%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00204_008111__11874910_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1346 ( 654 unt; 146 typ;   0 def)
%            Number of atoms       : 2884 (1797 equ;   0 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 10195 ( 308   ~;  73   |; 204   &;8441   @)
%                                         (   0 <=>;1169  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   6 avg)
%            Number of types       :   18 (  17 usr)
%            Number of type conns  :  421 ( 421   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  132 ( 129 usr;  12 con; 0-3 aty)
%            Number of variables   : 3217 ( 104   ^;2991   !; 122   ?;3217   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:13:50.218
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P7940050157051400743st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc1828647624359046049st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    list_P7736225833432154391st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P5364314822750548887at_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J,type,
    list_l5212752354702395664st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4575160907756185873st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc1540777390238407569at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    list_list_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    set_list_list_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

% Explicit typings (129)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__Nat__Onat_J,type,
    bNF_Gr9051742241863529473st_nat: set_list_list_nat > list_nat > set_list_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
    bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Nat__Onat_J,type,
    bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
    bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: nat > nat > nat ).

thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
    equiva7426478223624825838m_rgfs: nat > list_list_nat ).

thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
    equiva3371634703666331078on_rgf: list_nat > $o ).

thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit,type,
    equiva5889994315859557365_limit: list_nat > nat ).

thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit__rel,type,
    equiva5575797544161152836it_rel: list_nat > list_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    append_list_list_nat: list_list_list_nat > list_list_list_nat > list_list_list_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
    append_list_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    append2623875052807961020st_nat: list_P7940050157051400743st_nat > list_P7940050157051400743st_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    append6293600105956864812at_nat: list_P5364314822750548887at_nat > list_P5364314822750548887at_nat > list_P5364314822750548887at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    append104611586619867308st_nat: list_P7736225833432154391st_nat > list_P7736225833432154391st_nat > list_P7736225833432154391st_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).

thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
    butlast_list_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
    butlast_nat: list_nat > list_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
    concat_list_nat: list_list_list_nat > list_list_nat ).

thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

thf(sy_c_List_Ocount__list_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    count_8975921713275557620st_nat: list_list_list_nat > list_list_nat > nat ).

thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
    count_list_list_nat: list_list_nat > list_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
    count_list_nat: list_nat > nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    count_8697817428404038429st_nat: list_P7940050157051400743st_nat > produc1828647624359046049st_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    count_2704080706644019469at_nat: list_P5364314822750548887at_nat > produc1540777390238407569at_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    count_5738464224161797773st_nat: list_P7736225833432154391st_nat > produc4575160907756185873st_nat > nat ).

thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).

thf(sy_c_List_Odistinct__adj_001t__List__Olist_It__Nat__Onat_J,type,
    distin876741697294417026st_nat: list_list_nat > $o ).

thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
    distinct_adj_nat: list_nat > $o ).

thf(sy_c_List_Odrop_001t__List__Olist_It__Nat__Onat_J,type,
    drop_list_nat: nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Ofilter_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    filter_list_list_nat: ( list_list_nat > $o ) > list_list_list_nat > list_list_list_nat ).

thf(sy_c_List_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
    filter_list_nat: ( list_nat > $o ) > list_list_nat > list_list_nat ).

thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
    filter_nat: ( nat > $o ) > list_nat > list_nat ).

thf(sy_c_List_Ofilter_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    filter3449886728380977566st_nat: ( produc1828647624359046049st_nat > $o ) > list_P7940050157051400743st_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Ofilter_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    filter8517723620809974798at_nat: ( produc1540777390238407569at_nat > $o ) > list_P5364314822750548887at_nat > list_P5364314822750548887at_nat ).

thf(sy_c_List_Ofilter_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    filter2328735101472977294st_nat: ( produc4575160907756185873st_nat > $o ) > list_P7736225833432154391st_nat > list_P7736225833432154391st_nat ).

thf(sy_c_List_Ofilter_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    filter6372491115368938494at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
    gen_length_list_nat: nat > list_list_nat > nat ).

thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
    gen_length_nat: nat > list_nat > nat ).

thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
    last_list_nat: list_list_nat > list_nat ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    cons_l7310388135179752778st_nat: list_list_list_nat > list_l5212752354702395664st_nat > list_l5212752354702395664st_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
    cons_list_nat: list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    nil_li5455921165481563386st_nat: list_l5212752354702395664st_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    nil_list_list_nat: list_list_list_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
    nil_list_nat: list_list_nat ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    hd_list_list_nat: list_list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
    hd_list_nat: list_list_nat > list_nat ).

thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
    list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
    list_ex1_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
    list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).

thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    maps_l5785965478274863235st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_nat ).

thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    maps_list_nat_nat: ( list_nat > list_nat ) > list_list_nat > list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    maps_nat_list_nat: ( nat > list_list_nat ) > list_nat > list_list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
    maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).

thf(sy_c_List_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
    n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).

thf(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2861266219255159431st_nat: list_list_nat > list_list_nat > list_P7940050157051400743st_nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    product_list_nat_nat: list_list_nat > list_nat > list_P5364314822750548887at_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    product_nat_list_nat: list_nat > list_list_nat > list_P7736225833432154391st_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
    produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).

thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
    rotate1_list_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
    subseqs_list_nat: list_list_nat > list_list_list_nat ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Otake_001t__List__Olist_It__Nat__Onat_J,type,
    take_list_nat: nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s6248950052170075156st_nat: list_list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s7664791237847770771st_nat: list_P7940050157051400743st_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    size_s6663376490332876291at_nat: list_P5364314822750548887at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    size_s9035287501014481795st_nat: list_P7736225833432154391st_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: nat > nat > nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Stirling_OStirling,type,
    stirling: nat > nat > nat ).

thf(sy_c_Stirling_Ostirling,type,
    stirling2: nat > nat > nat ).

thf(sy_c_Stirling_Ostirling__row,type,
    stirling_row: nat > list_nat ).

thf(sy_c_Stirling_Ostirling__row__aux_001t__Nat__Onat,type,
    stirling_row_aux_nat: nat > nat > list_nat > list_nat ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__Nat__Onat_J,type,
    prefix_list_nat: list_list_nat > list_list_nat > $o ).

thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
    prefix_nat: list_nat > list_nat > $o ).

thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__Nat__Onat_J,type,
    prefixes_list_nat: list_list_nat > list_list_list_nat ).

thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
    prefixes_nat: list_nat > list_list_nat ).

thf(sy_c_Sublist_Osublists_001t__List__Olist_It__Nat__Onat_J,type,
    sublists_list_nat: list_list_nat > list_list_list_nat ).

thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
    sublists_nat: list_nat > list_list_nat ).

thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
    suffixes_list_nat: list_list_nat > list_list_list_nat ).

thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
    suffixes_nat: list_nat > list_list_nat ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    member_list_list_nat: list_list_nat > set_list_list_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_v_x,type,
    x: list_nat ).

thf(sy_v_xa____,type,
    xa: nat ).

thf(sy_v_xs____,type,
    xs: list_nat ).

% Relevant facts (1196)
thf(fact_0_assms,axiom,
    equiva3371634703666331078on_rgf @ x ).

% assms
thf(fact_1_snoc_Oprems,axiom,
    equiva3371634703666331078on_rgf @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ).

% snoc.prems
thf(fact_2_b,axiom,
    equiva3371634703666331078on_rgf @ xs ).

% b
thf(fact_3_rgf__limit_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X2: nat,Xs: list_nat] :
            ( X
           != ( cons_nat @ X2 @ Xs ) ) ) ).

