @ Xs_82 ) @ J_7 )
= ( nth_Ar125878976le_alt @ Xs_82 @ J_7 ) ) ) ).
thf(fact_289_list__update__length,axiom,
! [Xs_81: list_A2115238852le_alt,X_56: arrow_475358991le_alt,Ys_14: list_A2115238852le_alt,Y_27: arrow_475358991le_alt] :
( ( list_u62477863le_alt @ ( append179082452le_alt @ Xs_81 @ ( cons_A228743023le_alt @ X_56 @ Ys_14 ) ) @ ( size_s1858781230le_alt @ Xs_81 ) @ Y_27 )
= ( append179082452le_alt @ Xs_81 @ ( cons_A228743023le_alt @ Y_27 @ Ys_14 ) ) ) ).
thf(fact_290_max__0L,axiom,
! [N_5: nat] :
( ( ord_max_nat @ zero_zero_nat @ N_5 )
= N_5 ) ).
thf(fact_291_less__zeroE,axiom,
! [N_5: nat] :
~ ( ord_less_nat @ N_5 @ zero_zero_nat ) ).
thf(fact_292_list__update__overwrite,axiom,
! [Xs_80: list_A2115238852le_alt,I_26: nat,X_55: arrow_475358991le_alt,Y_26: arrow_475358991le_alt] :
( ( list_u62477863le_alt @ ( list_u62477863le_alt @ Xs_80 @ I_26 @ X_55 ) @ I_26 @ Y_26 )
= ( list_u62477863le_alt @ Xs_80 @ I_26 @ Y_26 ) ) ).
thf(fact_293_list__update__swap,axiom,
! [Xs_79: list_A2115238852le_alt,X_54: arrow_475358991le_alt,X_53: arrow_475358991le_alt,I_25: nat,I_24: nat] :
( ( I_25 != I_24 )
=> ( ( list_u62477863le_alt @ ( list_u62477863le_alt @ Xs_79 @ I_25 @ X_54 ) @ I_24 @ X_53 )
= ( list_u62477863le_alt @ ( list_u62477863le_alt @ Xs_79 @ I_24 @ X_53 ) @ I_25 @ X_54 ) ) ) ).
thf(fact_294_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N_5: nat] :
( ( ( ord_less_nat @ M @ N_5 )
=> ( P @ N_5 @ M ) )
=> ( ( ( M = N_5 )
=> ( P @ N_5 @ M ) )
=> ( ( ( ord_less_nat @ N_5 @ M )
=> ( P @ N_5 @ M ) )
=> ( P @ N_5 @ M ) ) ) ) ).
thf(fact_295_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
thf(fact_296_less__not__refl2,axiom,
! [N_5: nat,M: nat] :
( ( ord_less_nat @ N_5 @ M )
=> ( M != N_5 ) ) ).
thf(fact_297_less__irrefl__nat,axiom,
! [N_5: nat] :
~ ( ord_less_nat @ N_5 @ N_5 ) ).
thf(fact_298_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_299_nat__neq__iff,axiom,
! [M: nat,N_5: nat] :
( ( M != N_5 )
<=> ( ( ord_less_nat @ M @ N_5 )
| ( ord_less_nat @ N_5 @ M ) ) ) ).
thf(fact_300_less__not__refl,axiom,
! [N_5: nat] :
~ ( ord_less_nat @ N_5 @ N_5 ) ).
thf(fact_301_gr0I,axiom,
! [N_5: nat] :
( ( N_5 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N_5 ) ) ).
thf(fact_302_gr__implies__not0,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
=> ( N_5 != zero_zero_nat ) ) ).
thf(fact_303_less__nat__zero__code,axiom,
! [N_5: nat] :
~ ( ord_less_nat @ N_5 @ zero_zero_nat ) ).
thf(fact_304_neq0__conv,axiom,
! [N_5: nat] :
( ( N_5 != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N_5 ) ) ).
thf(fact_305_not__less0,axiom,
! [N_5: nat] :
~ ( ord_less_nat @ N_5 @ zero_zero_nat ) ).
thf(fact_306_order__less__irrefl,axiom,
! [X_52: int] :
~ ( ord_less_int @ X_52 @ X_52 ) ).
thf(fact_307_order__less__irrefl,axiom,
! [X_52: nat > $o] :
~ ( ord_less_nat_o @ X_52 @ X_52 ) ).
thf(fact_308_order__less__irrefl,axiom,
! [X_52: nat] :
~ ( ord_less_nat @ X_52 @ X_52 ) ).
thf(fact_309_linorder__neq__iff,axiom,
! [X_51: int,Y_25: int] :
( ( X_51 != Y_25 )
<=> ( ( ord_less_int @ X_51 @ Y_25 )
| ( ord_less_int @ Y_25 @ X_51 ) ) ) ).
thf(fact_310_linorder__neq__iff,axiom,
! [X_51: nat,Y_25: nat] :
( ( X_51 != Y_25 )
<=> ( ( ord_less_nat @ X_51 @ Y_25 )
| ( ord_less_nat @ Y_25 @ X_51 ) ) ) ).
thf(fact_311_not__less__iff__gr__or__eq,axiom,
! [X_50: int,Y_24: int] :
( ~ ( ord_less_int @ X_50 @ Y_24 )
<=> ( ( ord_less_int @ Y_24 @ X_50 )
| ( X_50 = Y_24 ) ) ) ).
thf(fact_312_not__less__iff__gr__or__eq,axiom,
! [X_50: nat,Y_24: nat] :
( ~ ( ord_less_nat @ X_50 @ Y_24 )
<=> ( ( ord_less_nat @ Y_24 @ X_50 )
| ( X_50 = Y_24 ) ) ) ).
thf(fact_313_linorder__less__linear,axiom,
! [X_49: int,Y_23: int] :
( ( ord_less_int @ X_49 @ Y_23 )
| ( X_49 = Y_23 )
| ( ord_less_int @ Y_23 @ X_49 ) ) ).
thf(fact_314_linorder__less__linear,axiom,
! [X_49: nat,Y_23: nat] :
( ( ord_less_nat @ X_49 @ Y_23 )
| ( X_49 = Y_23 )
| ( ord_less_nat @ Y_23 @ X_49 ) ) ).
thf(fact_315_linorder__antisym__conv3,axiom,
! [Y_22: int,X_48: int] :
( ~ ( ord_less_int @ Y_22 @ X_48 )
=> ( ~ ( ord_less_int @ X_48 @ Y_22 )
<=> ( X_48 = Y_22 ) ) ) ).
thf(fact_316_linorder__antisym__conv3,axiom,
! [Y_22: nat,X_48: nat] :
( ~ ( ord_less_nat @ Y_22 @ X_48 )
=> ( ~ ( ord_less_nat @ X_48 @ Y_22 )
<=> ( X_48 = Y_22 ) ) ) ).
thf(fact_317_linorder__neqE,axiom,
! [X_47: int,Y_21: int] :
( ( X_47 != Y_21 )
=> ( ~ ( ord_less_int @ X_47 @ Y_21 )
=> ( ord_less_int @ Y_21 @ X_47 ) ) ) ).
thf(fact_318_linorder__neqE,axiom,
! [X_47: nat,Y_21: nat] :
( ( X_47 != Y_21 )
=> ( ~ ( ord_less_nat @ X_47 @ Y_21 )
=> ( ord_less_nat @ Y_21 @ X_47 ) ) ) ).
thf(fact_319_less__imp__neq,axiom,
! [X_46: int,Y_20: int] :
( ( ord_less_int @ X_46 @ Y_20 )
=> ( X_46 != Y_20 ) ) ).
thf(fact_320_less__imp__neq,axiom,
! [X_46: nat > $o,Y_20: nat > $o] :
( ( ord_less_nat_o @ X_46 @ Y_20 )
=> ( X_46 != Y_20 ) ) ).
thf(fact_321_less__imp__neq,axiom,
! [X_46: nat,Y_20: nat] :
( ( ord_less_nat @ X_46 @ Y_20 )
=> ( X_46 != Y_20 ) ) ).
thf(fact_322_order__less__not__sym,axiom,
! [X_45: int,Y_19: int] :
( ( ord_less_int @ X_45 @ Y_19 )
=> ~ ( ord_less_int @ Y_19 @ X_45 ) ) ).
thf(fact_323_order__less__not__sym,axiom,
! [X_45: nat > $o,Y_19: nat > $o] :
( ( ord_less_nat_o @ X_45 @ Y_19 )
=> ~ ( ord_less_nat_o @ Y_19 @ X_45 ) ) ).
thf(fact_324_order__less__not__sym,axiom,
! [X_45: nat,Y_19: nat] :
( ( ord_less_nat @ X_45 @ Y_19 )
=> ~ ( ord_less_nat @ Y_19 @ X_45 ) ) ).
thf(fact_325_order__less__imp__not__less,axiom,
! [X_44: int,Y_18: int] :
( ( ord_less_int @ X_44 @ Y_18 )
=> ~ ( ord_less_int @ Y_18 @ X_44 ) ) ).
thf(fact_326_order__less__imp__not__less,axiom,
! [X_44: nat > $o,Y_18: nat > $o] :
( ( ord_less_nat_o @ X_44 @ Y_18 )
=> ~ ( ord_less_nat_o @ Y_18 @ X_44 ) ) ).
thf(fact_327_order__less__imp__not__less,axiom,
! [X_44: nat,Y_18: nat] :
( ( ord_less_nat @ X_44 @ Y_18 )
=> ~ ( ord_less_nat @ Y_18 @ X_44 ) ) ).
thf(fact_328_order__less__imp__not__eq,axiom,
! [X_43: int,Y_17: int] :
( ( ord_less_int @ X_43 @ Y_17 )
=> ( X_43 != Y_17 ) ) ).
thf(fact_329_order__less__imp__not__eq,axiom,
! [X_43: nat > $o,Y_17: nat > $o] :
( ( ord_less_nat_o @ X_43 @ Y_17 )
=> ( X_43 != Y_17 ) ) ).
thf(fact_330_order__less__imp__not__eq,axiom,
! [X_43: nat,Y_17: nat] :
( ( ord_less_nat @ X_43 @ Y_17 )
=> ( X_43 != Y_17 ) ) ).
thf(fact_331_order__less__imp__not__eq2,axiom,
! [X_42: int,Y_16: int] :
( ( ord_less_int @ X_42 @ Y_16 )
=> ( Y_16 != X_42 ) ) ).
thf(fact_332_order__less__imp__not__eq2,axiom,
! [X_42: nat > $o,Y_16: nat > $o] :
( ( ord_less_nat_o @ X_42 @ Y_16 )
=> ( Y_16 != X_42 ) ) ).
thf(fact_333_order__less__imp__not__eq2,axiom,
! [X_42: nat,Y_16: nat] :
( ( ord_less_nat @ X_42 @ Y_16 )
=> ( Y_16 != X_42 ) ) ).
thf(fact_334_order__less__imp__triv,axiom,
! [P_10: $o,X_41: int,Y_15: int] :
( ( ord_less_int @ X_41 @ Y_15 )
=> ( ( ord_less_int @ Y_15 @ X_41 )
=> P_10 ) ) ).
thf(fact_335_order__less__imp__triv,axiom,
! [P_10: $o,X_41: nat > $o,Y_15: nat > $o] :
( ( ord_less_nat_o @ X_41 @ Y_15 )
=> ( ( ord_less_nat_o @ Y_15 @ X_41 )
=> P_10 ) ) ).
thf(fact_336_order__less__imp__triv,axiom,
! [P_10: $o,X_41: nat,Y_15: nat] :
( ( ord_less_nat @ X_41 @ Y_15 )
=> ( ( ord_less_nat @ Y_15 @ X_41 )
=> P_10 ) ) ).
thf(fact_337_order__less__asym_H,axiom,
! [A_70: int,B_43: int] :
( ( ord_less_int @ A_70 @ B_43 )
=> ~ ( ord_less_int @ B_43 @ A_70 ) ) ).
thf(fact_338_order__less__asym_H,axiom,
! [A_70: nat > $o,B_43: nat > $o] :
( ( ord_less_nat_o @ A_70 @ B_43 )
=> ~ ( ord_less_nat_o @ B_43 @ A_70 ) ) ).
thf(fact_339_order__less__asym_H,axiom,
! [A_70: nat,B_43: nat] :
( ( ord_less_nat @ A_70 @ B_43 )
=> ~ ( ord_less_nat @ B_43 @ A_70 ) ) ).
thf(fact_340_xt1_I9_J,axiom,
! [B_42: int,A_69: int] :
( ( ord_less_int @ B_42 @ A_69 )
=> ~ ( ord_less_int @ A_69 @ B_42 ) ) ).
thf(fact_341_xt1_I9_J,axiom,
! [B_42: nat > $o,A_69: nat > $o] :
( ( ord_less_nat_o @ B_42 @ A_69 )
=> ~ ( ord_less_nat_o @ A_69 @ B_42 ) ) ).
thf(fact_342_xt1_I9_J,axiom,
! [B_42: nat,A_69: nat] :
( ( ord_less_nat @ B_42 @ A_69 )
=> ~ ( ord_less_nat @ A_69 @ B_42 ) ) ).
thf(fact_343_ord__eq__less__trans,axiom,
! [C_33: int,A_68: int,B_41: int] :
( ( A_68 = B_41 )
=> ( ( ord_less_int @ B_41 @ C_33 )
=> ( ord_less_int @ A_68 @ C_33 ) ) ) ).
thf(fact_344_ord__eq__less__trans,axiom,
! [C_33: nat > $o,A_68: nat > $o,B_41: nat > $o] :
( ( A_68 = B_41 )
=> ( ( ord_less_nat_o @ B_41 @ C_33 )
=> ( ord_less_nat_o @ A_68 @ C_33 ) ) ) ).
thf(fact_345_ord__eq__less__trans,axiom,
! [C_33: nat,A_68: nat,B_41: nat] :
( ( A_68 = B_41 )
=> ( ( ord_less_nat @ B_41 @ C_33 )
=> ( ord_less_nat @ A_68 @ C_33 ) ) ) ).
thf(fact_346_xt1_I1_J,axiom,
! [C_32: int,A_67: int,B_40: int] :
( ( A_67 = B_40 )
=> ( ( ord_less_int @ C_32 @ B_40 )
=> ( ord_less_int @ C_32 @ A_67 ) ) ) ).
thf(fact_347_xt1_I1_J,axiom,
! [C_32: nat > $o,A_67: nat > $o,B_40: nat > $o] :
( ( A_67 = B_40 )
=> ( ( ord_less_nat_o @ C_32 @ B_40 )
=> ( ord_less_nat_o @ C_32 @ A_67 ) ) ) ).