% rgf_limit.cases
thf(fact_4_count__list__expand,axiom,
    ( count_8975921713275557620st_nat
    = ( ^ [Xs2: list_list_list_nat,X3: list_list_nat] :
          ( size_s6248950052170075156st_nat
          @ ( filter_list_list_nat
            @ ( ^ [Y: list_list_nat,Z: list_list_nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_5_count__list__expand,axiom,
    ( count_4203492906077236349at_nat
    = ( ^ [Xs2: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
          ( size_s5460976970255530739at_nat
          @ ( filter6372491115368938494at_nat
            @ ( ^ [Y: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_6_count__list__expand,axiom,
    ( count_5738464224161797773st_nat
    = ( ^ [Xs2: list_P7736225833432154391st_nat,X3: produc4575160907756185873st_nat] :
          ( size_s9035287501014481795st_nat
          @ ( filter2328735101472977294st_nat
            @ ( ^ [Y: produc4575160907756185873st_nat,Z: produc4575160907756185873st_nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_7_count__list__expand,axiom,
    ( count_2704080706644019469at_nat
    = ( ^ [Xs2: list_P5364314822750548887at_nat,X3: produc1540777390238407569at_nat] :
          ( size_s6663376490332876291at_nat
          @ ( filter8517723620809974798at_nat
            @ ( ^ [Y: produc1540777390238407569at_nat,Z: produc1540777390238407569at_nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_8_count__list__expand,axiom,
    ( count_8697817428404038429st_nat
    = ( ^ [Xs2: list_P7940050157051400743st_nat,X3: produc1828647624359046049st_nat] :
          ( size_s7664791237847770771st_nat
          @ ( filter3449886728380977566st_nat
            @ ( ^ [Y: produc1828647624359046049st_nat,Z: produc1828647624359046049st_nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_9_count__list__expand,axiom,
    ( count_list_list_nat
    = ( ^ [Xs2: list_list_nat,X3: list_nat] :
          ( size_s3023201423986296836st_nat
          @ ( filter_list_nat
            @ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_10_count__list__expand,axiom,
    ( count_list_nat
    = ( ^ [Xs2: list_nat,X3: nat] :
          ( size_size_list_nat
          @ ( filter_nat
            @ ( ^ [Y: nat,Z: nat] : ( Y = Z )
              @ X3 )
            @ Xs2 ) ) ) ) ).

% count_list_expand
thf(fact_11_list__induct__2__rev,axiom,
    ! [X: list_nat,Y2: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ X )
        = ( size_size_list_nat @ Y2 ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_12_list__induct__2__rev,axiom,
    ! [X: list_list_nat,Y2: list_nat,P: list_list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ X )
        = ( size_size_list_nat @ Y2 ) )
     => ( ( P @ nil_list_nat @ nil_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_13_list__induct__2__rev,axiom,
    ! [X: list_nat,Y2: list_list_nat,P: list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ X )
        = ( size_s3023201423986296836st_nat @ Y2 ) )
     => ( ( P @ nil_nat @ nil_list_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_s3023201423986296836st_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_14_list__induct__2__rev,axiom,
    ! [X: list_list_nat,Y2: list_list_nat,P: list_list_nat > list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ X )
        = ( size_s3023201423986296836st_nat @ Y2 ) )
     => ( ( P @ nil_list_nat @ nil_list_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs )
                = ( size_s3023201423986296836st_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_15_list__induct__2__rev,axiom,
    ! [X: list_nat,Y2: list_list_list_nat,P: list_nat > list_list_list_nat > $o] :
      ( ( ( size_size_list_nat @ X )
        = ( size_s6248950052170075156st_nat @ Y2 ) )
     => ( ( P @ nil_nat @ nil_list_list_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: list_list_nat,Ys: list_list_list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_s6248950052170075156st_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_list_list_nat @ Ys @ ( cons_list_list_nat @ Y3 @ nil_list_list_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_16_list__induct__2__rev,axiom,
    ! [X: list_nat,Y2: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_size_list_nat @ X )
        = ( size_s5460976970255530739at_nat @ Y2 ) )
     => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_s5460976970255530739at_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y3 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_17_list__induct__2__rev,axiom,
    ! [X: list_list_list_nat,Y2: list_nat,P: list_list_list_nat > list_nat > $o] :
      ( ( ( size_s6248950052170075156st_nat @ X )
        = ( size_size_list_nat @ Y2 ) )
     => ( ( P @ nil_list_list_nat @ nil_nat )
       => ( ! [X2: list_list_nat,Xs: list_list_list_nat,Y3: nat,Ys: list_nat] :
              ( ( ( size_s6248950052170075156st_nat @ Xs )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_18_list__induct__2__rev,axiom,
    ! [X: list_P6011104703257516679at_nat,Y2: list_nat,P: list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ X )
        = ( size_size_list_nat @ Y2 ) )
     => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
       => ( ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y3: nat,Ys: list_nat] :
              ( ( ( size_s5460976970255530739at_nat @ Xs )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X2 @ nil_Pr5478986624290739719at_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_19_list__induct__2__rev,axiom,
    ! [X: list_list_nat,Y2: list_list_list_nat,P: list_list_nat > list_list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ X )
        = ( size_s6248950052170075156st_nat @ Y2 ) )
     => ( ( P @ nil_list_nat @ nil_list_list_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_list_nat,Ys: list_list_list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs )
                = ( size_s6248950052170075156st_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_list_list_nat @ Ys @ ( cons_list_list_nat @ Y3 @ nil_list_list_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_20_list__induct__2__rev,axiom,
    ! [X: list_list_nat,Y2: list_P6011104703257516679at_nat,P: list_list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ X )
        = ( size_s5460976970255530739at_nat @ Y2 ) )
     => ( ( P @ nil_list_nat @ nil_Pr5478986624290739719at_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs )
                = ( size_s5460976970255530739at_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y3 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
         => ( P @ X @ Y2 ) ) ) ) ).

% list_induct_2_rev
thf(fact_21_calculation,axiom,
    ( one_one_nat
    = ( size_s3023201423986296836st_nat
      @ ( filter_list_nat
        @ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
          @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) )
        @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ) ) ) ) ) ).

% calculation
thf(fact_22_append1__eq__conv,axiom,
    ! [Xs3: list_list_list_nat,X: list_list_nat,Ys2: list_list_list_nat,Y2: list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) )
        = ( append_list_list_nat @ Ys2 @ ( cons_list_list_nat @ Y2 @ nil_list_list_nat ) ) )
      = ( ( Xs3 = Ys2 )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_23_append1__eq__conv,axiom,
    ! [Xs3: list_nat,X: nat,Ys2: list_nat,Y2: nat] :
      ( ( ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) )
        = ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
      = ( ( Xs3 = Ys2 )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_24_append1__eq__conv,axiom,
    ! [Xs3: list_list_nat,X: list_nat,Ys2: list_list_nat,Y2: list_nat] :
      ( ( ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) )
        = ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
      = ( ( Xs3 = Ys2 )
        & ( X = Y2 ) ) ) ).

% append1_eq_conv
thf(fact_25_snoc_OIH,axiom,
    ( ( equiva3371634703666331078on_rgf @ xs )
   => ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) @ xs )
      = one_one_nat ) ) ).

% snoc.IH
thf(fact_26__092_060open_0621_A_061_Acount__list_A_Ienum__rgfs_A_Ilength_Axs_J_J_Axs_092_060close_062,axiom,
    ( one_one_nat
    = ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) @ xs ) ) ).

% \<open>1 = count_list (enum_rgfs (length xs)) xs\<close>
thf(fact_27_filter__append,axiom,
    ! [P: list_list_nat > $o,Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( filter_list_list_nat @ P @ ( append_list_list_nat @ Xs3 @ Ys2 ) )
      = ( append_list_list_nat @ ( filter_list_list_nat @ P @ Xs3 ) @ ( filter_list_list_nat @ P @ Ys2 ) ) ) ).

% filter_append
thf(fact_28_filter__append,axiom,
    ! [P: list_nat > $o,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( filter_list_nat @ P @ ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( append_list_nat @ ( filter_list_nat @ P @ Xs3 ) @ ( filter_list_nat @ P @ Ys2 ) ) ) ).

% filter_append
thf(fact_29_filter__append,axiom,
    ! [P: nat > $o,Xs3: list_nat,Ys2: list_nat] :
      ( ( filter_nat @ P @ ( append_nat @ Xs3 @ Ys2 ) )
      = ( append_nat @ ( filter_nat @ P @ Xs3 ) @ ( filter_nat @ P @ Ys2 ) ) ) ).

% filter_append
thf(fact_30_append__eq__append__conv,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat,Us: list_list_list_nat,Vs: list_list_list_nat] :
      ( ( ( ( size_s6248950052170075156st_nat @ Xs3 )
          = ( size_s6248950052170075156st_nat @ Ys2 ) )
        | ( ( size_s6248950052170075156st_nat @ Us )
          = ( size_s6248950052170075156st_nat @ Vs ) ) )
     => ( ( ( append_list_list_nat @ Xs3 @ Us )
          = ( append_list_list_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_31_append__eq__append__conv,axiom,
    ! [Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat,Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat] :
      ( ( ( ( size_s5460976970255530739at_nat @ Xs3 )
          = ( size_s5460976970255530739at_nat @ Ys2 ) )
        | ( ( size_s5460976970255530739at_nat @ Us )
          = ( size_s5460976970255530739at_nat @ Vs ) ) )
     => ( ( ( append985823374593552924at_nat @ Xs3 @ Us )
          = ( append985823374593552924at_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_32_append__eq__append__conv,axiom,
    ! [Xs3: list_P7736225833432154391st_nat,Ys2: list_P7736225833432154391st_nat,Us: list_P7736225833432154391st_nat,Vs: list_P7736225833432154391st_nat] :
      ( ( ( ( size_s9035287501014481795st_nat @ Xs3 )
          = ( size_s9035287501014481795st_nat @ Ys2 ) )
        | ( ( size_s9035287501014481795st_nat @ Us )
          = ( size_s9035287501014481795st_nat @ Vs ) ) )
     => ( ( ( append104611586619867308st_nat @ Xs3 @ Us )
          = ( append104611586619867308st_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_33_append__eq__append__conv,axiom,
    ! [Xs3: list_P5364314822750548887at_nat,Ys2: list_P5364314822750548887at_nat,Us: list_P5364314822750548887at_nat,Vs: list_P5364314822750548887at_nat] :
      ( ( ( ( size_s6663376490332876291at_nat @ Xs3 )
          = ( size_s6663376490332876291at_nat @ Ys2 ) )
        | ( ( size_s6663376490332876291at_nat @ Us )
          = ( size_s6663376490332876291at_nat @ Vs ) ) )
     => ( ( ( append6293600105956864812at_nat @ Xs3 @ Us )
          = ( append6293600105956864812at_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_34_append__eq__append__conv,axiom,
    ! [Xs3: list_P7940050157051400743st_nat,Ys2: list_P7940050157051400743st_nat,Us: list_P7940050157051400743st_nat,Vs: list_P7940050157051400743st_nat] :
      ( ( ( ( size_s7664791237847770771st_nat @ Xs3 )
          = ( size_s7664791237847770771st_nat @ Ys2 ) )
        | ( ( size_s7664791237847770771st_nat @ Us )
          = ( size_s7664791237847770771st_nat @ Vs ) ) )
     => ( ( ( append2623875052807961020st_nat @ Xs3 @ Us )
          = ( append2623875052807961020st_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_35_append__eq__append__conv,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
      ( ( ( ( size_s3023201423986296836st_nat @ Xs3 )
          = ( size_s3023201423986296836st_nat @ Ys2 ) )
        | ( ( size_s3023201423986296836st_nat @ Us )
          = ( size_s3023201423986296836st_nat @ Vs ) ) )
     => ( ( ( append_list_nat @ Xs3 @ Us )
          = ( append_list_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_36_append__eq__append__conv,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs3 )
          = ( size_size_list_nat @ Ys2 ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs3 @ Us )
          = ( append_nat @ Ys2 @ Vs ) )
        = ( ( Xs3 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_37_append_Oright__neutral,axiom,
    ! [A: list_list_list_nat] :
      ( ( append_list_list_nat @ A @ nil_list_list_nat )
      = A ) ).

% append.right_neutral
thf(fact_38_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_39_append_Oright__neutral,axiom,
    ! [A: list_list_nat] :
      ( ( append_list_nat @ A @ nil_list_nat )
      = A ) ).

% append.right_neutral
thf(fact_40_append__Nil2,axiom,
    ! [Xs3: list_list_list_nat] :
      ( ( append_list_list_nat @ Xs3 @ nil_list_list_nat )
      = Xs3 ) ).

% append_Nil2
thf(fact_41_append__Nil2,axiom,
    ! [Xs3: list_nat] :
      ( ( append_nat @ Xs3 @ nil_nat )
      = Xs3 ) ).

% append_Nil2
thf(fact_42_append__Nil2,axiom,
    ! [Xs3: list_list_nat] :
      ( ( append_list_nat @ Xs3 @ nil_list_nat )
      = Xs3 ) ).

% append_Nil2
thf(fact_43_append__self__conv,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ Ys2 )
        = Xs3 )
      = ( Ys2 = nil_list_list_nat ) ) ).

% append_self_conv
thf(fact_44_append__self__conv,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs3 @ Ys2 )
        = Xs3 )
      = ( Ys2 = nil_nat ) ) ).

% append_self_conv
thf(fact_45_append__self__conv,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Ys2 )
        = Xs3 )
      = ( Ys2 = nil_list_nat ) ) ).

% append_self_conv
thf(fact_46_self__append__conv,axiom,
    ! [Y2: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( Y2
        = ( append_list_list_nat @ Y2 @ Ys2 ) )
      = ( Ys2 = nil_list_list_nat ) ) ).

% self_append_conv
thf(fact_47_self__append__conv,axiom,
    ! [Y2: list_nat,Ys2: list_nat] :
      ( ( Y2
        = ( append_nat @ Y2 @ Ys2 ) )
      = ( Ys2 = nil_nat ) ) ).

% self_append_conv
thf(fact_48_self__append__conv,axiom,
    ! [Y2: list_list_nat,Ys2: list_list_nat] :
      ( ( Y2
        = ( append_list_nat @ Y2 @ Ys2 ) )
      = ( Ys2 = nil_list_nat ) ) ).

% self_append_conv
thf(fact_49_append__self__conv2,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ Ys2 )
        = Ys2 )
      = ( Xs3 = nil_list_list_nat ) ) ).

% append_self_conv2
thf(fact_50_append__self__conv2,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs3 @ Ys2 )
        = Ys2 )
      = ( Xs3 = nil_nat ) ) ).

% append_self_conv2
thf(fact_51_append__self__conv2,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Ys2 )
        = Ys2 )
      = ( Xs3 = nil_list_nat ) ) ).

% append_self_conv2
thf(fact_52_self__append__conv2,axiom,
    ! [Y2: list_list_list_nat,Xs3: list_list_list_nat] :
      ( ( Y2
        = ( append_list_list_nat @ Xs3 @ Y2 ) )
      = ( Xs3 = nil_list_list_nat ) ) ).

% self_append_conv2
thf(fact_53_self__append__conv2,axiom,
    ! [Y2: list_nat,Xs3: list_nat] :
      ( ( Y2
        = ( append_nat @ Xs3 @ Y2 ) )
      = ( Xs3 = nil_nat ) ) ).

% self_append_conv2
thf(fact_54_self__append__conv2,axiom,
    ! [Y2: list_list_nat,Xs3: list_list_nat] :
      ( ( Y2
        = ( append_list_nat @ Xs3 @ Y2 ) )
      = ( Xs3 = nil_list_nat ) ) ).

% self_append_conv2
thf(fact_55_list_Oinject,axiom,
    ! [X21: list_list_nat,X22: list_list_list_nat,Y21: list_list_nat,Y22: list_list_list_nat] :
      ( ( ( cons_list_list_nat @ X21 @ X22 )
        = ( cons_list_list_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_56_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_57_list_Oinject,axiom,
    ! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
      ( ( ( cons_list_nat @ X21 @ X22 )
        = ( cons_list_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_58_same__append__eq,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat,Zs: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ Ys2 )
        = ( append_list_list_nat @ Xs3 @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_59_same__append__eq,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs3 @ Ys2 )
        = ( append_nat @ Xs3 @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_60_same__append__eq,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Ys2 )
        = ( append_list_nat @ Xs3 @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_61_append__same__eq,axiom,
    ! [Ys2: list_list_list_nat,Xs3: list_list_list_nat,Zs: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Ys2 @ Xs3 )
        = ( append_list_list_nat @ Zs @ Xs3 ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_62_append__same__eq,axiom,
    ! [Ys2: list_nat,Xs3: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys2 @ Xs3 )
        = ( append_nat @ Zs @ Xs3 ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_63_append__same__eq,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_nat,Zs: list_list_nat] :
      ( ( ( append_list_nat @ Ys2 @ Xs3 )
        = ( append_list_nat @ Zs @ Xs3 ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_64_append__assoc,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat,Zs: list_list_list_nat] :
      ( ( append_list_list_nat @ ( append_list_list_nat @ Xs3 @ Ys2 ) @ Zs )
      = ( append_list_list_nat @ Xs3 @ ( append_list_list_nat @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_65_append__assoc,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs3 @ Ys2 ) @ Zs )
      = ( append_nat @ Xs3 @ ( append_nat @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_66_append__assoc,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( append_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ Zs )
      = ( append_list_nat @ Xs3 @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_67_append_Oassoc,axiom,
    ! [A: list_list_list_nat,B: list_list_list_nat,C: list_list_list_nat] :
      ( ( append_list_list_nat @ ( append_list_list_nat @ A @ B ) @ C )
      = ( append_list_list_nat @ A @ ( append_list_list_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_68_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C )
      = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_69_append_Oassoc,axiom,
    ! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
      ( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C )
      = ( append_list_nat @ A @ ( append_list_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_70_append__is__Nil__conv,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ Ys2 )
        = nil_list_list_nat )
      = ( ( Xs3 = nil_list_list_nat )
        & ( Ys2 = nil_list_list_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_71_append__is__Nil__conv,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs3 @ Ys2 )
        = nil_nat )
      = ( ( Xs3 = nil_nat )
        & ( Ys2 = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_72_append__is__Nil__conv,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Ys2 )
        = nil_list_nat )
      = ( ( Xs3 = nil_list_nat )
        & ( Ys2 = nil_list_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_73_Nil__is__append__conv,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( nil_list_list_nat
        = ( append_list_list_nat @ Xs3 @ Ys2 ) )
      = ( ( Xs3 = nil_list_list_nat )
        & ( Ys2 = nil_list_list_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_74_Nil__is__append__conv,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs3 @ Ys2 ) )
      = ( ( Xs3 = nil_nat )
        & ( Ys2 = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_75_Nil__is__append__conv,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( nil_list_nat
        = ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( ( Xs3 = nil_list_nat )
        & ( Ys2 = nil_list_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_76_not__Cons__self2,axiom,
    ! [X: list_list_nat,Xs3: list_list_list_nat] :
      ( ( cons_list_list_nat @ X @ Xs3 )
     != Xs3 ) ).