thf(fact_348_xt1_I1_J,axiom,
! [C_32: nat,A_67: nat,B_40: nat] :
( ( A_67 = B_40 )
=> ( ( ord_less_nat @ C_32 @ B_40 )
=> ( ord_less_nat @ C_32 @ A_67 ) ) ) ).
thf(fact_349_ord__less__eq__trans,axiom,
! [C_31: int,A_66: int,B_39: int] :
( ( ord_less_int @ A_66 @ B_39 )
=> ( ( B_39 = C_31 )
=> ( ord_less_int @ A_66 @ C_31 ) ) ) ).
thf(fact_350_ord__less__eq__trans,axiom,
! [C_31: nat > $o,A_66: nat > $o,B_39: nat > $o] :
( ( ord_less_nat_o @ A_66 @ B_39 )
=> ( ( B_39 = C_31 )
=> ( ord_less_nat_o @ A_66 @ C_31 ) ) ) ).
thf(fact_351_ord__less__eq__trans,axiom,
! [C_31: nat,A_66: nat,B_39: nat] :
( ( ord_less_nat @ A_66 @ B_39 )
=> ( ( B_39 = C_31 )
=> ( ord_less_nat @ A_66 @ C_31 ) ) ) ).
thf(fact_352_xt1_I2_J,axiom,
! [C_30: int,B_38: int,A_65: int] :
( ( ord_less_int @ B_38 @ A_65 )
=> ( ( B_38 = C_30 )
=> ( ord_less_int @ C_30 @ A_65 ) ) ) ).
thf(fact_353_xt1_I2_J,axiom,
! [C_30: nat > $o,B_38: nat > $o,A_65: nat > $o] :
( ( ord_less_nat_o @ B_38 @ A_65 )
=> ( ( B_38 = C_30 )
=> ( ord_less_nat_o @ C_30 @ A_65 ) ) ) ).
thf(fact_354_xt1_I2_J,axiom,
! [C_30: nat,B_38: nat,A_65: nat] :
( ( ord_less_nat @ B_38 @ A_65 )
=> ( ( B_38 = C_30 )
=> ( ord_less_nat @ C_30 @ A_65 ) ) ) ).
thf(fact_355_order__less__trans,axiom,
! [Z_6: int,X_40: int,Y_14: int] :
( ( ord_less_int @ X_40 @ Y_14 )
=> ( ( ord_less_int @ Y_14 @ Z_6 )
=> ( ord_less_int @ X_40 @ Z_6 ) ) ) ).
thf(fact_356_order__less__trans,axiom,
! [Z_6: nat > $o,X_40: nat > $o,Y_14: nat > $o] :
( ( ord_less_nat_o @ X_40 @ Y_14 )
=> ( ( ord_less_nat_o @ Y_14 @ Z_6 )
=> ( ord_less_nat_o @ X_40 @ Z_6 ) ) ) ).
thf(fact_357_order__less__trans,axiom,
! [Z_6: nat,X_40: nat,Y_14: nat] :
( ( ord_less_nat @ X_40 @ Y_14 )
=> ( ( ord_less_nat @ Y_14 @ Z_6 )
=> ( ord_less_nat @ X_40 @ Z_6 ) ) ) ).
thf(fact_358_xt1_I10_J,axiom,
! [Z_5: int,Y_13: int,X_39: int] :
( ( ord_less_int @ Y_13 @ X_39 )
=> ( ( ord_less_int @ Z_5 @ Y_13 )
=> ( ord_less_int @ Z_5 @ X_39 ) ) ) ).
thf(fact_359_xt1_I10_J,axiom,
! [Z_5: nat > $o,Y_13: nat > $o,X_39: nat > $o] :
( ( ord_less_nat_o @ Y_13 @ X_39 )
=> ( ( ord_less_nat_o @ Z_5 @ Y_13 )
=> ( ord_less_nat_o @ Z_5 @ X_39 ) ) ) ).
thf(fact_360_xt1_I10_J,axiom,
! [Z_5: nat,Y_13: nat,X_39: nat] :
( ( ord_less_nat @ Y_13 @ X_39 )
=> ( ( ord_less_nat @ Z_5 @ Y_13 )
=> ( ord_less_nat @ Z_5 @ X_39 ) ) ) ).
thf(fact_361_order__less__asym,axiom,
! [X_38: int,Y_12: int] :
( ( ord_less_int @ X_38 @ Y_12 )
=> ~ ( ord_less_int @ Y_12 @ X_38 ) ) ).
thf(fact_362_order__less__asym,axiom,
! [X_38: nat > $o,Y_12: nat > $o] :
( ( ord_less_nat_o @ X_38 @ Y_12 )
=> ~ ( ord_less_nat_o @ Y_12 @ X_38 ) ) ).
thf(fact_363_order__less__asym,axiom,
! [X_38: nat,Y_12: nat] :
( ( ord_less_nat @ X_38 @ Y_12 )
=> ~ ( ord_less_nat @ Y_12 @ X_38 ) ) ).
thf(fact_364_linorder__cases,axiom,
! [X_37: int,Y_11: int] :
( ~ ( ord_less_int @ X_37 @ Y_11 )
=> ( ( X_37 != Y_11 )
=> ( ord_less_int @ Y_11 @ X_37 ) ) ) ).
thf(fact_365_linorder__cases,axiom,
! [X_37: nat,Y_11: nat] :
( ~ ( ord_less_nat @ X_37 @ Y_11 )
=> ( ( X_37 != Y_11 )
=> ( ord_less_nat @ Y_11 @ X_37 ) ) ) ).
thf(fact_366_list__update__append1,axiom,
! [Ys_13: list_A2115238852le_alt,X_36: arrow_475358991le_alt,I_23: nat,Xs_78: list_A2115238852le_alt] :
( ( ord_less_nat @ I_23 @ ( size_s1858781230le_alt @ Xs_78 ) )
=> ( ( list_u62477863le_alt @ ( append179082452le_alt @ Xs_78 @ Ys_13 ) @ I_23 @ X_36 )
= ( append179082452le_alt @ ( list_u62477863le_alt @ Xs_78 @ I_23 @ X_36 ) @ Ys_13 ) ) ) ).
thf(fact_367_nth__list__update,axiom,
! [X_35: arrow_475358991le_alt,J_6: nat,I_22: nat,Xs_77: list_A2115238852le_alt] :
( ( ord_less_nat @ I_22 @ ( size_s1858781230le_alt @ Xs_77 ) )
=> ( ( ( I_22 = J_6 )
=> ( ( nth_Ar125878976le_alt @ ( list_u62477863le_alt @ Xs_77 @ I_22 @ X_35 ) @ J_6 )
= X_35 ) )
& ( ( I_22 != J_6 )
=> ( ( nth_Ar125878976le_alt @ ( list_u62477863le_alt @ Xs_77 @ I_22 @ X_35 ) @ J_6 )
= ( nth_Ar125878976le_alt @ Xs_77 @ J_6 ) ) ) ) ) ).
thf(fact_368_nth__list__update,axiom,
! [X_35: nat,J_6: nat,I_22: nat,Xs_77: list_nat] :
( ( ord_less_nat @ I_22 @ ( size_size_list_nat @ Xs_77 ) )
=> ( ( ( I_22 = J_6 )
=> ( ( nth_nat @ ( list_update_nat @ Xs_77 @ I_22 @ X_35 ) @ J_6 )
= X_35 ) )
& ( ( I_22 != J_6 )
=> ( ( nth_nat @ ( list_update_nat @ Xs_77 @ I_22 @ X_35 ) @ J_6 )
= ( nth_nat @ Xs_77 @ J_6 ) ) ) ) ) ).
thf(fact_369_list__update__same__conv,axiom,
! [X_34: arrow_475358991le_alt,I_21: nat,Xs_76: list_A2115238852le_alt] :
( ( ord_less_nat @ I_21 @ ( size_s1858781230le_alt @ Xs_76 ) )
=> ( ( ( list_u62477863le_alt @ Xs_76 @ I_21 @ X_34 )
= Xs_76 )
<=> ( ( nth_Ar125878976le_alt @ Xs_76 @ I_21 )
= X_34 ) ) ) ).
thf(fact_370_list__update__same__conv,axiom,
! [X_34: nat,I_21: nat,Xs_76: list_nat] :
( ( ord_less_nat @ I_21 @ ( size_size_list_nat @ Xs_76 ) )
=> ( ( ( list_update_nat @ Xs_76 @ I_21 @ X_34 )
= Xs_76 )
<=> ( ( nth_nat @ Xs_76 @ I_21 )
= X_34 ) ) ) ).
thf(fact_371_nth__list__update__eq,axiom,
! [X_33: arrow_475358991le_alt,I_20: nat,Xs_75: list_A2115238852le_alt] :
( ( ord_less_nat @ I_20 @ ( size_s1858781230le_alt @ Xs_75 ) )
=> ( ( nth_Ar125878976le_alt @ ( list_u62477863le_alt @ Xs_75 @ I_20 @ X_33 ) @ I_20 )
= X_33 ) ) ).
thf(fact_372_nth__list__update__eq,axiom,
! [X_33: nat,I_20: nat,Xs_75: list_nat] :
( ( ord_less_nat @ I_20 @ ( size_size_list_nat @ Xs_75 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs_75 @ I_20 @ X_33 ) @ I_20 )
= X_33 ) ) ).
thf(fact_373_last__drop,axiom,
! [N_43: nat,Xs_74: list_A2115238852le_alt] :
( ( ord_less_nat @ N_43 @ ( size_s1858781230le_alt @ Xs_74 ) )
=> ( ( last_A1217315288le_alt @ ( drop_A1346709759le_alt @ N_43 @ Xs_74 ) )
= ( last_A1217315288le_alt @ Xs_74 ) ) ) ).
thf(fact_374_not__less__bot,axiom,
! [A_64: $o] :
~ ( ord_less_o @ A_64 @ bot_bot_o ) ).
thf(fact_375_not__less__bot,axiom,
! [A_64: nat] :
~ ( ord_less_nat @ A_64 @ bot_bot_nat ) ).
thf(fact_376_not__less__bot,axiom,
! [A_64: nat > $o] :
~ ( ord_less_nat_o @ A_64 @ bot_bot_nat_o ) ).
thf(fact_377_bot__less,axiom,
! [A_63: $o] :
( ~ ( A_63
<=> bot_bot_o )
<=> ( ord_less_o @ bot_bot_o @ A_63 ) ) ).
thf(fact_378_bot__less,axiom,
! [A_63: nat] :
( ( A_63 != bot_bot_nat )
<=> ( ord_less_nat @ bot_bot_nat @ A_63 ) ) ).
thf(fact_379_bot__less,axiom,
! [A_63: nat > $o] :
( ( A_63 != bot_bot_nat_o )
<=> ( ord_less_nat_o @ bot_bot_nat_o @ A_63 ) ) ).
thf(fact_380_drop__0,axiom,
! [Xs_73: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ zero_zero_nat @ Xs_73 )
= Xs_73 ) ).
thf(fact_381_drop__Nil,axiom,
! [N_42: nat] :
( ( drop_nat @ N_42 @ nil_nat )
= nil_nat ) ).
thf(fact_382_drop__Nil,axiom,
! [N_42: nat] :
( ( drop_A1346709759le_alt @ N_42 @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_383_less__max__iff__disj,axiom,
! [Z_4: int,X_32: int,Y_10: int] :
( ( ord_less_int @ Z_4 @ ( ord_max_int @ X_32 @ Y_10 ) )
<=> ( ( ord_less_int @ Z_4 @ X_32 )
| ( ord_less_int @ Z_4 @ Y_10 ) ) ) ).
thf(fact_384_less__max__iff__disj,axiom,
! [Z_4: nat,X_32: nat,Y_10: nat] :
( ( ord_less_nat @ Z_4 @ ( ord_max_nat @ X_32 @ Y_10 ) )
<=> ( ( ord_less_nat @ Z_4 @ X_32 )
| ( ord_less_nat @ Z_4 @ Y_10 ) ) ) ).
thf(fact_385_max__less__iff__conj,axiom,
! [X_31: int,Y_9: int,Z_3: int] :
( ( ord_less_int @ ( ord_max_int @ X_31 @ Y_9 ) @ Z_3 )
<=> ( ( ord_less_int @ X_31 @ Z_3 )
& ( ord_less_int @ Y_9 @ Z_3 ) ) ) ).
thf(fact_386_max__less__iff__conj,axiom,
! [X_31: nat,Y_9: nat,Z_3: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X_31 @ Y_9 ) @ Z_3 )
<=> ( ( ord_less_nat @ X_31 @ Z_3 )
& ( ord_less_nat @ Y_9 @ Z_3 ) ) ) ).
thf(fact_387_min__max_Oless__supI1,axiom,
! [B_37: int,X_30: int,A_62: int] :
( ( ord_less_int @ X_30 @ A_62 )
=> ( ord_less_int @ X_30 @ ( ord_max_int @ A_62 @ B_37 ) ) ) ).
thf(fact_388_min__max_Oless__supI1,axiom,
! [B_37: nat,X_30: nat,A_62: nat] :
( ( ord_less_nat @ X_30 @ A_62 )
=> ( ord_less_nat @ X_30 @ ( ord_max_nat @ A_62 @ B_37 ) ) ) ).
thf(fact_389_min__max_Oless__supI2,axiom,
! [A_61: int,X_29: int,B_36: int] :
( ( ord_less_int @ X_29 @ B_36 )
=> ( ord_less_int @ X_29 @ ( ord_max_int @ A_61 @ B_36 ) ) ) ).
thf(fact_390_min__max_Oless__supI2,axiom,
! [A_61: nat,X_29: nat,B_36: nat] :
( ( ord_less_nat @ X_29 @ B_36 )
=> ( ord_less_nat @ X_29 @ ( ord_max_nat @ A_61 @ B_36 ) ) ) ).
thf(fact_391_distinct__drop,axiom,
! [I_19: nat,Xs_72: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_72 )
=> ( distin236324274le_alt @ ( drop_A1346709759le_alt @ I_19 @ Xs_72 ) ) ) ).
thf(fact_392_sorted__drop,axiom,
! [N_41: nat,Xs_71: list_nat] :
( ( linorder_sorted_nat @ Xs_71 )
=> ( linorder_sorted_nat @ ( drop_nat @ N_41 @ Xs_71 ) ) ) ).