% not_Cons_self2
thf(fact_77_not__Cons__self2,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( cons_nat @ X @ Xs3 )
     != Xs3 ) ).

% not_Cons_self2
thf(fact_78_not__Cons__self2,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( cons_list_nat @ X @ Xs3 )
     != Xs3 ) ).

% not_Cons_self2
thf(fact_79_neq__if__length__neq,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( ( size_s6248950052170075156st_nat @ Xs3 )
       != ( size_s6248950052170075156st_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_80_neq__if__length__neq,axiom,
    ! [Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs3 )
       != ( size_s5460976970255530739at_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_81_neq__if__length__neq,axiom,
    ! [Xs3: list_P7736225833432154391st_nat,Ys2: list_P7736225833432154391st_nat] :
      ( ( ( size_s9035287501014481795st_nat @ Xs3 )
       != ( size_s9035287501014481795st_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_82_neq__if__length__neq,axiom,
    ! [Xs3: list_P5364314822750548887at_nat,Ys2: list_P5364314822750548887at_nat] :
      ( ( ( size_s6663376490332876291at_nat @ Xs3 )
       != ( size_s6663376490332876291at_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_83_neq__if__length__neq,axiom,
    ! [Xs3: list_P7940050157051400743st_nat,Ys2: list_P7940050157051400743st_nat] :
      ( ( ( size_s7664791237847770771st_nat @ Xs3 )
       != ( size_s7664791237847770771st_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_84_neq__if__length__neq,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
       != ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_85_neq__if__length__neq,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs3 )
       != ( size_size_list_nat @ Ys2 ) )
     => ( Xs3 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_86_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_list_list_nat] :
      ( ( size_s6248950052170075156st_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_87_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P6011104703257516679at_nat] :
      ( ( size_s5460976970255530739at_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_88_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P7736225833432154391st_nat] :
      ( ( size_s9035287501014481795st_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_89_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P5364314822750548887at_nat] :
      ( ( size_s6663376490332876291at_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_90_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P7940050157051400743st_nat] :
      ( ( size_s7664791237847770771st_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_91_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_92_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_nat] :
      ( ( size_size_list_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_93_append__eq__append__conv2,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat,Zs: list_list_list_nat,Ts: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ Ys2 )
        = ( append_list_list_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_list_list_nat] :
            ( ( ( Xs3
                = ( append_list_list_nat @ Zs @ Us2 ) )
              & ( ( append_list_list_nat @ Us2 @ Ys2 )
                = Ts ) )
            | ( ( ( append_list_list_nat @ Xs3 @ Us2 )
                = Zs )
              & ( Ys2
                = ( append_list_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_94_append__eq__append__conv2,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs3 @ Ys2 )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs3
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys2 )
                = Ts ) )
            | ( ( ( append_nat @ Xs3 @ Us2 )
                = Zs )
              & ( Ys2
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_95_append__eq__append__conv2,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Ys2 )
        = ( append_list_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_list_nat] :
            ( ( ( Xs3
                = ( append_list_nat @ Zs @ Us2 ) )
              & ( ( append_list_nat @ Us2 @ Ys2 )
                = Ts ) )
            | ( ( ( append_list_nat @ Xs3 @ Us2 )
                = Zs )
              & ( Ys2
                = ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_96_append__eq__appendI,axiom,
    ! [Xs3: list_list_list_nat,Xs1: list_list_list_nat,Zs: list_list_list_nat,Ys2: list_list_list_nat,Us: list_list_list_nat] :
      ( ( ( append_list_list_nat @ Xs3 @ Xs1 )
        = Zs )
     => ( ( Ys2
          = ( append_list_list_nat @ Xs1 @ Us ) )
       => ( ( append_list_list_nat @ Xs3 @ Ys2 )
          = ( append_list_list_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_97_append__eq__appendI,axiom,
    ! [Xs3: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs3 @ Xs1 )
        = Zs )
     => ( ( Ys2
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs3 @ Ys2 )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_98_append__eq__appendI,axiom,
    ! [Xs3: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys2: list_list_nat,Us: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Xs1 )
        = Zs )
     => ( ( Ys2
          = ( append_list_nat @ Xs1 @ Us ) )
       => ( ( append_list_nat @ Xs3 @ Ys2 )
          = ( append_list_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_99_list__nonempty__induct,axiom,
    ! [Xs3: list_list_list_nat,P: list_list_list_nat > $o] :
      ( ( Xs3 != nil_list_list_nat )
     => ( ! [X2: list_list_nat] : ( P @ ( cons_list_list_nat @ X2 @ nil_list_list_nat ) )
       => ( ! [X2: list_list_nat,Xs: list_list_list_nat] :
              ( ( Xs != nil_list_list_nat )
             => ( ( P @ Xs )
               => ( P @ ( cons_list_list_nat @ X2 @ Xs ) ) ) )
         => ( P @ Xs3 ) ) ) ) ).

% list_nonempty_induct
thf(fact_100_list__nonempty__induct,axiom,
    ! [Xs3: list_nat,P: list_nat > $o] :
      ( ( Xs3 != nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X2: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( ( P @ Xs )
               => ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
         => ( P @ Xs3 ) ) ) ) ).

% list_nonempty_induct
thf(fact_101_list__nonempty__induct,axiom,
    ! [Xs3: list_list_nat,P: list_list_nat > $o] :
      ( ( Xs3 != nil_list_nat )
     => ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
       => ( ! [X2: list_nat,Xs: list_list_nat] :
              ( ( Xs != nil_list_nat )
             => ( ( P @ Xs )
               => ( P @ ( cons_list_nat @ X2 @ Xs ) ) ) )
         => ( P @ Xs3 ) ) ) ) ).