thf(fact_393_list__update__nonempty,axiom,
! [Xs_70: list_nat,K_7: nat,X_28: nat] :
( ( ( list_update_nat @ Xs_70 @ K_7 @ X_28 )
= nil_nat )
<=> ( Xs_70 = nil_nat ) ) ).
thf(fact_394_list__update__nonempty,axiom,
! [Xs_70: list_A2115238852le_alt,K_7: nat,X_28: arrow_475358991le_alt] :
( ( ( list_u62477863le_alt @ Xs_70 @ K_7 @ X_28 )
= nil_Ar1286194111le_alt )
<=> ( Xs_70 = nil_Ar1286194111le_alt ) ) ).
thf(fact_395_list__update_Osimps_I1_J,axiom,
! [I_18: nat,V_3: nat] :
( ( list_update_nat @ nil_nat @ I_18 @ V_3 )
= nil_nat ) ).
thf(fact_396_list__update_Osimps_I1_J,axiom,
! [I_18: nat,V_3: arrow_475358991le_alt] :
( ( list_u62477863le_alt @ nil_Ar1286194111le_alt @ I_18 @ V_3 )
= nil_Ar1286194111le_alt ) ).
thf(fact_397_list__update__code_I1_J,axiom,
! [I_17: nat,Y_8: nat] :
( ( list_update_nat @ nil_nat @ I_17 @ Y_8 )
= nil_nat ) ).
thf(fact_398_list__update__code_I1_J,axiom,
! [I_17: nat,Y_8: arrow_475358991le_alt] :
( ( list_u62477863le_alt @ nil_Ar1286194111le_alt @ I_17 @ Y_8 )
= nil_Ar1286194111le_alt ) ).
thf(fact_399_length__list__update,axiom,
! [Xs_69: list_A2115238852le_alt,I_16: nat,X_27: arrow_475358991le_alt] :
( ( size_s1858781230le_alt @ ( list_u62477863le_alt @ Xs_69 @ I_16 @ X_27 ) )
= ( size_s1858781230le_alt @ Xs_69 ) ) ).
thf(fact_400_list__update__id,axiom,
! [Xs_68: list_A2115238852le_alt,I_15: nat] :
( ( list_u62477863le_alt @ Xs_68 @ I_15 @ ( nth_Ar125878976le_alt @ Xs_68 @ I_15 ) )
= Xs_68 ) ).
thf(fact_401_list__update__id,axiom,
! [Xs_68: list_nat,I_15: nat] :
( ( list_update_nat @ Xs_68 @ I_15 @ ( nth_nat @ Xs_68 @ I_15 ) )
= Xs_68 ) ).
thf(fact_402_nth__list__update__neq,axiom,
! [Xs_67: list_A2115238852le_alt,X_26: arrow_475358991le_alt,I_14: nat,J_5: nat] :
( ( I_14 != J_5 )
=> ( ( nth_Ar125878976le_alt @ ( list_u62477863le_alt @ Xs_67 @ I_14 @ X_26 ) @ J_5 )
= ( nth_Ar125878976le_alt @ Xs_67 @ J_5 ) ) ) ).
thf(fact_403_nth__list__update__neq,axiom,
! [Xs_67: list_nat,X_26: nat,I_14: nat,J_5: nat] :
( ( I_14 != J_5 )
=> ( ( nth_nat @ ( list_update_nat @ Xs_67 @ I_14 @ X_26 ) @ J_5 )
= ( nth_nat @ Xs_67 @ J_5 ) ) ) ).
thf(fact_404_drop__butlast,axiom,
! [N_40: nat,Xs_66: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ N_40 @ ( butlas274947851le_alt @ Xs_66 ) )
= ( butlas274947851le_alt @ ( drop_A1346709759le_alt @ N_40 @ Xs_66 ) ) ) ).
thf(fact_405_butlast__drop,axiom,
! [N_39: nat,Xs_65: list_A2115238852le_alt] :
( ( butlas274947851le_alt @ ( drop_A1346709759le_alt @ N_39 @ Xs_65 ) )
= ( drop_A1346709759le_alt @ N_39 @ ( butlas274947851le_alt @ Xs_65 ) ) ) ).
thf(fact_406_tl__drop,axiom,
! [N_38: nat,Xs_64: list_A2115238852le_alt] :
( ( tl_Arr465451158le_alt @ ( drop_A1346709759le_alt @ N_38 @ Xs_64 ) )
= ( drop_A1346709759le_alt @ N_38 @ ( tl_Arr465451158le_alt @ Xs_64 ) ) ) ).
thf(fact_407_drop__tl,axiom,
! [N_37: nat,Xs_63: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ N_37 @ ( tl_Arr465451158le_alt @ Xs_63 ) )
= ( tl_Arr465451158le_alt @ ( drop_A1346709759le_alt @ N_37 @ Xs_63 ) ) ) ).
thf(fact_408_hd__drop__conv__nth,axiom,
! [N_36: nat,Xs_62: list_nat] :
( ( Xs_62 != nil_nat )
=> ( ( ord_less_nat @ N_36 @ ( size_size_list_nat @ Xs_62 ) )
=> ( ( hd_nat @ ( drop_nat @ N_36 @ Xs_62 ) )
= ( nth_nat @ Xs_62 @ N_36 ) ) ) ) ).
thf(fact_409_hd__drop__conv__nth,axiom,
! [N_36: nat,Xs_62: list_A2115238852le_alt] :
( ( Xs_62 != nil_Ar1286194111le_alt )
=> ( ( ord_less_nat @ N_36 @ ( size_s1858781230le_alt @ Xs_62 ) )
=> ( ( hd_Arr1965683346le_alt @ ( drop_A1346709759le_alt @ N_36 @ Xs_62 ) )
= ( nth_Ar125878976le_alt @ Xs_62 @ N_36 ) ) ) ) ).
thf(fact_410_nth__via__drop,axiom,
! [N_35: nat,Xs_61: list_nat,Y_7: nat,Ys_12: list_nat] :
( ( ( drop_nat @ N_35 @ Xs_61 )
= ( cons_nat @ Y_7 @ Ys_12 ) )
=> ( ( nth_nat @ Xs_61 @ N_35 )
= Y_7 ) ) ).
thf(fact_411_nth__via__drop,axiom,
! [N_35: nat,Xs_61: list_A2115238852le_alt,Y_7: arrow_475358991le_alt,Ys_12: list_A2115238852le_alt] :
( ( ( drop_A1346709759le_alt @ N_35 @ Xs_61 )
= ( cons_A228743023le_alt @ Y_7 @ Ys_12 ) )
=> ( ( nth_Ar125878976le_alt @ Xs_61 @ N_35 )
= Y_7 ) ) ).
thf(fact_412_list__eq__iff__nth__eq,axiom,
! [Xs_60: list_A2115238852le_alt,Ys_11: list_A2115238852le_alt] :
( ( Xs_60 = Ys_11 )
<=> ( ( ( size_s1858781230le_alt @ Xs_60 )
= ( size_s1858781230le_alt @ Ys_11 ) )
& ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ ( size_s1858781230le_alt @ Xs_60 ) )
=> ( ( nth_Ar125878976le_alt @ Xs_60 @ I_4 )
= ( nth_Ar125878976le_alt @ Ys_11 @ I_4 ) ) ) ) ) ).
thf(fact_413_list__eq__iff__nth__eq,axiom,
! [Xs_60: list_nat,Ys_11: list_nat] :
( ( Xs_60 = Ys_11 )
<=> ( ( ( size_size_list_nat @ Xs_60 )
= ( size_size_list_nat @ Ys_11 ) )
& ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ ( size_size_list_nat @ Xs_60 ) )
=> ( ( nth_nat @ Xs_60 @ I_4 )
= ( nth_nat @ Ys_11 @ I_4 ) ) ) ) ) ).
thf(fact_414_list__update__code_I2_J,axiom,
! [X_25: arrow_475358991le_alt,Xs_59: list_A2115238852le_alt,Y_6: arrow_475358991le_alt] :
( ( list_u62477863le_alt @ ( cons_A228743023le_alt @ X_25 @ Xs_59 ) @ zero_zero_nat @ Y_6 )
= ( cons_A228743023le_alt @ Y_6 @ Xs_59 ) ) ).
thf(fact_415_nth__replicate,axiom,
! [X_24: nat,I_13: nat,N_34: nat] :
( ( ord_less_nat @ I_13 @ N_34 )
=> ( ( nth_nat @ ( replicate_nat @ N_34 @ X_24 ) @ I_13 )
= X_24 ) ) ).
thf(fact_416_nth__replicate,axiom,
! [X_24: arrow_475358991le_alt,I_13: nat,N_34: nat] :
( ( ord_less_nat @ I_13 @ N_34 )
=> ( ( nth_Ar125878976le_alt @ ( replic1511538809le_alt @ N_34 @ X_24 ) @ I_13 )
= X_24 ) ) ).
thf(fact_417_length__greater__0__conv,axiom,
! [Xs_58: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs_58 ) )
<=> ( Xs_58 != nil_nat ) ) ).
thf(fact_418_length__greater__0__conv,axiom,
! [Xs_58: list_A2115238852le_alt] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1858781230le_alt @ Xs_58 ) )
<=> ( Xs_58 != nil_Ar1286194111le_alt ) ) ).
thf(fact_419_distinct__conv__nth,axiom,
! [Xs_57: list_nat] :
( ( distinct_nat @ Xs_57 )
<=> ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ ( size_size_list_nat @ Xs_57 ) )
=> ! [J_2: nat] :
( ( ord_less_nat @ J_2 @ ( size_size_list_nat @ Xs_57 ) )
=> ( ( I_4 != J_2 )
=> ( ( nth_nat @ Xs_57 @ I_4 )
!= ( nth_nat @ Xs_57 @ J_2 ) ) ) ) ) ) ).
thf(fact_420_distinct__conv__nth,axiom,
! [Xs_57: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_57 )
<=> ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ ( size_s1858781230le_alt @ Xs_57 ) )
=> ! [J_2: nat] :
( ( ord_less_nat @ J_2 @ ( size_s1858781230le_alt @ Xs_57 ) )
=> ( ( I_4 != J_2 )
=> ( ( nth_Ar125878976le_alt @ Xs_57 @ I_4 )
!= ( nth_Ar125878976le_alt @ Xs_57 @ J_2 ) ) ) ) ) ) ).
thf(fact_421_nth__eq__iff__index__eq,axiom,
! [J_4: nat,I_12: nat,Xs_56: list_nat] :
( ( distinct_nat @ Xs_56 )
=> ( ( ord_less_nat @ I_12 @ ( size_size_list_nat @ Xs_56 ) )
=> ( ( ord_less_nat @ J_4 @ ( size_size_list_nat @ Xs_56 ) )
=> ( ( ( nth_nat @ Xs_56 @ I_12 )
= ( nth_nat @ Xs_56 @ J_4 ) )
<=> ( I_12 = J_4 ) ) ) ) ) ).
thf(fact_422_nth__eq__iff__index__eq,axiom,
! [J_4: nat,I_12: nat,Xs_56: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_56 )
=> ( ( ord_less_nat @ I_12 @ ( size_s1858781230le_alt @ Xs_56 ) )
=> ( ( ord_less_nat @ J_4 @ ( size_s1858781230le_alt @ Xs_56 ) )
=> ( ( ( nth_Ar125878976le_alt @ Xs_56 @ I_12 )
= ( nth_Ar125878976le_alt @ Xs_56 @ J_4 ) )
<=> ( I_12 = J_4 ) ) ) ) ) ).
thf(fact_423_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
thf(fact_424_max__0R,axiom,
! [N_5: nat] :
( ( ord_max_nat @ N_5 @ zero_zero_nat )
= N_5 ) ).
thf(fact_425_nth__equalityI,axiom,
! [Xs_55: list_A2115238852le_alt,Ys_10: list_A2115238852le_alt] :
( ( ( size_s1858781230le_alt @ Xs_55 )
= ( size_s1858781230le_alt @ Ys_10 ) )
=> ( ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ ( size_s1858781230le_alt @ Xs_55 ) )
=> ( ( nth_Ar125878976le_alt @ Xs_55 @ I_4 )
= ( nth_Ar125878976le_alt @ Ys_10 @ I_4 ) ) )
=> ( Xs_55 = Ys_10 ) ) ) ).
thf(fact_426_nth__equalityI,axiom,
! [Xs_55: list_nat,Ys_10: list_nat] :
( ( ( size_size_list_nat @ Xs_55 )
= ( size_size_list_nat @ Ys_10 ) )
=> ( ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ ( size_size_list_nat @ Xs_55 ) )
=> ( ( nth_nat @ Xs_55 @ I_4 )
= ( nth_nat @ Ys_10 @ I_4 ) ) )
=> ( Xs_55 = Ys_10 ) ) ) ).
thf(fact_427_Skolem__list__nth,axiom,
! [P_9: nat > arrow_475358991le_alt > $o,K_6: nat] :
( ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ K_6 )
=> ( ex2 @ ( P_9 @ I_4 ) ) )
<=> ? [Xs_54: list_A2115238852le_alt] :
( ( ( size_s1858781230le_alt @ Xs_54 )
= K_6 )
& ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ K_6 )
=> ( P_9 @ I_4 @ ( nth_Ar125878976le_alt @ Xs_54 @ I_4 ) ) ) ) ) ).
thf(fact_428_Skolem__list__nth,axiom,
! [P_9: nat > nat > $o,K_6: nat] :
( ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ K_6 )
=> ( ex1 @ ( P_9 @ I_4 ) ) )
<=> ? [Xs_54: list_nat] :
( ( ( size_size_list_nat @ Xs_54 )
= K_6 )
& ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ K_6 )
=> ( P_9 @ I_4 @ ( nth_nat @ Xs_54 @ I_4 ) ) ) ) ) ).
thf(fact_429_dropWhile__nth,axiom,
! [J_3: nat,P_8: nat > $o,Xs_53: list_nat] :
( ( ord_less_nat @ J_3 @ ( size_size_list_nat @ ( dropWhile_nat @ P_8 @ Xs_53 ) ) )
=> ( ( nth_nat @ ( dropWhile_nat @ P_8 @ Xs_53 ) @ J_3 )
= ( nth_nat @ Xs_53 @ ( plus_plus_nat @ J_3 @ ( size_size_list_nat @ ( takeWhile_nat @ P_8 @ Xs_53 ) ) ) ) ) ) ).
thf(fact_430_dropWhile__nth,axiom,
! [J_3: nat,P_8: arrow_475358991le_alt > $o,Xs_53: list_A2115238852le_alt] :
( ( ord_less_nat @ J_3 @ ( size_s1858781230le_alt @ ( dropWh1316781920le_alt @ P_8 @ Xs_53 ) ) )
=> ( ( nth_Ar125878976le_alt @ ( dropWh1316781920le_alt @ P_8 @ Xs_53 ) @ J_3 )
= ( nth_Ar125878976le_alt @ Xs_53 @ ( plus_plus_nat @ J_3 @ ( size_s1858781230le_alt @ ( takeWh1696291512le_alt @ P_8 @ Xs_53 ) ) ) ) ) ) ).
thf(fact_431_nth__drop_H,axiom,
! [I_11: nat,Xs_52: list_nat] :
( ( ord_less_nat @ I_11 @ ( size_size_list_nat @ Xs_52 ) )
=> ( ( cons_nat @ ( nth_nat @ Xs_52 @ I_11 ) @ ( drop_nat @ ( suc @ I_11 ) @ Xs_52 ) )
= ( drop_nat @ I_11 @ Xs_52 ) ) ) ).
thf(fact_432_nth__drop_H,axiom,
! [I_11: nat,Xs_52: list_A2115238852le_alt] :
( ( ord_less_nat @ I_11 @ ( size_s1858781230le_alt @ Xs_52 ) )
=> ( ( cons_A228743023le_alt @ ( nth_Ar125878976le_alt @ Xs_52 @ I_11 ) @ ( drop_A1346709759le_alt @ ( suc @ I_11 ) @ Xs_52 ) )
= ( drop_A1346709759le_alt @ I_11 @ Xs_52 ) ) ) ).
thf(fact_433_list__ex__length,axiom,
! [P_7: arrow_475358991le_alt > $o,Xs_51: list_A2115238852le_alt] :
( ( list_e1252802557le_alt @ P_7 @ Xs_51 )
<=> ? [N_7: nat] :
( ( ord_less_nat @ N_7 @ ( size_s1858781230le_alt @ Xs_51 ) )
& ( P_7 @ ( nth_Ar125878976le_alt @ Xs_51 @ N_7 ) ) ) ) ).
thf(fact_434_list__ex__length,axiom,
! [P_7: nat > $o,Xs_51: list_nat] :
( ( list_ex_nat @ P_7 @ Xs_51 )
<=> ? [N_7: nat] :
( ( ord_less_nat @ N_7 @ ( size_size_list_nat @ Xs_51 ) )
& ( P_7 @ ( nth_nat @ Xs_51 @ N_7 ) ) ) ) ).
thf(fact_435_lessI,axiom,
! [N_5: nat] : ( ord_less_nat @ N_5 @ ( suc @ N_5 ) ) ).
thf(fact_436_Suc__mono,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_5 ) ) ) ).
thf(fact_437_zero__less__Suc,axiom,
! [N_5: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N_5 ) ) ).
thf(fact_438_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_439_nat__add__right__cancel,axiom,
! [M: nat,K_3: nat,N_5: nat] :
( ( ( plus_plus_nat @ M @ K_3 )
= ( plus_plus_nat @ N_5 @ K_3 ) )
<=> ( M = N_5 ) ) ).
thf(fact_440_nat__add__left__cancel,axiom,
! [K_3: nat,M: nat,N_5: nat] :
( ( ( plus_plus_nat @ K_3 @ M )
= ( plus_plus_nat @ K_3 @ N_5 ) )
<=> ( M = N_5 ) ) ).
thf(fact_441_nat__add__assoc,axiom,
! [M: nat,N_5: nat,K_3: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N_5 ) @ K_3 )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N_5 @ K_3 ) ) ) ).
thf(fact_442_nat_Oinject,axiom,
! [Nat_3: nat,Nat_2: nat] :
( ( ( suc @ Nat_3 )
= ( suc @ Nat_2 ) )
<=> ( Nat_3 = Nat_2 ) ) ).
thf(fact_443_add__Suc__shift,axiom,
! [M: nat,N_5: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_5 )
= ( plus_plus_nat @ M @ ( suc @ N_5 ) ) ) ).
thf(fact_444_add__Suc,axiom,
! [M: nat,N_5: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N_5 )
= ( suc @ ( plus_plus_nat @ M @ N_5 ) ) ) ).
thf(fact_445_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_1: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z_1 ) ) ) ).
thf(fact_446_add__Suc__right,axiom,
! [M: nat,N_5: nat] :
( ( plus_plus_nat @ M @ ( suc @ N_5 ) )
= ( suc @ ( plus_plus_nat @ M @ N_5 ) ) ) ).
thf(fact_447_nat__add__commute,axiom,
! [M: nat,N_5: nat] :
( ( plus_plus_nat @ M @ N_5 )
= ( plus_plus_nat @ N_5 @ M ) ) ).
thf(fact_448_less__iff__Suc__add,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
<=> ? [K_5: nat] :
( N_5
= ( suc @ ( plus_plus_nat @ M @ K_5 ) ) ) ) ).
thf(fact_449_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_450_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_451_Suc__n__not__n,axiom,
! [N_5: nat] :
( ( suc @ N_5 )
!= N_5 ) ).
thf(fact_452_n__not__Suc__n,axiom,
! [N_5: nat] :
( N_5
!= ( suc @ N_5 ) ) ).
thf(fact_453_add__is__1,axiom,
! [M: nat,N_5: nat] :
( ( ( plus_plus_nat @ M @ N_5 )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_5 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_5
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_454_one__is__add,axiom,
! [M: nat,N_5: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N_5 ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N_5 = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N_5
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_455_add__right__imp__eq,axiom,
! [B_35: nat,A_60: nat,C_29: nat] :
( ( ( plus_plus_nat @ B_35 @ A_60 )
= ( plus_plus_nat @ C_29 @ A_60 ) )
=> ( B_35 = C_29 ) ) ).
thf(fact_456_add__right__imp__eq,axiom,
! [B_35: int,A_60: int,C_29: int] :
( ( ( plus_plus_int @ B_35 @ A_60 )
= ( plus_plus_int @ C_29 @ A_60 ) )
=> ( B_35 = C_29 ) ) ).
thf(fact_457_add__imp__eq,axiom,
! [A_59: nat,B_34: nat,C_28: nat] :
( ( ( plus_plus_nat @ A_59 @ B_34 )
= ( plus_plus_nat @ A_59 @ C_28 ) )
=> ( B_34 = C_28 ) ) ).
thf(fact_458_add__imp__eq,axiom,
! [A_59: int,B_34: int,C_28: int] :
( ( ( plus_plus_int @ A_59 @ B_34 )
= ( plus_plus_int @ A_59 @ C_28 ) )
=> ( B_34 = C_28 ) ) ).
thf(fact_459_add__left__imp__eq,axiom,
! [A_58: nat,B_33: nat,C_27: nat] :
( ( ( plus_plus_nat @ A_58 @ B_33 )
= ( plus_plus_nat @ A_58 @ C_27 ) )
=> ( B_33 = C_27 ) ) ).
thf(fact_460_add__left__imp__eq,axiom,
! [A_58: int,B_33: int,C_27: int] :
( ( ( plus_plus_int @ A_58 @ B_33 )
= ( plus_plus_int @ A_58 @ C_27 ) )
=> ( B_33 = C_27 ) ) ).
thf(fact_461_add__right__cancel,axiom,
! [B_32: nat,A_57: nat,C_26: nat] :
( ( ( plus_plus_nat @ B_32 @ A_57 )
= ( plus_plus_nat @ C_26 @ A_57 ) )
<=> ( B_32 = C_26 ) ) ).
thf(fact_462_add__right__cancel,axiom,
! [B_32: int,A_57: int,C_26: int] :
( ( ( plus_plus_int @ B_32 @ A_57 )
= ( plus_plus_int @ C_26 @ A_57 ) )
<=> ( B_32 = C_26 ) ) ).
thf(fact_463_add__left__cancel,axiom,
! [A_56: nat,B_31: nat,C_25: nat] :
( ( ( plus_plus_nat @ A_56 @ B_31 )
= ( plus_plus_nat @ A_56 @ C_25 ) )
<=> ( B_31 = C_25 ) ) ).
thf(fact_464_add__left__cancel,axiom,
! [A_56: int,B_31: int,C_25: int] :
( ( ( plus_plus_int @ A_56 @ B_31 )
= ( plus_plus_int @ A_56 @ C_25 ) )
<=> ( B_31 = C_25 ) ) ).
thf(fact_465_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_55: nat,B_30: nat,C_24: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_55 @ B_30 ) @ C_24 )
= ( plus_plus_nat @ A_55 @ ( plus_plus_nat @ B_30 @ C_24 ) ) ) ).
thf(fact_466_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_55: int,B_30: int,C_24: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_55 @ B_30 ) @ C_24 )
= ( plus_plus_int @ A_55 @ ( plus_plus_int @ B_30 @ C_24 ) ) ) ).
thf(fact_467_add_Ocomm__neutral,axiom,
! [A_54: nat] :
( ( plus_plus_nat @ A_54 @ zero_zero_nat )
= A_54 ) ).
thf(fact_468_add_Ocomm__neutral,axiom,
! [A_54: int] :
( ( plus_plus_int @ A_54 @ zero_zero_int )
= A_54 ) ).
thf(fact_469_add__0__right,axiom,
! [A_53: nat] :
( ( plus_plus_nat @ A_53 @ zero_zero_nat )
= A_53 ) ).
thf(fact_470_add__0__right,axiom,
! [A_53: int] :
( ( plus_plus_int @ A_53 @ zero_zero_int )
= A_53 ) ).
thf(fact_471_double__zero__sym,axiom,
! [A_52: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_52 @ A_52 ) )
<=> ( A_52 = zero_zero_int ) ) ).
thf(fact_472_add__0,axiom,
! [A_51: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_51 )
= A_51 ) ).
thf(fact_473_add__0,axiom,
! [A_51: int] :
( ( plus_plus_int @ zero_zero_int @ A_51 )
= A_51 ) ).
thf(fact_474_add__0__left,axiom,
! [A_50: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_50 )
= A_50 ) ).
thf(fact_475_add__0__left,axiom,
! [A_50: int] :
( ( plus_plus_int @ zero_zero_int @ A_50 )
= A_50 ) ).
thf(fact_476_add__less__imp__less__left,axiom,
! [C_23: nat,A_49: nat,B_29: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_23 @ A_49 ) @ ( plus_plus_nat @ C_23 @ B_29 ) )
=> ( ord_less_nat @ A_49 @ B_29 ) ) ).
thf(fact_477_add__less__imp__less__left,axiom,
! [C_23: int,A_49: int,B_29: int] :
( ( ord_less_int @ ( plus_plus_int @ C_23 @ A_49 ) @ ( plus_plus_int @ C_23 @ B_29 ) )
=> ( ord_less_int @ A_49 @ B_29 ) ) ).
thf(fact_478_add__less__imp__less__right,axiom,
! [A_48: nat,C_22: nat,B_28: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_48 @ C_22 ) @ ( plus_plus_nat @ B_28 @ C_22 ) )
=> ( ord_less_nat @ A_48 @ B_28 ) ) ).
thf(fact_479_add__less__imp__less__right,axiom,
! [A_48: int,C_22: int,B_28: int] :
( ( ord_less_int @ ( plus_plus_int @ A_48 @ C_22 ) @ ( plus_plus_int @ B_28 @ C_22 ) )
=> ( ord_less_int @ A_48 @ B_28 ) ) ).
thf(fact_480_add__strict__mono,axiom,
! [C_21: nat,D_6: nat,A_47: nat,B_27: nat] :
( ( ord_less_nat @ A_47 @ B_27 )
=> ( ( ord_less_nat @ C_21 @ D_6 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_47 @ C_21 ) @ ( plus_plus_nat @ B_27 @ D_6 ) ) ) ) ).
thf(fact_481_add__strict__mono,axiom,
! [C_21: int,D_6: int,A_47: int,B_27: int] :
( ( ord_less_int @ A_47 @ B_27 )
=> ( ( ord_less_int @ C_21 @ D_6 )
=> ( ord_less_int @ ( plus_plus_int @ A_47 @ C_21 ) @ ( plus_plus_int @ B_27 @ D_6 ) ) ) ) ).
thf(fact_482_add__strict__left__mono,axiom,
! [C_20: nat,A_46: nat,B_26: nat] :
( ( ord_less_nat @ A_46 @ B_26 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_20 @ A_46 ) @ ( plus_plus_nat @ C_20 @ B_26 ) ) ) ).
thf(fact_483_add__strict__left__mono,axiom,
! [C_20: int,A_46: int,B_26: int] :
( ( ord_less_int @ A_46 @ B_26 )
=> ( ord_less_int @ ( plus_plus_int @ C_20 @ A_46 ) @ ( plus_plus_int @ C_20 @ B_26 ) ) ) ).
thf(fact_484_add__strict__right__mono,axiom,
! [C_19: nat,A_45: nat,B_25: nat] :
( ( ord_less_nat @ A_45 @ B_25 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_45 @ C_19 ) @ ( plus_plus_nat @ B_25 @ C_19 ) ) ) ).
thf(fact_485_add__strict__right__mono,axiom,
! [C_19: int,A_45: int,B_25: int] :
( ( ord_less_int @ A_45 @ B_25 )
=> ( ord_less_int @ ( plus_plus_int @ A_45 @ C_19 ) @ ( plus_plus_int @ B_25 @ C_19 ) ) ) ).
thf(fact_486_add__less__cancel__left,axiom,
! [C_18: nat,A_44: nat,B_24: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_18 @ A_44 ) @ ( plus_plus_nat @ C_18 @ B_24 ) )
<=> ( ord_less_nat @ A_44 @ B_24 ) ) ).
thf(fact_487_add__less__cancel__left,axiom,
! [C_18: int,A_44: int,B_24: int] :
( ( ord_less_int @ ( plus_plus_int @ C_18 @ A_44 ) @ ( plus_plus_int @ C_18 @ B_24 ) )
<=> ( ord_less_int @ A_44 @ B_24 ) ) ).
thf(fact_488_add__less__cancel__right,axiom,
! [A_43: nat,C_17: nat,B_23: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_43 @ C_17 ) @ ( plus_plus_nat @ B_23 @ C_17 ) )
<=> ( ord_less_nat @ A_43 @ B_23 ) ) ).
thf(fact_489_add__less__cancel__right,axiom,
! [A_43: int,C_17: int,B_23: int] :
( ( ord_less_int @ ( plus_plus_int @ A_43 @ C_17 ) @ ( plus_plus_int @ B_23 @ C_17 ) )
<=> ( ord_less_int @ A_43 @ B_23 ) ) ).