% list_nonempty_induct
thf(fact_102_list__induct2_H,axiom,
    ! [P: list_nat > list_list_list_nat > $o,Xs3: list_nat,Ys2: list_list_list_nat] :
      ( ( P @ nil_nat @ nil_list_list_nat )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_list_nat )
       => ( ! [Y3: list_list_nat,Ys: list_list_list_nat] : ( P @ nil_nat @ ( cons_list_list_nat @ Y3 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y3: list_list_nat,Ys: list_list_list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_list_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_103_list__induct2_H,axiom,
    ! [P: list_list_nat > list_list_list_nat > $o,Xs3: list_list_nat,Ys2: list_list_list_nat] :
      ( ( P @ nil_list_nat @ nil_list_list_nat )
     => ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_list_nat )
       => ( ! [Y3: list_list_nat,Ys: list_list_list_nat] : ( P @ nil_list_nat @ ( cons_list_list_nat @ Y3 @ Ys ) )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_list_nat,Ys: list_list_list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_list_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_104_list__induct2_H,axiom,
    ! [P: list_list_list_nat > list_nat > $o,Xs3: list_list_list_nat,Ys2: list_nat] :
      ( ( P @ nil_list_list_nat @ nil_nat )
     => ( ! [X2: list_list_nat,Xs: list_list_list_nat] : ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_list_list_nat @ ( cons_nat @ Y3 @ Ys ) )
         => ( ! [X2: list_list_nat,Xs: list_list_list_nat,Y3: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_105_list__induct2_H,axiom,
    ! [P: list_list_list_nat > list_list_nat > $o,Xs3: list_list_list_nat,Ys2: list_list_nat] :
      ( ( P @ nil_list_list_nat @ nil_list_nat )
     => ( ! [X2: list_list_nat,Xs: list_list_list_nat] : ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ nil_list_nat )
       => ( ! [Y3: list_nat,Ys: list_list_nat] : ( P @ nil_list_list_nat @ ( cons_list_nat @ Y3 @ Ys ) )
         => ( ! [X2: list_list_nat,Xs: list_list_list_nat,Y3: list_nat,Ys: list_list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_106_list__induct2_H,axiom,
    ! [P: list_list_list_nat > list_list_list_nat > $o,Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( P @ nil_list_list_nat @ nil_list_list_nat )
     => ( ! [X2: list_list_nat,Xs: list_list_list_nat] : ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ nil_list_list_nat )
       => ( ! [Y3: list_list_nat,Ys: list_list_list_nat] : ( P @ nil_list_list_nat @ ( cons_list_list_nat @ Y3 @ Ys ) )
         => ( ! [X2: list_list_nat,Xs: list_list_list_nat,Y3: list_list_nat,Ys: list_list_list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_list_nat @ X2 @ Xs ) @ ( cons_list_list_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_107_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs3: list_nat,Ys2: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_108_list__induct2_H,axiom,
    ! [P: list_nat > list_list_nat > $o,Xs3: list_nat,Ys2: list_list_nat] :
      ( ( P @ nil_nat @ nil_list_nat )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat )
       => ( ! [Y3: list_nat,Ys: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y3 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_109_list__induct2_H,axiom,
    ! [P: list_list_nat > list_nat > $o,Xs3: list_list_nat,Ys2: list_nat] :
      ( ( P @ nil_list_nat @ nil_nat )
     => ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y3: nat,Ys: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y3 @ Ys ) )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_110_list__induct2_H,axiom,
    ! [P: list_list_nat > list_list_nat > $o,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( P @ nil_list_nat @ nil_list_nat )
     => ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
       => ( ! [Y3: list_nat,Ys: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y3 @ Ys ) )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) )
           => ( P @ Xs3 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_111_neq__Nil__conv,axiom,
    ! [Xs3: list_list_list_nat] :
      ( ( Xs3 != nil_list_list_nat )
      = ( ? [Y4: list_list_nat,Ys3: list_list_list_nat] :
            ( Xs3
            = ( cons_list_list_nat @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_112_neq__Nil__conv,axiom,
    ! [Xs3: list_nat] :
      ( ( Xs3 != nil_nat )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( Xs3
            = ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_113_neq__Nil__conv,axiom,
    ! [Xs3: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
      = ( ? [Y4: list_nat,Ys3: list_list_nat] :
            ( Xs3
            = ( cons_list_nat @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_114_remdups__adj_Ocases,axiom,
    ! [X: list_list_list_nat] :
      ( ( X != nil_list_list_nat )
     => ( ! [X2: list_list_nat] :
            ( X
           != ( cons_list_list_nat @ X2 @ nil_list_list_nat ) )
       => ~ ! [X2: list_list_nat,Y3: list_list_nat,Xs: list_list_list_nat] :
              ( X
             != ( cons_list_list_nat @ X2 @ ( cons_list_list_nat @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_115_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X2: nat] :
            ( X
           != ( cons_nat @ X2 @ nil_nat ) )
       => ~ ! [X2: nat,Y3: nat,Xs: list_nat] :
              ( X
             != ( cons_nat @ X2 @ ( cons_nat @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_116_remdups__adj_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [X2: list_nat] :
            ( X
           != ( cons_list_nat @ X2 @ nil_list_nat ) )
       => ~ ! [X2: list_nat,Y3: list_nat,Xs: list_list_nat] :
              ( X
             != ( cons_list_nat @ X2 @ ( cons_list_nat @ Y3 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_117_transpose_Ocases,axiom,
    ! [X: list_l5212752354702395664st_nat] :
      ( ( X != nil_li5455921165481563386st_nat )
     => ( ! [Xss: list_l5212752354702395664st_nat] :
            ( X
           != ( cons_l7310388135179752778st_nat @ nil_list_list_nat @ Xss ) )
       => ~ ! [X2: list_list_nat,Xs: list_list_list_nat,Xss: list_l5212752354702395664st_nat] :
              ( X
             != ( cons_l7310388135179752778st_nat @ ( cons_list_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_118_transpose_Ocases,axiom,
    ! [X: list_list_list_nat] :
      ( ( X != nil_list_list_nat )
     => ( ! [Xss: list_list_list_nat] :
            ( X
           != ( cons_list_list_nat @ nil_list_nat @ Xss ) )
       => ~ ! [X2: list_nat,Xs: list_list_nat,Xss: list_list_list_nat] :
              ( X
             != ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_119_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_120_mem__Collect__eq,axiom,
    ! [A: list_list_nat,P: list_list_nat > $o] :
      ( ( member_list_list_nat @ A @ ( collec5989764272469232197st_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_123_Collect__mem__eq,axiom,
    ! [A2: set_list_list_nat] :
      ( ( collec5989764272469232197st_nat
        @ ^ [X3: list_list_nat] : ( member_list_list_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_125_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_126_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X2: nat,Xs: list_nat] :
          ( X
         != ( cons_nat @ X2 @ Xs ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_127_list_Oexhaust,axiom,
    ! [Y2: list_list_list_nat] :
      ( ( Y2 != nil_list_list_nat )
     => ~ ! [X212: list_list_nat,X222: list_list_list_nat] :
            ( Y2
           != ( cons_list_list_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_128_list_Oexhaust,axiom,
    ! [Y2: list_nat] :
      ( ( Y2 != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y2
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_129_list_Oexhaust,axiom,
    ! [Y2: list_list_nat] :
      ( ( Y2 != nil_list_nat )
     => ~ ! [X212: list_nat,X222: list_list_nat] :
            ( Y2
           != ( cons_list_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_130_list_OdiscI,axiom,
    ! [List: list_list_list_nat,X21: list_list_nat,X22: list_list_list_nat] :
      ( ( List
        = ( cons_list_list_nat @ X21 @ X22 ) )
     => ( List != nil_list_list_nat ) ) ).

% list.discI
thf(fact_131_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_132_list_OdiscI,axiom,
    ! [List: list_list_nat,X21: list_nat,X22: list_list_nat] :
      ( ( List
        = ( cons_list_nat @ X21 @ X22 ) )
     => ( List != nil_list_nat ) ) ).

% list.discI
thf(fact_133_list_Odistinct_I1_J,axiom,
    ! [X21: list_list_nat,X22: list_list_list_nat] :
      ( nil_list_list_nat
     != ( cons_list_list_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_134_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_135_list_Odistinct_I1_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] :
      ( nil_list_nat
     != ( cons_list_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_136_Cons__eq__appendI,axiom,
    ! [X: list_list_nat,Xs1: list_list_list_nat,Ys2: list_list_list_nat,Xs3: list_list_list_nat,Zs: list_list_list_nat] :
      ( ( ( cons_list_list_nat @ X @ Xs1 )
        = Ys2 )
     => ( ( Xs3
          = ( append_list_list_nat @ Xs1 @ Zs ) )
       => ( ( cons_list_list_nat @ X @ Xs3 )
          = ( append_list_list_nat @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_137_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs3: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs1 )
        = Ys2 )
     => ( ( Xs3
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X @ Xs3 )
          = ( append_nat @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_138_Cons__eq__appendI,axiom,
    ! [X: list_nat,Xs1: list_list_nat,Ys2: list_list_nat,Xs3: list_list_nat,Zs: list_list_nat] :
      ( ( ( cons_list_nat @ X @ Xs1 )
        = Ys2 )
     => ( ( Xs3
          = ( append_list_nat @ Xs1 @ Zs ) )
       => ( ( cons_list_nat @ X @ Xs3 )
          = ( append_list_nat @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_139_append__Cons,axiom,
    ! [X: list_list_nat,Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( append_list_list_nat @ ( cons_list_list_nat @ X @ Xs3 ) @ Ys2 )
      = ( cons_list_list_nat @ X @ ( append_list_list_nat @ Xs3 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_140_append__Cons,axiom,
    ! [X: nat,Xs3: list_nat,Ys2: list_nat] :
      ( ( append_nat @ ( cons_nat @ X @ Xs3 ) @ Ys2 )
      = ( cons_nat @ X @ ( append_nat @ Xs3 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_141_append__Cons,axiom,
    ! [X: list_nat,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( append_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ Ys2 )
      = ( cons_list_nat @ X @ ( append_list_nat @ Xs3 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_142_eq__Nil__appendI,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( Xs3 = Ys2 )
     => ( Xs3
        = ( append_list_list_nat @ nil_list_list_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_143_eq__Nil__appendI,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( Xs3 = Ys2 )
     => ( Xs3
        = ( append_nat @ nil_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_144_eq__Nil__appendI,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( Xs3 = Ys2 )
     => ( Xs3
        = ( append_list_nat @ nil_list_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_145_append_Oleft__neutral,axiom,
    ! [A: list_list_list_nat] :
      ( ( append_list_list_nat @ nil_list_list_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_146_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_147_append_Oleft__neutral,axiom,
    ! [A: list_list_nat] :
      ( ( append_list_nat @ nil_list_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_148_append__Nil,axiom,
    ! [Ys2: list_list_list_nat] :
      ( ( append_list_list_nat @ nil_list_list_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_149_append__Nil,axiom,
    ! [Ys2: list_nat] :
      ( ( append_nat @ nil_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_150_append__Nil,axiom,
    ! [Ys2: list_list_nat] :
      ( ( append_list_nat @ nil_list_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_151_filter_Osimps_I2_J,axiom,
    ! [P: list_list_nat > $o,X: list_list_nat,Xs3: list_list_list_nat] :
      ( ( ( P @ X )
       => ( ( filter_list_list_nat @ P @ ( cons_list_list_nat @ X @ Xs3 ) )
          = ( cons_list_list_nat @ X @ ( filter_list_list_nat @ P @ Xs3 ) ) ) )
      & ( ~ ( P @ X )
       => ( ( filter_list_list_nat @ P @ ( cons_list_list_nat @ X @ Xs3 ) )
          = ( filter_list_list_nat @ P @ Xs3 ) ) ) ) ).