thf(fact_490_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_491_nat_Osimps_I2_J,axiom,
! [Nat_2: nat] :
( zero_zero_nat
!= ( suc @ Nat_2 ) ) ).
thf(fact_492_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_493_nat_Osimps_I3_J,axiom,
! [Nat_1: nat] :
( ( suc @ Nat_1 )
!= zero_zero_nat ) ).
thf(fact_494_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_495_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_496_not__less__eq,axiom,
! [M: nat,N_5: nat] :
( ~ ( ord_less_nat @ M @ N_5 )
<=> ( ord_less_nat @ N_5 @ ( suc @ M ) ) ) ).
thf(fact_497_less__Suc__eq,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ ( suc @ N_5 ) )
<=> ( ( ord_less_nat @ M @ N_5 )
| ( M = N_5 ) ) ) ).
thf(fact_498_Suc__less__eq,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_5 ) )
<=> ( ord_less_nat @ M @ N_5 ) ) ).
thf(fact_499_not__less__less__Suc__eq,axiom,
! [N_5: nat,M: nat] :
( ~ ( ord_less_nat @ N_5 @ M )
=> ( ( ord_less_nat @ N_5 @ ( suc @ M ) )
<=> ( N_5 = M ) ) ) ).
thf(fact_500_less__antisym,axiom,
! [N_5: nat,M: nat] :
( ~ ( ord_less_nat @ N_5 @ M )
=> ( ( ord_less_nat @ N_5 @ ( suc @ M ) )
=> ( M = N_5 ) ) ) ).
thf(fact_501_less__SucI,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
=> ( ord_less_nat @ M @ ( suc @ N_5 ) ) ) ).
thf(fact_502_Suc__lessI,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
=> ( ( ( suc @ M )
!= N_5 )
=> ( ord_less_nat @ ( suc @ M ) @ N_5 ) ) ) ).
thf(fact_503_less__trans__Suc,axiom,
! [K_3: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K_3 )
=> ( ord_less_nat @ ( suc @ I ) @ K_3 ) ) ) ).
thf(fact_504_less__SucE,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ ( suc @ N_5 ) )
=> ( ~ ( ord_less_nat @ M @ N_5 )
=> ( M = N_5 ) ) ) ).
thf(fact_505_Suc__lessD,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N_5 )
=> ( ord_less_nat @ M @ N_5 ) ) ).
thf(fact_506_Suc__less__SucD,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N_5 ) )
=> ( ord_less_nat @ M @ N_5 ) ) ).
thf(fact_507_not__psubset__empty,axiom,
! [A_42: nat > $o] :
~ ( ord_less_nat_o @ A_42 @ bot_bot_nat_o ) ).
thf(fact_508_plus__nat_Oadd__0,axiom,
! [N_5: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N_5 )
= N_5 ) ).
thf(fact_509_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_510_add__is__0,axiom,
! [M: nat,N_5: nat] :
( ( ( plus_plus_nat @ M @ N_5 )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N_5 = zero_zero_nat ) ) ) ).
thf(fact_511_add__eq__self__zero,axiom,
! [M: nat,N_5: nat] :
( ( ( plus_plus_nat @ M @ N_5 )
= M )
=> ( N_5 = zero_zero_nat ) ) ).
thf(fact_512_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_513_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_514_nat__add__left__cancel__less,axiom,
! [K_3: nat,M: nat,N_5: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K_3 @ M ) @ ( plus_plus_nat @ K_3 @ N_5 ) )
<=> ( ord_less_nat @ M @ N_5 ) ) ).
thf(fact_515_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_516_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_517_add__less__mono1,axiom,
! [K_3: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_3 ) @ ( plus_plus_nat @ J @ K_3 ) ) ) ).
thf(fact_518_add__less__mono,axiom,
! [K_3: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K_3 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_519_less__add__eq__less,axiom,
! [M: nat,N_5: nat,K_3: nat,L: nat] :
( ( ord_less_nat @ K_3 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K_3 @ N_5 ) )
=> ( ord_less_nat @ M @ N_5 ) ) ) ).
thf(fact_520_add__lessD1,axiom,
! [I: nat,J: nat,K_3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K_3 )
=> ( ord_less_nat @ I @ K_3 ) ) ).
thf(fact_521_max__add__distrib__left,axiom,
! [X_23: nat,Y_5: nat,Z_2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X_23 @ Y_5 ) @ Z_2 )
= ( ord_max_nat @ ( plus_plus_nat @ X_23 @ Z_2 ) @ ( plus_plus_nat @ Y_5 @ Z_2 ) ) ) ).
thf(fact_522_max__add__distrib__left,axiom,
! [X_23: int,Y_5: int,Z_2: int] :
( ( plus_plus_int @ ( ord_max_int @ X_23 @ Y_5 ) @ Z_2 )
= ( ord_max_int @ ( plus_plus_int @ X_23 @ Z_2 ) @ ( plus_plus_int @ Y_5 @ Z_2 ) ) ) ).
thf(fact_523_drop__drop,axiom,
! [N_33: nat,M_4: nat,Xs_50: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ N_33 @ ( drop_A1346709759le_alt @ M_4 @ Xs_50 ) )
= ( drop_A1346709759le_alt @ ( plus_plus_nat @ N_33 @ M_4 ) @ Xs_50 ) ) ).
thf(fact_524_max__Suc__Suc,axiom,
! [M: nat,N_5: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N_5 ) )
= ( suc @ ( ord_max_nat @ M @ N_5 ) ) ) ).
thf(fact_525_list_Osize_I4_J,axiom,
! [A_41: arrow_475358991le_alt,List_1: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( cons_A228743023le_alt @ A_41 @ List_1 ) )
= ( plus_plus_nat @ ( size_s1858781230le_alt @ List_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_526_list_Osize_I2_J,axiom,
! [Fa: arrow_475358991le_alt > nat,A_40: arrow_475358991le_alt,List: list_A2115238852le_alt] :
( ( list_s250295873le_alt @ Fa @ ( cons_A228743023le_alt @ A_40 @ List ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( Fa @ A_40 ) @ ( list_s250295873le_alt @ Fa @ List ) ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_527_list__ex__simps_I1_J,axiom,
! [P_6: arrow_475358991le_alt > $o,X_22: arrow_475358991le_alt,Xs_49: list_A2115238852le_alt] :
( ( list_e1252802557le_alt @ P_6 @ ( cons_A228743023le_alt @ X_22 @ Xs_49 ) )
<=> ( ( P_6 @ X_22 )
| ( list_e1252802557le_alt @ P_6 @ Xs_49 ) ) ) ).
thf(fact_528_list__ex__simps_I2_J,axiom,
! [P_5: nat > $o] :
~ ( list_ex_nat @ P_5 @ nil_nat ) ).
thf(fact_529_list__ex__simps_I2_J,axiom,
! [P_5: arrow_475358991le_alt > $o] :
~ ( list_e1252802557le_alt @ P_5 @ nil_Ar1286194111le_alt ) ).
thf(fact_530_list__ex__append,axiom,
! [P_4: arrow_475358991le_alt > $o,Xs_48: list_A2115238852le_alt,Ys_9: list_A2115238852le_alt] :
( ( list_e1252802557le_alt @ P_4 @ ( append179082452le_alt @ Xs_48 @ Ys_9 ) )
<=> ( ( list_e1252802557le_alt @ P_4 @ Xs_48 )
| ( list_e1252802557le_alt @ P_4 @ Ys_9 ) ) ) ).
thf(fact_531_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_39: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_39 @ A_39 ) )
<=> ( ord_less_int @ zero_zero_int @ A_39 ) ) ).
thf(fact_532_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_38: int] :
( ( ord_less_int @ ( plus_plus_int @ A_38 @ A_38 ) @ zero_zero_int )
<=> ( ord_less_int @ A_38 @ zero_zero_int ) ) ).
thf(fact_533_add__pos__pos,axiom,
! [B_22: nat,A_37: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_37 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_22 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_37 @ B_22 ) ) ) ) ).
thf(fact_534_add__pos__pos,axiom,
! [B_22: int,A_37: int] :
( ( ord_less_int @ zero_zero_int @ A_37 )
=> ( ( ord_less_int @ zero_zero_int @ B_22 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_37 @ B_22 ) ) ) ) ).
thf(fact_535_add__neg__neg,axiom,
! [B_21: nat,A_36: nat] :
( ( ord_less_nat @ A_36 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_21 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_36 @ B_21 ) @ zero_zero_nat ) ) ) ).
thf(fact_536_add__neg__neg,axiom,
! [B_21: int,A_36: int] :
( ( ord_less_int @ A_36 @ zero_zero_int )
=> ( ( ord_less_int @ B_21 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_36 @ B_21 ) @ zero_zero_int ) ) ) ).
thf(fact_537_list__ex__rev,axiom,
! [P_3: nat > $o,Xs_47: list_nat] :
( ( list_ex_nat @ P_3 @ ( rev_nat @ Xs_47 ) )
<=> ( list_ex_nat @ P_3 @ Xs_47 ) ) ).
thf(fact_538_list__ex__rev,axiom,
! [P_3: arrow_475358991le_alt > $o,Xs_47: list_A2115238852le_alt] :
( ( list_e1252802557le_alt @ P_3 @ ( rev_Ar1106406943le_alt @ Xs_47 ) )
<=> ( list_e1252802557le_alt @ P_3 @ Xs_47 ) ) ).
thf(fact_539_gr0__conv__Suc,axiom,
! [N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
<=> ? [M_1: nat] :
( N_5
= ( suc @ M_1 ) ) ) ).
thf(fact_540_less__Suc0,axiom,
! [N_5: nat] :
( ( ord_less_nat @ N_5 @ ( suc @ zero_zero_nat ) )
<=> ( N_5 = zero_zero_nat ) ) ).
thf(fact_541_less__Suc__eq__0__disj,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ ( suc @ N_5 ) )
<=> ( ( M = zero_zero_nat )
| ? [J_2: nat] :
( ( M
= ( suc @ J_2 ) )
& ( ord_less_nat @ J_2 @ N_5 ) ) ) ) ).
thf(fact_542_add__gr__0,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N_5 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N_5 ) ) ) ).
thf(fact_543_nth__Cons__Suc,axiom,
! [X_21: nat,Xs_46: list_nat,N_32: nat] :
( ( nth_nat @ ( cons_nat @ X_21 @ Xs_46 ) @ ( suc @ N_32 ) )
= ( nth_nat @ Xs_46 @ N_32 ) ) ).
thf(fact_544_nth__Cons__Suc,axiom,
! [X_21: arrow_475358991le_alt,Xs_46: list_A2115238852le_alt,N_32: nat] :
( ( nth_Ar125878976le_alt @ ( cons_A228743023le_alt @ X_21 @ Xs_46 ) @ ( suc @ N_32 ) )
= ( nth_Ar125878976le_alt @ Xs_46 @ N_32 ) ) ).
thf(fact_545_length__append,axiom,
! [Xs_45: list_A2115238852le_alt,Ys_8: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( append179082452le_alt @ Xs_45 @ Ys_8 ) )
= ( plus_plus_nat @ ( size_s1858781230le_alt @ Xs_45 ) @ ( size_s1858781230le_alt @ Ys_8 ) ) ) ).
thf(fact_546_drop__Suc__Cons,axiom,
! [N_31: nat,X_20: arrow_475358991le_alt,Xs_44: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ ( suc @ N_31 ) @ ( cons_A228743023le_alt @ X_20 @ Xs_44 ) )
= ( drop_A1346709759le_alt @ N_31 @ Xs_44 ) ) ).
thf(fact_547_list__update__code_I3_J,axiom,
! [X_19: arrow_475358991le_alt,Xs_43: list_A2115238852le_alt,I_10: nat,Y_4: arrow_475358991le_alt] :
( ( list_u62477863le_alt @ ( cons_A228743023le_alt @ X_19 @ Xs_43 ) @ ( suc @ I_10 ) @ Y_4 )
= ( cons_A228743023le_alt @ X_19 @ ( list_u62477863le_alt @ Xs_43 @ I_10 @ Y_4 ) ) ) ).
thf(fact_548_replicate__Suc,axiom,
! [N_30: nat,X_18: arrow_475358991le_alt] :
( ( replic1511538809le_alt @ ( suc @ N_30 ) @ X_18 )
= ( cons_A228743023le_alt @ X_18 @ ( replic1511538809le_alt @ N_30 @ X_18 ) ) ) ).
thf(fact_549_replicate__add,axiom,
! [N_29: nat,M_3: nat,X_17: arrow_475358991le_alt] :
( ( replic1511538809le_alt @ ( plus_plus_nat @ N_29 @ M_3 ) @ X_17 )
= ( append179082452le_alt @ ( replic1511538809le_alt @ N_29 @ X_17 ) @ ( replic1511538809le_alt @ M_3 @ X_17 ) ) ) ).
thf(fact_550_drop__Suc,axiom,
! [N_28: nat,Xs_42: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ ( suc @ N_28 ) @ Xs_42 )
= ( drop_A1346709759le_alt @ N_28 @ ( tl_Arr465451158le_alt @ Xs_42 ) ) ) ).
thf(fact_551_length__splice,axiom,
! [Xs_41: list_A2115238852le_alt,Ys_7: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( splice1520898450le_alt @ Xs_41 @ Ys_7 ) )
= ( plus_plus_nat @ ( size_s1858781230le_alt @ Xs_41 ) @ ( size_s1858781230le_alt @ Ys_7 ) ) ) ).
thf(fact_552_nth__append__length__plus,axiom,
! [Xs_40: list_nat,Ys_6: list_nat,N_27: nat] :
( ( nth_nat @ ( append_nat @ Xs_40 @ Ys_6 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs_40 ) @ N_27 ) )
= ( nth_nat @ Ys_6 @ N_27 ) ) ).
thf(fact_553_nth__append__length__plus,axiom,
! [Xs_40: list_A2115238852le_alt,Ys_6: list_A2115238852le_alt,N_27: nat] :
( ( nth_Ar125878976le_alt @ ( append179082452le_alt @ Xs_40 @ Ys_6 ) @ ( plus_plus_nat @ ( size_s1858781230le_alt @ Xs_40 ) @ N_27 ) )
= ( nth_Ar125878976le_alt @ Ys_6 @ N_27 ) ) ).
thf(fact_554_less__imp__Suc__add,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
=> ? [K_5: nat] :
( N_5
= ( suc @ ( plus_plus_nat @ M @ K_5 ) ) ) ) ).
thf(fact_555_length__Suc__conv,axiom,
! [Xs_39: list_A2115238852le_alt,N_26: nat] :
( ( ( size_s1858781230le_alt @ Xs_39 )
= ( suc @ N_26 ) )
<=> ? [Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] :
( ( Xs_39
= ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) )
& ( ( size_s1858781230le_alt @ Ys_5 )
= N_26 ) ) ) ).
thf(fact_556_Suc__length__conv,axiom,
! [N_25: nat,Xs_38: list_A2115238852le_alt] :
( ( ( suc @ N_25 )
= ( size_s1858781230le_alt @ Xs_38 ) )
<=> ? [Y_3: arrow_475358991le_alt,Ys_5: list_A2115238852le_alt] :
( ( Xs_38
= ( cons_A228743023le_alt @ Y_3 @ Ys_5 ) )
& ( ( size_s1858781230le_alt @ Ys_5 )
= N_25 ) ) ) ).
thf(fact_557_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_5 )
& ( ( plus_plus_nat @ I @ K_5 )
= J ) ) ) ).
thf(fact_558_psubsetD,axiom,
! [C_16: int,A_35: int > $o,B_20: int > $o] :
( ( ord_less_int_o @ A_35 @ B_20 )
=> ( ( member_int @ C_16 @ A_35 )
=> ( member_int @ C_16 @ B_20 ) ) ) ).
thf(fact_559_psubsetD,axiom,
! [C_16: nat,A_35: nat > $o,B_20: nat > $o] :
( ( ord_less_nat_o @ A_35 @ B_20 )
=> ( ( member_nat @ C_16 @ A_35 )
=> ( member_nat @ C_16 @ B_20 ) ) ) ).
thf(fact_560_gr0__implies__Suc,axiom,
! [N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ? [M_1: nat] :
( N_5
= ( suc @ M_1 ) ) ) ).
thf(fact_561_lift__Suc__mono__less,axiom,
! [N_24: nat,N_23: nat,F_1: nat > nat] :
( ! [N_7: nat] : ( ord_less_nat @ ( F_1 @ N_7 ) @ ( F_1 @ ( suc @ N_7 ) ) )
=> ( ( ord_less_nat @ N_24 @ N_23 )
=> ( ord_less_nat @ ( F_1 @ N_24 ) @ ( F_1 @ N_23 ) ) ) ) ).
thf(fact_562_lift__Suc__mono__less__iff,axiom,
! [N_22: nat,M_2: nat,F: nat > nat] :
( ! [N_7: nat] : ( ord_less_nat @ ( F @ N_7 ) @ ( F @ ( suc @ N_7 ) ) )
=> ( ( ord_less_nat @ ( F @ N_22 ) @ ( F @ M_2 ) )
<=> ( ord_less_nat @ N_22 @ M_2 ) ) ) ).
thf(fact_563_even__less__0__iff,axiom,
! [A_34: int] :
( ( ord_less_int @ ( plus_plus_int @ A_34 @ A_34 ) @ zero_zero_int )
<=> ( ord_less_int @ A_34 @ zero_zero_int ) ) ).
thf(fact_564_pos__add__strict,axiom,
! [B_19: int,C_15: int,A_33: int] :
( ( ord_less_int @ zero_zero_int @ A_33 )
=> ( ( ord_less_int @ B_19 @ C_15 )
=> ( ord_less_int @ B_19 @ ( plus_plus_int @ A_33 @ C_15 ) ) ) ) ).
thf(fact_565_pos__add__strict,axiom,
! [B_19: nat,C_15: nat,A_33: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_33 )
=> ( ( ord_less_nat @ B_19 @ C_15 )
=> ( ord_less_nat @ B_19 @ ( plus_plus_nat @ A_33 @ C_15 ) ) ) ) ).
thf(fact_566_double__eq__0__iff,axiom,
! [A_32: int] :
( ( ( plus_plus_int @ A_32 @ A_32 )
= zero_zero_int )
<=> ( A_32 = zero_zero_int ) ) ).
thf(fact_567_lessE,axiom,
! [I: nat,K_3: nat] :
( ( ord_less_nat @ I @ K_3 )
=> ( ( K_3
!= ( suc @ I ) )
=> ~ ! [J_2: nat] :
( ( ord_less_nat @ I @ J_2 )
=> ( K_3
!= ( suc @ J_2 ) ) ) ) ) ).
thf(fact_568_Suc__lessE,axiom,
! [I: nat,K_3: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K_3 )
=> ~ ! [J_2: nat] :
( ( ord_less_nat @ I @ J_2 )
=> ( K_3
!= ( suc @ J_2 ) ) ) ) ).
thf(fact_569_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y
!= ( suc @ Nat ) ) ) ).
thf(fact_570_not0__implies__Suc,axiom,
! [N_5: nat] :
( ( N_5 != zero_zero_nat )
=> ? [M_1: nat] :
( N_5
= ( suc @ M_1 ) ) ) ).
thf(fact_571_nat__induct,axiom,
! [N_5: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N_7: nat] :
( ( P @ N_7 )
=> ( P @ ( suc @ N_7 ) ) )
=> ( P @ N_5 ) ) ) ).
thf(fact_572_zero__induct,axiom,
! [P: nat > $o,K_3: nat] :
( ( P @ K_3 )
=> ( ! [N_7: nat] :
( ( P @ ( suc @ N_7 ) )
=> ( P @ N_7 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_573_take__hd__drop,axiom,
! [N_21: nat,Xs_37: list_A2115238852le_alt] :
( ( ord_less_nat @ N_21 @ ( size_s1858781230le_alt @ Xs_37 ) )
=> ( ( append179082452le_alt @ ( take_A614047655le_alt @ N_21 @ Xs_37 ) @ ( cons_A228743023le_alt @ ( hd_Arr1965683346le_alt @ ( drop_A1346709759le_alt @ N_21 @ Xs_37 ) ) @ nil_Ar1286194111le_alt ) )
= ( take_A614047655le_alt @ ( suc @ N_21 ) @ Xs_37 ) ) ) ).
thf(fact_574_take__tl,axiom,
! [N_20: nat,Xs_36: list_A2115238852le_alt] :
( ( take_A614047655le_alt @ N_20 @ ( tl_Arr465451158le_alt @ Xs_36 ) )
= ( tl_Arr465451158le_alt @ ( take_A614047655le_alt @ ( suc @ N_20 ) @ Xs_36 ) ) ) ).
thf(fact_575_take__Nil,axiom,
! [N_19: nat] :
( ( take_A614047655le_alt @ N_19 @ nil_Ar1286194111le_alt )
= nil_Ar1286194111le_alt ) ).
thf(fact_576_distinct__take,axiom,
! [I_9: nat,Xs_35: list_A2115238852le_alt] :
( ( distin236324274le_alt @ Xs_35 )
=> ( distin236324274le_alt @ ( take_A614047655le_alt @ I_9 @ Xs_35 ) ) ) ).
thf(fact_577_takeWhile__eq__take,axiom,
! [P_2: arrow_475358991le_alt > $o,Xs_34: list_A2115238852le_alt] :
( ( takeWh1696291512le_alt @ P_2 @ Xs_34 )
= ( take_A614047655le_alt @ ( size_s1858781230le_alt @ ( takeWh1696291512le_alt @ P_2 @ Xs_34 ) ) @ Xs_34 ) ) ).
thf(fact_578_append__take__drop__id,axiom,
! [N_18: nat,Xs_33: list_A2115238852le_alt] :
( ( append179082452le_alt @ ( take_A614047655le_alt @ N_18 @ Xs_33 ) @ ( drop_A1346709759le_alt @ N_18 @ Xs_33 ) )
= Xs_33 ) ).
thf(fact_579_take__Suc__Cons,axiom,
! [N_17: nat,X_16: arrow_475358991le_alt,Xs_32: list_A2115238852le_alt] :
( ( take_A614047655le_alt @ ( suc @ N_17 ) @ ( cons_A228743023le_alt @ X_16 @ Xs_32 ) )
= ( cons_A228743023le_alt @ X_16 @ ( take_A614047655le_alt @ N_17 @ Xs_32 ) ) ) ).
thf(fact_580_take__eq__Nil,axiom,
! [N_16: nat,Xs_31: list_A2115238852le_alt] :
( ( ( take_A614047655le_alt @ N_16 @ Xs_31 )
= nil_Ar1286194111le_alt )
<=> ( ( N_16 = zero_zero_nat )
| ( Xs_31 = nil_Ar1286194111le_alt ) ) ) ).
thf(fact_581_take__0,axiom,
! [Xs_30: list_A2115238852le_alt] :
( ( take_A614047655le_alt @ zero_zero_nat @ Xs_30 )
= nil_Ar1286194111le_alt ) ).
thf(fact_582_append__eq__conv__conj,axiom,
! [Xs_29: list_A2115238852le_alt,Ys_4: list_A2115238852le_alt,Zs: list_A2115238852le_alt] :
( ( ( append179082452le_alt @ Xs_29 @ Ys_4 )
= Zs )
<=> ( ( Xs_29
= ( take_A614047655le_alt @ ( size_s1858781230le_alt @ Xs_29 ) @ Zs ) )
& ( Ys_4
= ( drop_A1346709759le_alt @ ( size_s1858781230le_alt @ Xs_29 ) @ Zs ) ) ) ) ).
thf(fact_583_take__add,axiom,
! [I_8: nat,J_1: nat,Xs_28: list_A2115238852le_alt] :
( ( take_A614047655le_alt @ ( plus_plus_nat @ I_8 @ J_1 ) @ Xs_28 )
= ( append179082452le_alt @ ( take_A614047655le_alt @ I_8 @ Xs_28 ) @ ( take_A614047655le_alt @ J_1 @ ( drop_A1346709759le_alt @ I_8 @ Xs_28 ) ) ) ) ).
thf(fact_584_take__butlast,axiom,
! [N_15: nat,Xs_27: list_A2115238852le_alt] :
( ( ord_less_nat @ N_15 @ ( size_s1858781230le_alt @ Xs_27 ) )
=> ( ( take_A614047655le_alt @ N_15 @ ( butlas274947851le_alt @ Xs_27 ) )
= ( take_A614047655le_alt @ N_15 @ Xs_27 ) ) ) ).
thf(fact_585_take__Suc,axiom,
! [N_14: nat,Xs_26: list_A2115238852le_alt] :
( ( Xs_26 != nil_Ar1286194111le_alt )
=> ( ( take_A614047655le_alt @ ( suc @ N_14 ) @ Xs_26 )
= ( cons_A228743023le_alt @ ( hd_Arr1965683346le_alt @ Xs_26 ) @ ( take_A614047655le_alt @ N_14 @ ( tl_Arr465451158le_alt @ Xs_26 ) ) ) ) ) ).
thf(fact_586_take__Suc__conv__app__nth,axiom,
! [I_7: nat,Xs_25: list_A2115238852le_alt] :
( ( ord_less_nat @ I_7 @ ( size_s1858781230le_alt @ Xs_25 ) )
=> ( ( take_A614047655le_alt @ ( suc @ I_7 ) @ Xs_25 )
= ( append179082452le_alt @ ( take_A614047655le_alt @ I_7 @ Xs_25 ) @ ( cons_A228743023le_alt @ ( nth_Ar125878976le_alt @ Xs_25 @ I_7 ) @ nil_Ar1286194111le_alt ) ) ) ) ).
thf(fact_587_id__take__nth__drop,axiom,
! [I_6: nat,Xs_24: list_A2115238852le_alt] :
( ( ord_less_nat @ I_6 @ ( size_s1858781230le_alt @ Xs_24 ) )
=> ( Xs_24
= ( append179082452le_alt @ ( take_A614047655le_alt @ I_6 @ Xs_24 ) @ ( cons_A228743023le_alt @ ( nth_Ar125878976le_alt @ Xs_24 @ I_6 ) @ ( drop_A1346709759le_alt @ ( suc @ I_6 ) @ Xs_24 ) ) ) ) ) ).
thf(fact_588_upd__conv__take__nth__drop,axiom,
! [A_31: arrow_475358991le_alt,I_5: nat,Xs_23: list_A2115238852le_alt] :
( ( ord_less_nat @ I_5 @ ( size_s1858781230le_alt @ Xs_23 ) )
=> ( ( list_u62477863le_alt @ Xs_23 @ I_5 @ A_31 )
= ( append179082452le_alt @ ( take_A614047655le_alt @ I_5 @ Xs_23 ) @ ( cons_A228743023le_alt @ A_31 @ ( drop_A1346709759le_alt @ ( suc @ I_5 ) @ Xs_23 ) ) ) ) ) ).
thf(fact_589_takeWhile__eq__take__P__nth,axiom,
! [P_1: arrow_475358991le_alt > $o,Xs_22: list_A2115238852le_alt,N_13: nat] :
( ! [I_4: nat] :
( ( ord_less_nat @ I_4 @ N_13 )
=> ( ( ord_less_nat @ I_4 @ ( size_s1858781230le_alt @ Xs_22 ) )
=> ( P_1 @ ( nth_Ar125878976le_alt @ Xs_22 @ I_4 ) ) ) )
=> ( ( ( ord_less_nat @ N_13 @ ( size_s1858781230le_alt @ Xs_22 ) )
=> ~ ( P_1 @ ( nth_Ar125878976le_alt @ Xs_22 @ N_13 ) ) )
=> ( ( takeWh1696291512le_alt @ P_1 @ Xs_22 )
= ( take_A614047655le_alt @ N_13 @ Xs_22 ) ) ) ) ).
thf(fact_590_termination__basic__simps_I1_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_591_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( ( ord_less_nat @ X @ Z_1 )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_30: int] :
( ( plus_plus_int @ zero_zero_int @ A_30 )
= A_30 ) ).
thf(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_30: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_30 )
= A_30 ) ).
thf(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_29: int] :
( ( plus_plus_int @ A_29 @ zero_zero_int )
= A_29 ) ).
thf(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_29: nat] :
( ( plus_plus_nat @ A_29 @ zero_zero_nat )
= A_29 ) ).
thf(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_28: int,B_18: int,C_14: int,D_5: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_28 @ B_18 ) @ ( plus_plus_int @ C_14 @ D_5 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_28 @ C_14 ) @ ( plus_plus_int @ B_18 @ D_5 ) ) ) ).
thf(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_28: nat,B_18: nat,C_14: nat,D_5: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_28 @ B_18 ) @ ( plus_plus_nat @ C_14 @ D_5 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_28 @ C_14 ) @ ( plus_plus_nat @ B_18 @ D_5 ) ) ) ).
thf(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_27: int,B_17: int,C_13: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_27 @ B_17 ) @ C_13 )
= ( plus_plus_int @ ( plus_plus_int @ A_27 @ C_13 ) @ B_17 ) ) ).