% filter.simps(2)
thf(fact_152_filter_Osimps_I2_J,axiom,
    ! [P: nat > $o,X: nat,Xs3: list_nat] :
      ( ( ( P @ X )
       => ( ( filter_nat @ P @ ( cons_nat @ X @ Xs3 ) )
          = ( cons_nat @ X @ ( filter_nat @ P @ Xs3 ) ) ) )
      & ( ~ ( P @ X )
       => ( ( filter_nat @ P @ ( cons_nat @ X @ Xs3 ) )
          = ( filter_nat @ P @ Xs3 ) ) ) ) ).

% filter.simps(2)
thf(fact_153_filter_Osimps_I2_J,axiom,
    ! [P: list_nat > $o,X: list_nat,Xs3: list_list_nat] :
      ( ( ( P @ X )
       => ( ( filter_list_nat @ P @ ( cons_list_nat @ X @ Xs3 ) )
          = ( cons_list_nat @ X @ ( filter_list_nat @ P @ Xs3 ) ) ) )
      & ( ~ ( P @ X )
       => ( ( filter_list_nat @ P @ ( cons_list_nat @ X @ Xs3 ) )
          = ( filter_list_nat @ P @ Xs3 ) ) ) ) ).

% filter.simps(2)
thf(fact_154_filter_Osimps_I1_J,axiom,
    ! [P: list_list_nat > $o] :
      ( ( filter_list_list_nat @ P @ nil_list_list_nat )
      = nil_list_list_nat ) ).

% filter.simps(1)
thf(fact_155_filter_Osimps_I1_J,axiom,
    ! [P: nat > $o] :
      ( ( filter_nat @ P @ nil_nat )
      = nil_nat ) ).

% filter.simps(1)
thf(fact_156_filter_Osimps_I1_J,axiom,
    ! [P: list_nat > $o] :
      ( ( filter_list_nat @ P @ nil_list_nat )
      = nil_list_nat ) ).

% filter.simps(1)
thf(fact_157_list__induct4,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_158_list__induct4,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_list_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s3023201423986296836st_nat @ Xs )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_159_list__induct4,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_list_nat @ nil_nat @ nil_nat )
           => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_s3023201423986296836st_nat @ Ys ) )
                 => ( ( ( size_s3023201423986296836st_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_160_list__induct4,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_list_nat,Ws: list_nat,P: list_nat > list_nat > list_list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( ( size_s3023201423986296836st_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_list_nat @ nil_nat )
           => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z2: list_nat,Zs2: list_list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_s3023201423986296836st_nat @ Zs2 ) )
                   => ( ( ( size_s3023201423986296836st_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_161_list__induct4,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_list_nat,P: list_nat > list_nat > list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s3023201423986296836st_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_list_nat )
           => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: list_nat,Ws2: list_list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s3023201423986296836st_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_162_list__induct4,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_nat,Ws: list_nat,P: list_list_nat > list_list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_list_nat @ nil_list_nat @ nil_nat @ nil_nat )
           => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s3023201423986296836st_nat @ Xs )
                    = ( size_s3023201423986296836st_nat @ Ys ) )
                 => ( ( ( size_s3023201423986296836st_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_163_list__induct4,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat,Zs: list_list_nat,Ws: list_nat,P: list_list_nat > list_nat > list_list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( ( size_s3023201423986296836st_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_list_nat @ nil_nat @ nil_list_nat @ nil_nat )
           => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat,Z2: list_nat,Zs2: list_list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s3023201423986296836st_nat @ Xs )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_s3023201423986296836st_nat @ Zs2 ) )
                   => ( ( ( size_s3023201423986296836st_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_164_list__induct4,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_list_nat,P: list_list_nat > list_nat > list_nat > list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s3023201423986296836st_nat @ Ws ) )
         => ( ( P @ nil_list_nat @ nil_nat @ nil_nat @ nil_list_nat )
           => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: list_nat,Ws2: list_list_nat] :
                  ( ( ( size_s3023201423986296836st_nat @ Xs )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s3023201423986296836st_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_165_list__induct4,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat,Zs: list_list_nat,Ws: list_nat,P: list_nat > list_list_nat > list_list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( ( size_s3023201423986296836st_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat @ nil_nat )
           => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat,Z2: list_nat,Zs2: list_list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_s3023201423986296836st_nat @ Ys ) )
                 => ( ( ( size_s3023201423986296836st_nat @ Ys )
                      = ( size_s3023201423986296836st_nat @ Zs2 ) )
                   => ( ( ( size_s3023201423986296836st_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_166_list__induct4,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat,Zs: list_nat,Ws: list_list_nat,P: list_nat > list_list_nat > list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s3023201423986296836st_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_list_nat @ nil_nat @ nil_list_nat )
           => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat,Z2: nat,Zs2: list_nat,W: list_nat,Ws2: list_list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = ( size_s3023201423986296836st_nat @ Ys ) )
                 => ( ( ( size_s3023201423986296836st_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s3023201423986296836st_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs3 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_167_list__induct3,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat,P: list_list_nat > list_list_nat > list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( P @ nil_list_nat @ nil_list_nat @ nil_list_nat )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat,Z2: list_nat,Zs2: list_list_nat] :
                ( ( ( size_s3023201423986296836st_nat @ Xs )
                  = ( size_s3023201423986296836st_nat @ Ys ) )
               => ( ( ( size_s3023201423986296836st_nat @ Ys )
                    = ( size_s3023201423986296836st_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_168_list__induct3,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_nat,P: list_list_nat > list_list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_list_nat @ nil_list_nat @ nil_nat )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_s3023201423986296836st_nat @ Xs )
                  = ( size_s3023201423986296836st_nat @ Ys ) )
               => ( ( ( size_s3023201423986296836st_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_169_list__induct3,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat,Zs: list_list_nat,P: list_list_nat > list_nat > list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( P @ nil_list_nat @ nil_nat @ nil_list_nat )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat,Z2: list_nat,Zs2: list_list_nat] :
                ( ( ( size_s3023201423986296836st_nat @ Xs )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_s3023201423986296836st_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_170_list__induct3,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat,Zs: list_nat,P: list_list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_list_nat @ nil_nat @ nil_nat )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_s3023201423986296836st_nat @ Xs )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_171_list__induct3,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat,Zs: list_list_nat,P: list_nat > list_list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat )
         => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat,Z2: list_nat,Zs2: list_list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_s3023201423986296836st_nat @ Ys ) )
               => ( ( ( size_s3023201423986296836st_nat @ Ys )
                    = ( size_s3023201423986296836st_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_172_list__induct3,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat,Zs: list_nat,P: list_nat > list_list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_list_nat @ nil_nat )
         => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_s3023201423986296836st_nat @ Ys ) )
               => ( ( ( size_s3023201423986296836st_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_173_list__induct3,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_list_nat,P: list_nat > list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_s3023201423986296836st_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_list_nat )
         => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z2: list_nat,Zs2: list_list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_s3023201423986296836st_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_list_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_174_list__induct3,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs3 @ Ys2 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_175_list__induct2,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,P: list_list_nat > list_list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( P @ nil_list_nat @ nil_list_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs )
                = ( size_s3023201423986296836st_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) ) )
         => ( P @ Xs3 @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_176_list__induct2,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat,P: list_list_nat > list_nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( P @ nil_list_nat @ nil_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: nat,Ys: list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
         => ( P @ Xs3 @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_177_list__induct2,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat,P: list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ( P @ nil_nat @ nil_list_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: list_nat,Ys: list_list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_s3023201423986296836st_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) ) )
         => ( P @ Xs3 @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_178_list__induct2,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) )
         => ( P @ Xs3 @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_179_rev__nonempty__induct,axiom,
    ! [Xs3: list_nat,P: list_nat > $o] :
      ( ( Xs3 != nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X2: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( ( P @ Xs )
               => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
         => ( P @ Xs3 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_180_rev__nonempty__induct,axiom,
    ! [Xs3: list_list_nat,P: list_list_nat > $o] :
      ( ( Xs3 != nil_list_nat )
     => ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
       => ( ! [X2: list_nat,Xs: list_list_nat] :
              ( ( Xs != nil_list_nat )
             => ( ( P @ Xs )
               => ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) ) )
         => ( P @ Xs3 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_181_append__eq__Cons__conv,axiom,
    ! [Ys2: list_nat,Zs: list_nat,X: nat,Xs3: list_nat] :
      ( ( ( append_nat @ Ys2 @ Zs )
        = ( cons_nat @ X @ Xs3 ) )
      = ( ( ( Ys2 = nil_nat )
          & ( Zs
            = ( cons_nat @ X @ Xs3 ) ) )
        | ? [Ys4: list_nat] :
            ( ( Ys2
              = ( cons_nat @ X @ Ys4 ) )
            & ( ( append_nat @ Ys4 @ Zs )
              = Xs3 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_182_append__eq__Cons__conv,axiom,
    ! [Ys2: list_list_nat,Zs: list_list_nat,X: list_nat,Xs3: list_list_nat] :
      ( ( ( append_list_nat @ Ys2 @ Zs )
        = ( cons_list_nat @ X @ Xs3 ) )
      = ( ( ( Ys2 = nil_list_nat )
          & ( Zs
            = ( cons_list_nat @ X @ Xs3 ) ) )
        | ? [Ys4: list_list_nat] :
            ( ( Ys2
              = ( cons_list_nat @ X @ Ys4 ) )
            & ( ( append_list_nat @ Ys4 @ Zs )
              = Xs3 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_183_Cons__eq__append__conv,axiom,
    ! [X: nat,Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs3 )
        = ( append_nat @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_nat )
          & ( ( cons_nat @ X @ Xs3 )
            = Zs ) )
        | ? [Ys4: list_nat] :
            ( ( ( cons_nat @ X @ Ys4 )
              = Ys2 )
            & ( Xs3
              = ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_184_Cons__eq__append__conv,axiom,
    ! [X: list_nat,Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( ( cons_list_nat @ X @ Xs3 )
        = ( append_list_nat @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_list_nat )
          & ( ( cons_list_nat @ X @ Xs3 )
            = Zs ) )
        | ? [Ys4: list_list_nat] :
            ( ( ( cons_list_nat @ X @ Ys4 )
              = Ys2 )
            & ( Xs3
              = ( append_list_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_185_rev__exhaust,axiom,
    ! [Xs3: list_nat] :
      ( ( Xs3 != nil_nat )
     => ~ ! [Ys: list_nat,Y3: nat] :
            ( Xs3
           != ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_186_rev__exhaust,axiom,
    ! [Xs3: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ~ ! [Ys: list_list_nat,Y3: list_nat] :
            ( Xs3
           != ( append_list_nat @ Ys @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) ).