thf(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_27: nat,B_17: nat,C_13: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_27 @ B_17 ) @ C_13 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_27 @ C_13 ) @ B_17 ) ) ).
thf(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_26: int,B_16: int,C_12: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_26 @ B_16 ) @ C_12 )
= ( plus_plus_int @ A_26 @ ( plus_plus_int @ B_16 @ C_12 ) ) ) ).
thf(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_26: nat,B_16: nat,C_12: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_26 @ B_16 ) @ C_12 )
= ( plus_plus_nat @ A_26 @ ( plus_plus_nat @ B_16 @ C_12 ) ) ) ).
thf(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_25: int,C_11: int,D_4: int] :
( ( plus_plus_int @ A_25 @ ( plus_plus_int @ C_11 @ D_4 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_25 @ C_11 ) @ D_4 ) ) ).
thf(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_25: nat,C_11: nat,D_4: nat] :
( ( plus_plus_nat @ A_25 @ ( plus_plus_nat @ C_11 @ D_4 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_25 @ C_11 ) @ D_4 ) ) ).
thf(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_24: int,C_10: int,D_3: int] :
( ( plus_plus_int @ A_24 @ ( plus_plus_int @ C_10 @ D_3 ) )
= ( plus_plus_int @ C_10 @ ( plus_plus_int @ A_24 @ D_3 ) ) ) ).
thf(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_24: nat,C_10: nat,D_3: nat] :
( ( plus_plus_nat @ A_24 @ ( plus_plus_nat @ C_10 @ D_3 ) )
= ( plus_plus_nat @ C_10 @ ( plus_plus_nat @ A_24 @ D_3 ) ) ) ).
thf(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_23: int,C_9: int] :
( ( plus_plus_int @ A_23 @ C_9 )
= ( plus_plus_int @ C_9 @ A_23 ) ) ).
thf(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_23: nat,C_9: nat] :
( ( plus_plus_nat @ A_23 @ C_9 )
= ( plus_plus_nat @ C_9 @ A_23 ) ) ).
thf(fact_608_add__0__iff,axiom,
! [B_15: int,A_22: int] :
( ( B_15
= ( plus_plus_int @ B_15 @ A_22 ) )
<=> ( A_22 = zero_zero_int ) ) ).
thf(fact_609_add__0__iff,axiom,
! [B_15: nat,A_22: nat] :
( ( B_15
= ( plus_plus_nat @ B_15 @ A_22 ) )
<=> ( A_22 = zero_zero_nat ) ) ).
thf(fact_610_rev__nth,axiom,
! [N_12: nat,Xs_21: list_A2115238852le_alt] :
( ( ord_less_nat @ N_12 @ ( size_s1858781230le_alt @ Xs_21 ) )
=> ( ( nth_Ar125878976le_alt @ ( rev_Ar1106406943le_alt @ Xs_21 ) @ N_12 )
= ( nth_Ar125878976le_alt @ Xs_21 @ ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_21 ) @ ( suc @ N_12 ) ) ) ) ) ).
thf(fact_611_drop__replicate,axiom,
! [I_3: nat,K_4: nat,X_15: arrow_475358991le_alt] :
( ( drop_A1346709759le_alt @ I_3 @ ( replic1511538809le_alt @ K_4 @ X_15 ) )
= ( replic1511538809le_alt @ ( minus_minus_nat @ K_4 @ I_3 ) @ X_15 ) ) ).
thf(fact_612_diff__cancel2,axiom,
! [M: nat,K_3: nat,N_5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K_3 ) @ ( plus_plus_nat @ N_5 @ K_3 ) )
= ( minus_minus_nat @ M @ N_5 ) ) ).
thf(fact_613_diff__cancel,axiom,
! [K_3: nat,M: nat,N_5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K_3 @ M ) @ ( plus_plus_nat @ K_3 @ N_5 ) )
= ( minus_minus_nat @ M @ N_5 ) ) ).
thf(fact_614_diff__diff__left,axiom,
! [I: nat,J: nat,K_3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K_3 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K_3 ) ) ) ).
thf(fact_615_diff__add__inverse,axiom,
! [N_5: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N_5 @ M ) @ N_5 )
= M ) ).
thf(fact_616_diff__add__inverse2,axiom,
! [M: nat,N_5: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N_5 ) @ N_5 )
= M ) ).
thf(fact_617_diff__eq__diff__less,axiom,
! [A_21: int,B_14: int,C_8: int,D_2: int] :
( ( ( minus_minus_int @ A_21 @ B_14 )
= ( minus_minus_int @ C_8 @ D_2 ) )
=> ( ( ord_less_int @ A_21 @ B_14 )
<=> ( ord_less_int @ C_8 @ D_2 ) ) ) ).
thf(fact_618_add__diff__cancel,axiom,
! [A_20: int,B_13: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_20 @ B_13 ) @ B_13 )
= A_20 ) ).
thf(fact_619_diff__add__cancel,axiom,
! [A_19: int,B_12: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_19 @ B_12 ) @ B_12 )
= A_19 ) ).
thf(fact_620_less__imp__diff__less,axiom,
! [N_5: nat,J: nat,K_3: nat] :
( ( ord_less_nat @ J @ K_3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N_5 ) @ K_3 ) ) ).
thf(fact_621_diff__less__mono2,axiom,
! [L: nat,M: nat,N_5: nat] :
( ( ord_less_nat @ M @ N_5 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N_5 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_622_less__iff__diff__less__0,axiom,
! [A_18: int,B_11: int] :
( ( ord_less_int @ A_18 @ B_11 )
<=> ( ord_less_int @ ( minus_minus_int @ A_18 @ B_11 ) @ zero_zero_int ) ) ).
thf(fact_623_diff__less,axiom,
! [M: nat,N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N_5 ) @ M ) ) ) ).
thf(fact_624_zero__less__diff,axiom,
! [N_5: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N_5 @ M ) )
<=> ( ord_less_nat @ M @ N_5 ) ) ).
thf(fact_625_diff__less__Suc,axiom,
! [M: nat,N_5: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N_5 ) @ ( suc @ M ) ) ).
thf(fact_626_diff__add__0,axiom,
! [N_5: nat,M: nat] :
( ( minus_minus_nat @ N_5 @ ( plus_plus_nat @ N_5 @ M ) )
= zero_zero_nat ) ).
thf(fact_627_less__diff__conv,axiom,
! [I: nat,J: nat,K_3: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K_3 ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_3 ) @ J ) ) ).
thf(fact_628_add__diff__inverse,axiom,
! [M: nat,N_5: nat] :
( ~ ( ord_less_nat @ M @ N_5 )
=> ( ( plus_plus_nat @ N_5 @ ( minus_minus_nat @ M @ N_5 ) )
= M ) ) ).
thf(fact_629_diff__commute,axiom,
! [I: nat,J: nat,K_3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K_3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K_3 ) @ J ) ) ).
thf(fact_630_max__diff__distrib__left,axiom,
! [X_14: int,Y_2: int,Z: int] :
( ( minus_minus_int @ ( ord_max_int @ X_14 @ Y_2 ) @ Z )
= ( ord_max_int @ ( minus_minus_int @ X_14 @ Z ) @ ( minus_minus_int @ Y_2 @ Z ) ) ) ).
thf(fact_631_diff__eq__diff__eq,axiom,
! [A_17: int,B_10: int,C_7: int,D_1: int] :
( ( ( minus_minus_int @ A_17 @ B_10 )
= ( minus_minus_int @ C_7 @ D_1 ) )
=> ( ( A_17 = B_10 )
<=> ( C_7 = D_1 ) ) ) ).
thf(fact_632_diff__0__right,axiom,
! [A_16: int] :
( ( minus_minus_int @ A_16 @ zero_zero_int )
= A_16 ) ).
thf(fact_633_diff__self,axiom,
! [A_15: int] :
( ( minus_minus_int @ A_15 @ A_15 )
= zero_zero_int ) ).
thf(fact_634_eq__iff__diff__eq__0,axiom,
! [A_14: int,B_9: int] :
( ( A_14 = B_9 )
<=> ( ( minus_minus_int @ A_14 @ B_9 )
= zero_zero_int ) ) ).
thf(fact_635_right__minus__eq,axiom,
! [A_13: int,B_8: int] :
( ( ( minus_minus_int @ A_13 @ B_8 )
= zero_zero_int )
<=> ( A_13 = B_8 ) ) ).
thf(fact_636_diffs0__imp__equal,axiom,
! [M: nat,N_5: nat] :
( ( ( minus_minus_nat @ M @ N_5 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N_5 @ M )
= zero_zero_nat )
=> ( M = N_5 ) ) ) ).
thf(fact_637_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_638_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_639_diff__0__eq__0,axiom,
! [N_5: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N_5 )
= zero_zero_nat ) ).
thf(fact_640_diff__Suc__Suc,axiom,
! [M: nat,N_5: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N_5 ) )
= ( minus_minus_nat @ M @ N_5 ) ) ).
thf(fact_641_Suc__diff__diff,axiom,
! [M: nat,N_5: nat,K_3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N_5 ) @ ( suc @ K_3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N_5 ) @ K_3 ) ) ).
thf(fact_642_nat__minus__add__max,axiom,
! [N_5: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N_5 @ M ) @ M )
= ( ord_max_nat @ N_5 @ M ) ) ).
thf(fact_643_diff__Suc__less,axiom,
! [I: nat,N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ord_less_nat @ ( minus_minus_nat @ N_5 @ ( suc @ I ) ) @ N_5 ) ) ).
thf(fact_644_Suc__pred,axiom,
! [N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ( suc @ ( minus_minus_nat @ N_5 @ ( suc @ zero_zero_nat ) ) )
= N_5 ) ) ).
thf(fact_645_nat__diff__split,axiom,
! [P: nat > $o,A_12: nat,B_7: nat] :
( ( P @ ( minus_minus_nat @ A_12 @ B_7 ) )
<=> ( ( ( ord_less_nat @ A_12 @ B_7 )
=> ( P @ zero_zero_nat ) )
& ! [D: nat] :
( ( A_12
= ( plus_plus_nat @ B_7 @ D ) )
=> ( P @ D ) ) ) ) ).
thf(fact_646_nat__diff__split__asm,axiom,
! [P: nat > $o,A_12: nat,B_7: nat] :
( ( P @ ( minus_minus_nat @ A_12 @ B_7 ) )
<=> ~ ( ( ( ord_less_nat @ A_12 @ B_7 )
& ~ ( P @ zero_zero_nat ) )
| ? [D: nat] :
( ( A_12
= ( plus_plus_nat @ B_7 @ D ) )
& ~ ( P @ D ) ) ) ) ).
thf(fact_647_take__append,axiom,
! [N_11: nat,Xs_20: list_A2115238852le_alt,Ys_3: list_A2115238852le_alt] :
( ( take_A614047655le_alt @ N_11 @ ( append179082452le_alt @ Xs_20 @ Ys_3 ) )
= ( append179082452le_alt @ ( take_A614047655le_alt @ N_11 @ Xs_20 ) @ ( take_A614047655le_alt @ ( minus_minus_nat @ N_11 @ ( size_s1858781230le_alt @ Xs_20 ) ) @ Ys_3 ) ) ) ).
thf(fact_648_drop__append,axiom,
! [N_10: nat,Xs_19: list_A2115238852le_alt,Ys_2: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ N_10 @ ( append179082452le_alt @ Xs_19 @ Ys_2 ) )
= ( append179082452le_alt @ ( drop_A1346709759le_alt @ N_10 @ Xs_19 ) @ ( drop_A1346709759le_alt @ ( minus_minus_nat @ N_10 @ ( size_s1858781230le_alt @ Xs_19 ) ) @ Ys_2 ) ) ) ).
thf(fact_649_nth__append,axiom,
! [Ys_1: list_A2115238852le_alt,N_9: nat,Xs_18: list_A2115238852le_alt] :
( ( ( ord_less_nat @ N_9 @ ( size_s1858781230le_alt @ Xs_18 ) )
=> ( ( nth_Ar125878976le_alt @ ( append179082452le_alt @ Xs_18 @ Ys_1 ) @ N_9 )
= ( nth_Ar125878976le_alt @ Xs_18 @ N_9 ) ) )
& ( ~ ( ord_less_nat @ N_9 @ ( size_s1858781230le_alt @ Xs_18 ) )
=> ( ( nth_Ar125878976le_alt @ ( append179082452le_alt @ Xs_18 @ Ys_1 ) @ N_9 )
= ( nth_Ar125878976le_alt @ Ys_1 @ ( minus_minus_nat @ N_9 @ ( size_s1858781230le_alt @ Xs_18 ) ) ) ) ) ) ).
thf(fact_650_list__update__append,axiom,
! [Ys: list_A2115238852le_alt,X_13: arrow_475358991le_alt,N_8: nat,Xs_17: list_A2115238852le_alt] :
( ( ( ord_less_nat @ N_8 @ ( size_s1858781230le_alt @ Xs_17 ) )
=> ( ( list_u62477863le_alt @ ( append179082452le_alt @ Xs_17 @ Ys ) @ N_8 @ X_13 )
= ( append179082452le_alt @ ( list_u62477863le_alt @ Xs_17 @ N_8 @ X_13 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N_8 @ ( size_s1858781230le_alt @ Xs_17 ) )
=> ( ( list_u62477863le_alt @ ( append179082452le_alt @ Xs_17 @ Ys ) @ N_8 @ X_13 )
= ( append179082452le_alt @ Xs_17 @ ( list_u62477863le_alt @ Ys @ ( minus_minus_nat @ N_8 @ ( size_s1858781230le_alt @ Xs_17 ) ) @ X_13 ) ) ) ) ) ).
thf(fact_651_rev__drop,axiom,
! [I_2: nat,Xs_16: list_A2115238852le_alt] :
( ( rev_Ar1106406943le_alt @ ( drop_A1346709759le_alt @ I_2 @ Xs_16 ) )
= ( take_A614047655le_alt @ ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_16 ) @ I_2 ) @ ( rev_Ar1106406943le_alt @ Xs_16 ) ) ) ).
thf(fact_652_rev__take,axiom,
! [I_1: nat,Xs_15: list_A2115238852le_alt] :
( ( rev_Ar1106406943le_alt @ ( take_A614047655le_alt @ I_1 @ Xs_15 ) )
= ( drop_A1346709759le_alt @ ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_15 ) @ I_1 ) @ ( rev_Ar1106406943le_alt @ Xs_15 ) ) ) ).