% rev_exhaust
thf(fact_187_rev__induct,axiom,
    ! [P: list_nat > $o,Xs3: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X2: nat,Xs: list_nat] :
            ( ( P @ Xs )
           => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) )
       => ( P @ Xs3 ) ) ) ).

% rev_induct
thf(fact_188_rev__induct,axiom,
    ! [P: list_list_nat > $o,Xs3: list_list_nat] :
      ( ( P @ nil_list_nat )
     => ( ! [X2: list_nat,Xs: list_list_nat] :
            ( ( P @ Xs )
           => ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) )
       => ( P @ Xs3 ) ) ) ).

% rev_induct
thf(fact_189_same__length__different,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( Xs3 != Ys2 )
     => ( ( ( size_s3023201423986296836st_nat @ Xs3 )
          = ( size_s3023201423986296836st_nat @ Ys2 ) )
       => ? [Pre: list_list_nat,X2: list_nat,Xs4: list_list_nat,Y3: list_nat,Ys5: list_list_nat] :
            ( ( X2 != Y3 )
            & ( Xs3
              = ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ X2 @ nil_list_nat ) @ Xs4 ) ) )
            & ( Ys2
              = ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ Y3 @ nil_list_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_190_same__length__different,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( Xs3 != Ys2 )
     => ( ( ( size_size_list_nat @ Xs3 )
          = ( size_size_list_nat @ Ys2 ) )
       => ? [Pre: list_nat,X2: nat,Xs4: list_nat,Y3: nat,Ys5: list_nat] :
            ( ( X2 != Y3 )
            & ( Xs3
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs4 ) ) )
            & ( Ys2
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_191_bind__simps_I2_J,axiom,
    ! [X: nat,Xs3: list_nat,F: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X @ Xs3 ) @ F )
      = ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs3 @ F ) ) ) ).

% bind_simps(2)
thf(fact_192_bind__simps_I2_J,axiom,
    ! [X: nat,Xs3: list_nat,F: nat > list_list_nat] :
      ( ( bind_nat_list_nat @ ( cons_nat @ X @ Xs3 ) @ F )
      = ( append_list_nat @ ( F @ X ) @ ( bind_nat_list_nat @ Xs3 @ F ) ) ) ).

% bind_simps(2)
thf(fact_193_bind__simps_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat,F: list_nat > list_nat] :
      ( ( bind_list_nat_nat @ ( cons_list_nat @ X @ Xs3 ) @ F )
      = ( append_nat @ ( F @ X ) @ ( bind_list_nat_nat @ Xs3 @ F ) ) ) ).

% bind_simps(2)
thf(fact_194_bind__simps_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat,F: list_nat > list_list_nat] :
      ( ( bind_l7796378977173581257st_nat @ ( cons_list_nat @ X @ Xs3 ) @ F )
      = ( append_list_nat @ ( F @ X ) @ ( bind_l7796378977173581257st_nat @ Xs3 @ F ) ) ) ).

% bind_simps(2)
thf(fact_195_maps__simps_I1_J,axiom,
    ! [F: nat > list_nat,X: nat,Xs3: list_nat] :
      ( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs3 ) )
      = ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs3 ) ) ) ).

% maps_simps(1)
thf(fact_196_maps__simps_I1_J,axiom,
    ! [F: nat > list_list_nat,X: nat,Xs3: list_nat] :
      ( ( maps_nat_list_nat @ F @ ( cons_nat @ X @ Xs3 ) )
      = ( append_list_nat @ ( F @ X ) @ ( maps_nat_list_nat @ F @ Xs3 ) ) ) ).

% maps_simps(1)
thf(fact_197_maps__simps_I1_J,axiom,
    ! [F: list_nat > list_nat,X: list_nat,Xs3: list_list_nat] :
      ( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X @ Xs3 ) )
      = ( append_nat @ ( F @ X ) @ ( maps_list_nat_nat @ F @ Xs3 ) ) ) ).

% maps_simps(1)
thf(fact_198_maps__simps_I1_J,axiom,
    ! [F: list_nat > list_list_nat,X: list_nat,Xs3: list_list_nat] :
      ( ( maps_l5785965478274863235st_nat @ F @ ( cons_list_nat @ X @ Xs3 ) )
      = ( append_list_nat @ ( F @ X ) @ ( maps_l5785965478274863235st_nat @ F @ Xs3 ) ) ) ).

% maps_simps(1)
thf(fact_199_insert__Nil,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ nil_nat )
      = ( cons_nat @ X @ nil_nat ) ) ).