thf(fact_653_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K_3: nat] :
( ( P @ K_3 )
=> ( ! [N_7: nat] :
( ( P @ ( suc @ N_7 ) )
=> ( P @ N_7 ) )
=> ( P @ ( minus_minus_nat @ K_3 @ I ) ) ) ) ).
thf(fact_654_last__list__update,axiom,
! [X_12: arrow_475358991le_alt,K_2: nat,Xs_14: list_A2115238852le_alt] :
( ( Xs_14 != nil_Ar1286194111le_alt )
=> ( ( ( K_2
= ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_14 ) @ one_one_nat ) )
=> ( ( last_A1217315288le_alt @ ( list_u62477863le_alt @ Xs_14 @ K_2 @ X_12 ) )
= X_12 ) )
& ( ( K_2
!= ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_14 ) @ one_one_nat ) )
=> ( ( last_A1217315288le_alt @ ( list_u62477863le_alt @ Xs_14 @ K_2 @ X_12 ) )
= ( last_A1217315288le_alt @ Xs_14 ) ) ) ) ) ).
thf(fact_655_empty__Diff,axiom,
! [A_11: nat > $o] :
( ( minus_minus_nat_o @ bot_bot_nat_o @ A_11 )
= bot_bot_nat_o ) ).
thf(fact_656_Diff__empty,axiom,
! [A_10: nat > $o] :
( ( minus_minus_nat_o @ A_10 @ bot_bot_nat_o )
= A_10 ) ).
thf(fact_657_Diff__cancel,axiom,
! [A_9: nat > $o] :
( ( minus_minus_nat_o @ A_9 @ A_9 )
= bot_bot_nat_o ) ).
thf(fact_658_diff__Suc__1,axiom,
! [N_5: nat] :
( ( minus_minus_nat @ ( suc @ N_5 ) @ one_one_nat )
= N_5 ) ).
thf(fact_659_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N_5: nat] :
( ( minus_minus_nat @ M @ ( suc @ N_5 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_5 ) ) ).
thf(fact_660_Suc__eq__plus1__left,axiom,
! [N_5: nat] :
( ( suc @ N_5 )
= ( plus_plus_nat @ one_one_nat @ N_5 ) ) ).
thf(fact_661_Suc__eq__plus1,axiom,
! [N_5: nat] :
( ( suc @ N_5 )
= ( plus_plus_nat @ N_5 @ one_one_nat ) ) ).
thf(fact_662_drop__1__Cons,axiom,
! [X_11: arrow_475358991le_alt,Xs_13: list_A2115238852le_alt] :
( ( drop_A1346709759le_alt @ one_one_nat @ ( cons_A228743023le_alt @ X_11 @ Xs_13 ) )
= Xs_13 ) ).
thf(fact_663_one__reorient,axiom,
! [X_10: nat] :
( ( one_one_nat = X_10 )
<=> ( X_10 = one_one_nat ) ) ).
thf(fact_664_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_665_less__add__one,axiom,
! [A_8: nat] : ( ord_less_nat @ A_8 @ ( plus_plus_nat @ A_8 @ one_one_nat ) ) ).
thf(fact_666_less__add__one,axiom,
! [A_8: int] : ( ord_less_int @ A_8 @ ( plus_plus_int @ A_8 @ one_one_int ) ) ).
thf(fact_667_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_668_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_669_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_670_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_671_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_672_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_673_take__1__Cons,axiom,
! [X_9: arrow_475358991le_alt,Xs_12: list_A2115238852le_alt] :
( ( take_A614047655le_alt @ one_one_nat @ ( cons_A228743023le_alt @ X_9 @ Xs_12 ) )
= ( cons_A228743023le_alt @ X_9 @ nil_Ar1286194111le_alt ) ) ).
thf(fact_674_length__butlast,axiom,
! [Xs_11: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( butlas274947851le_alt @ Xs_11 ) )
= ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_11 ) @ one_one_nat ) ) ).
thf(fact_675_length__tl,axiom,
! [Xs_10: list_A2115238852le_alt] :
( ( size_s1858781230le_alt @ ( tl_Arr465451158le_alt @ Xs_10 ) )
= ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_10 ) @ one_one_nat ) ) ).
thf(fact_676_tl__take,axiom,
! [N_6: nat,Xs_9: list_A2115238852le_alt] :
( ( tl_Arr465451158le_alt @ ( take_A614047655le_alt @ N_6 @ Xs_9 ) )
= ( take_A614047655le_alt @ ( minus_minus_nat @ N_6 @ one_one_nat ) @ ( tl_Arr465451158le_alt @ Xs_9 ) ) ) ).
thf(fact_677_Suc__pred_H,axiom,
! [N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( N_5
= ( suc @ ( minus_minus_nat @ N_5 @ one_one_nat ) ) ) ) ).
thf(fact_678_Suc__diff__1,axiom,
! [N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ( suc @ ( minus_minus_nat @ N_5 @ one_one_nat ) )
= N_5 ) ) ).
thf(fact_679_add__eq__if,axiom,
! [N_5: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_5 )
= N_5 ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N_5 )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N_5 ) ) ) ) ) ).
thf(fact_680_nth__Cons_H,axiom,
! [X_8: arrow_475358991le_alt,Xs_8: list_A2115238852le_alt,N_4: nat] :
( ( ( N_4 = zero_zero_nat )
=> ( ( nth_Ar125878976le_alt @ ( cons_A228743023le_alt @ X_8 @ Xs_8 ) @ N_4 )
= X_8 ) )
& ( ( N_4 != zero_zero_nat )
=> ( ( nth_Ar125878976le_alt @ ( cons_A228743023le_alt @ X_8 @ Xs_8 ) @ N_4 )
= ( nth_Ar125878976le_alt @ Xs_8 @ ( minus_minus_nat @ N_4 @ one_one_nat ) ) ) ) ) ).
thf(fact_681_drop__Cons_H,axiom,
! [X_7: arrow_475358991le_alt,Xs_7: list_A2115238852le_alt,N_3: nat] :
( ( ( N_3 = zero_zero_nat )
=> ( ( drop_A1346709759le_alt @ N_3 @ ( cons_A228743023le_alt @ X_7 @ Xs_7 ) )
= ( cons_A228743023le_alt @ X_7 @ Xs_7 ) ) )
& ( ( N_3 != zero_zero_nat )
=> ( ( drop_A1346709759le_alt @ N_3 @ ( cons_A228743023le_alt @ X_7 @ Xs_7 ) )
= ( drop_A1346709759le_alt @ ( minus_minus_nat @ N_3 @ one_one_nat ) @ Xs_7 ) ) ) ) ).
thf(fact_682_butlast__conv__take,axiom,
! [Xs_6: list_A2115238852le_alt] :
( ( butlas274947851le_alt @ Xs_6 )
= ( take_A614047655le_alt @ ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_6 ) @ one_one_nat ) @ Xs_6 ) ) ).
thf(fact_683_butlast__list__update,axiom,
! [X_6: arrow_475358991le_alt,K_1: nat,Xs_5: list_A2115238852le_alt] :
( ( ( K_1
= ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_5 ) @ one_one_nat ) )
=> ( ( butlas274947851le_alt @ ( list_u62477863le_alt @ Xs_5 @ K_1 @ X_6 ) )
= ( butlas274947851le_alt @ Xs_5 ) ) )
& ( ( K_1
!= ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_5 ) @ one_one_nat ) )
=> ( ( butlas274947851le_alt @ ( list_u62477863le_alt @ Xs_5 @ K_1 @ X_6 ) )
= ( list_u62477863le_alt @ ( butlas274947851le_alt @ Xs_5 ) @ K_1 @ X_6 ) ) ) ) ).
thf(fact_684_tl__replicate,axiom,
! [X_5: arrow_475358991le_alt,N_2: nat] :
( ( N_2 != zero_zero_nat )
=> ( ( tl_Arr465451158le_alt @ ( replic1511538809le_alt @ N_2 @ X_5 ) )
= ( replic1511538809le_alt @ ( minus_minus_nat @ N_2 @ one_one_nat ) @ X_5 ) ) ) ).
thf(fact_685_nth__Cons__pos,axiom,
! [X_4: arrow_475358991le_alt,Xs_4: list_A2115238852le_alt,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( nth_Ar125878976le_alt @ ( cons_A228743023le_alt @ X_4 @ Xs_4 ) @ N_1 )
= ( nth_Ar125878976le_alt @ Xs_4 @ ( minus_minus_nat @ N_1 @ one_one_nat ) ) ) ) ).
thf(fact_686_take__Cons_H,axiom,
! [X_3: arrow_475358991le_alt,Xs_3: list_A2115238852le_alt,N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_A614047655le_alt @ N @ ( cons_A228743023le_alt @ X_3 @ Xs_3 ) )
= nil_Ar1286194111le_alt ) )
& ( ( N != zero_zero_nat )
=> ( ( take_A614047655le_alt @ N @ ( cons_A228743023le_alt @ X_3 @ Xs_3 ) )
= ( cons_A228743023le_alt @ X_3 @ ( take_A614047655le_alt @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs_3 ) ) ) ) ) ).
thf(fact_687_rev__update,axiom,
! [Y_1: arrow_475358991le_alt,K: nat,Xs_2: list_A2115238852le_alt] :
( ( ord_less_nat @ K @ ( size_s1858781230le_alt @ Xs_2 ) )
=> ( ( rev_Ar1106406943le_alt @ ( list_u62477863le_alt @ Xs_2 @ K @ Y_1 ) )
= ( list_u62477863le_alt @ ( rev_Ar1106406943le_alt @ Xs_2 ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_2 ) @ K ) @ one_one_nat ) @ Y_1 ) ) ) ).
thf(fact_688_last__conv__nth,axiom,
! [Xs_1: list_A2115238852le_alt] :
( ( Xs_1 != nil_Ar1286194111le_alt )
=> ( ( last_A1217315288le_alt @ Xs_1 )
= ( nth_Ar125878976le_alt @ Xs_1 @ ( minus_minus_nat @ ( size_s1858781230le_alt @ Xs_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_689_take__Cons__number__of,axiom,
! [X_2: arrow_475358991le_alt,Xs: list_A2115238852le_alt,V_2: int] :
( ( ( ( number_number_of_nat @ V_2 )
= zero_zero_nat )
=> ( ( take_A614047655le_alt @ ( number_number_of_nat @ V_2 ) @ ( cons_A228743023le_alt @ X_2 @ Xs ) )
= nil_Ar1286194111le_alt ) )
& ( ( ( number_number_of_nat @ V_2 )
!= zero_zero_nat )
=> ( ( take_A614047655le_alt @ ( number_number_of_nat @ V_2 ) @ ( cons_A228743023le_alt @ X_2 @ Xs ) )
= ( cons_A228743023le_alt @ X_2 @ ( take_A614047655le_alt @ ( minus_minus_nat @ ( number_number_of_nat @ V_2 ) @ one_one_nat ) @ Xs ) ) ) ) ) ).
thf(fact_690_Ints__odd__less__0,axiom,
! [A_7: int] :
( ( member_int @ A_7 @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A_7 ) @ A_7 ) @ zero_zero_int )
<=> ( ord_less_int @ A_7 @ zero_zero_int ) ) ) ).
thf(fact_691_DiffE,axiom,
! [C_6: nat,A_6: nat > $o,B_6: nat > $o] :
( ( member_nat @ C_6 @ ( minus_minus_nat_o @ A_6 @ B_6 ) )
=> ~ ( ( member_nat @ C_6 @ A_6 )
=> ( member_nat @ C_6 @ B_6 ) ) ) ).
thf(fact_692_DiffI,axiom,
! [B_5: nat > $o,C_5: nat,A_5: nat > $o] :
( ( member_nat @ C_5 @ A_5 )
=> ( ~ ( member_nat @ C_5 @ B_5 )
=> ( member_nat @ C_5 @ ( minus_minus_nat_o @ A_5 @ B_5 ) ) ) ) ).
thf(fact_693_Ints__diff,axiom,
! [B_4: int,A_4: int] :
( ( member_int @ A_4 @ ring_1_Ints_int )
=> ( ( member_int @ B_4 @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A_4 @ B_4 ) @ ring_1_Ints_int ) ) ) ).
thf(fact_694_number__of__diff,axiom,
! [V_1: int,W_1: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_1 @ W_1 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_1 ) @ ( number_number_of_int @ W_1 ) ) ) ).
thf(fact_695_add__number__of__diff1,axiom,
! [V: int,W: int,C_4: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( minus_minus_int @ ( number_number_of_int @ W ) @ C_4 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) @ C_4 ) ) ).
thf(fact_696_DiffD2,axiom,
! [C_3: nat,A_3: nat > $o,B_3: nat > $o] :
( ( member_nat @ C_3 @ ( minus_minus_nat_o @ A_3 @ B_3 ) )
=> ~ ( member_nat @ C_3 @ B_3 ) ) ).
thf(fact_697_DiffD1,axiom,
! [C_2: nat,A_2: nat > $o,B_2: nat > $o] :
( ( member_nat @ C_2 @ ( minus_minus_nat_o @ A_2 @ B_2 ) )
=> ( member_nat @ C_2 @ A_2 ) ) ).
thf(fact_698_Diff__iff,axiom,
! [C_1: nat,A_1: nat > $o,B_1: nat > $o] :
( ( member_nat @ C_1 @ ( minus_minus_nat_o @ A_1 @ B_1 ) )
<=> ( ( member_nat @ C_1 @ A_1 )
& ~ ( member_nat @ C_1 @ B_1 ) ) ) ).
thf(fact_699_set__diff__eq,axiom,
! [A: nat > $o,B: nat > $o] :
( ( minus_minus_nat_o @ A @ B )
= ( collect_nat
@ ^ [X_1: nat] : ( (&) @ ( member_nat @ X_1 @ A ) @ ( (~) @ ( member_nat @ X_1 @ B ) ) ) ) ) ).
%----Helper facts (2)
thf(help_fequal_1_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____lc,axiom,
! [X: list_A2115238852le_alt,Y: list_A2115238852le_alt] :
( ~ ( fequal781288069le_alt @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____lc,axiom,
! [X: list_A2115238852le_alt,Y: list_A2115238852le_alt] :
( ( X != Y )
| ( fequal781288069le_alt @ X @ Y ) ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
a != b ).
thf(conj_1,conjecture,
? [C: arrow_475358991le_alt] : ( distin236324274le_alt @ ( cons_A228743023le_alt @ a @ ( cons_A228743023le_alt @ b @ ( cons_A228743023le_alt @ C @ nil_Ar1286194111le_alt ) ) ) ) ).
%------------------------------------------------------------------------------