% insert_Nil
thf(fact_200_insert__Nil,axiom,
    ! [X: list_nat] :
      ( ( insert_list_nat @ X @ nil_list_nat )
      = ( cons_list_nat @ X @ nil_list_nat ) ) ).

% insert_Nil
thf(fact_201_list__ex1__simps_I1_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex1_nat @ P @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_202_list__ex1__simps_I1_J,axiom,
    ! [P: list_nat > $o] :
      ~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).

% list_ex1_simps(1)
thf(fact_203_butlast__snoc,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( butlast_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) )
      = Xs3 ) ).

% butlast_snoc
thf(fact_204_butlast__snoc,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( butlast_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = Xs3 ) ).

% butlast_snoc
thf(fact_205_last__snoc,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( last_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% last_snoc
thf(fact_206_last__snoc,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( last_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = X ) ).

% last_snoc
thf(fact_207_rotate1_Osimps_I2_J,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( rotate1_nat @ ( cons_nat @ X @ Xs3 ) )
      = ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) ) ).

% rotate1.simps(2)
thf(fact_208_rotate1_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( rotate1_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
      = ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) ) ).

% rotate1.simps(2)
thf(fact_209_stirling__row__aux_Osimps_I1_J,axiom,
    ! [N: nat,Y2: nat] :
      ( ( stirling_row_aux_nat @ N @ Y2 @ nil_nat )
      = ( cons_nat @ one_one_nat @ nil_nat ) ) ).

% stirling_row_aux.simps(1)
thf(fact_210_length__Suc__conv__rev,axiom,
    ! [Xs3: list_list_nat,N: nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( suc @ N ) )
      = ( ? [Y4: list_nat,Ys3: list_list_nat] :
            ( ( Xs3
              = ( append_list_nat @ Ys3 @ ( cons_list_nat @ Y4 @ nil_list_nat ) ) )
            & ( ( size_s3023201423986296836st_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_211_length__Suc__conv__rev,axiom,
    ! [Xs3: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs3
              = ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_212_list__update__length,axiom,
    ! [Xs3: list_list_nat,X: list_nat,Ys2: list_list_nat,Y2: list_nat] :
      ( ( list_update_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs3 ) @ Y2 )
      = ( append_list_nat @ Xs3 @ ( cons_list_nat @ Y2 @ Ys2 ) ) ) ).

% list_update_length
thf(fact_213_list__update__length,axiom,
    ! [Xs3: list_nat,X: nat,Ys2: list_nat,Y2: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs3 ) @ Y2 )
      = ( append_nat @ Xs3 @ ( cons_nat @ Y2 @ Ys2 ) ) ) ).

% list_update_length
thf(fact_214_list__update__nonempty,axiom,
    ! [Xs3: list_nat,K: nat,X: nat] :
      ( ( ( list_update_nat @ Xs3 @ K @ X )
        = nil_nat )
      = ( Xs3 = nil_nat ) ) ).

% list_update_nonempty
thf(fact_215_list__update__nonempty,axiom,
    ! [Xs3: list_list_nat,K: nat,X: list_nat] :
      ( ( ( list_update_list_nat @ Xs3 @ K @ X )
        = nil_list_nat )
      = ( Xs3 = nil_list_nat ) ) ).

% list_update_nonempty
thf(fact_216_length__list__update,axiom,
    ! [Xs3: list_list_nat,I: nat,X: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( list_update_list_nat @ Xs3 @ I @ X ) )
      = ( size_s3023201423986296836st_nat @ Xs3 ) ) ).

% length_list_update
thf(fact_217_length__list__update,axiom,
    ! [Xs3: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs3 @ I @ X ) )
      = ( size_size_list_nat @ Xs3 ) ) ).

% length_list_update
thf(fact_218_rotate1__is__Nil__conv,axiom,
    ! [Xs3: list_nat] :
      ( ( ( rotate1_nat @ Xs3 )
        = nil_nat )
      = ( Xs3 = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_219_rotate1__is__Nil__conv,axiom,
    ! [Xs3: list_list_nat] :
      ( ( ( rotate1_list_nat @ Xs3 )
        = nil_list_nat )
      = ( Xs3 = nil_list_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_220_length__rotate1,axiom,
    ! [Xs3: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( rotate1_list_nat @ Xs3 ) )
      = ( size_s3023201423986296836st_nat @ Xs3 ) ) ).

% length_rotate1
thf(fact_221_length__rotate1,axiom,
    ! [Xs3: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs3 ) )
      = ( size_size_list_nat @ Xs3 ) ) ).

% length_rotate1
thf(fact_222_bind__simps_I1_J,axiom,
    ! [F: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_223_bind__simps_I1_J,axiom,
    ! [F: nat > list_list_nat] :
      ( ( bind_nat_list_nat @ nil_nat @ F )
      = nil_list_nat ) ).

% bind_simps(1)
thf(fact_224_bind__simps_I1_J,axiom,
    ! [F: list_nat > list_nat] :
      ( ( bind_list_nat_nat @ nil_list_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_225_bind__simps_I1_J,axiom,
    ! [F: list_nat > list_list_nat] :
      ( ( bind_l7796378977173581257st_nat @ nil_list_nat @ F )
      = nil_list_nat ) ).

% bind_simps(1)
thf(fact_226_last__appendL,axiom,
    ! [Ys2: list_nat,Xs3: list_nat] :
      ( ( Ys2 = nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs3 @ Ys2 ) )
        = ( last_nat @ Xs3 ) ) ) ).

% last_appendL
thf(fact_227_last__appendL,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_nat] :
      ( ( Ys2 = nil_list_nat )
     => ( ( last_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
        = ( last_list_nat @ Xs3 ) ) ) ).

% last_appendL
thf(fact_228_last__appendR,axiom,
    ! [Ys2: list_nat,Xs3: list_nat] :
      ( ( Ys2 != nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs3 @ Ys2 ) )
        = ( last_nat @ Ys2 ) ) ) ).

% last_appendR
thf(fact_229_last__appendR,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_nat] :
      ( ( Ys2 != nil_list_nat )
     => ( ( last_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
        = ( last_list_nat @ Ys2 ) ) ) ).

% last_appendR
thf(fact_230_append__butlast__last__id,axiom,
    ! [Xs3: list_nat] :
      ( ( Xs3 != nil_nat )
     => ( ( append_nat @ ( butlast_nat @ Xs3 ) @ ( cons_nat @ ( last_nat @ Xs3 ) @ nil_nat ) )
        = Xs3 ) ) ).

% append_butlast_last_id
thf(fact_231_append__butlast__last__id,axiom,
    ! [Xs3: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( append_list_nat @ ( butlast_list_nat @ Xs3 ) @ ( cons_list_nat @ ( last_list_nat @ Xs3 ) @ nil_list_nat ) )
        = Xs3 ) ) ).

% append_butlast_last_id
thf(fact_232_list__update__code_I3_J,axiom,
    ! [X: nat,Xs3: list_nat,I: nat,Y2: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs3 ) @ ( suc @ I ) @ Y2 )
      = ( cons_nat @ X @ ( list_update_nat @ Xs3 @ I @ Y2 ) ) ) ).

% list_update_code(3)
thf(fact_233_list__update__code_I3_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat,I: nat,Y2: list_nat] :
      ( ( list_update_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ ( suc @ I ) @ Y2 )
      = ( cons_list_nat @ X @ ( list_update_list_nat @ Xs3 @ I @ Y2 ) ) ) ).

% list_update_code(3)
thf(fact_234_list__update__code_I1_J,axiom,
    ! [I: nat,Y2: nat] :
      ( ( list_update_nat @ nil_nat @ I @ Y2 )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_235_list__update__code_I1_J,axiom,
    ! [I: nat,Y2: list_nat] :
      ( ( list_update_list_nat @ nil_list_nat @ I @ Y2 )
      = nil_list_nat ) ).

% list_update_code(1)
thf(fact_236_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_237_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: list_nat] :
      ( ( list_update_list_nat @ nil_list_nat @ I @ V )
      = nil_list_nat ) ).

% list_update.simps(1)
thf(fact_238_butlast_Osimps_I1_J,axiom,
    ( ( butlast_nat @ nil_nat )
    = nil_nat ) ).

% butlast.simps(1)
thf(fact_239_butlast_Osimps_I1_J,axiom,
    ( ( butlast_list_nat @ nil_list_nat )
    = nil_list_nat ) ).

% butlast.simps(1)
thf(fact_240_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_241_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_list_nat @ nil_list_nat )
    = nil_list_nat ) ).

% rotate1.simps(1)
thf(fact_242_snoc__eq__iff__butlast,axiom,
    ! [Xs3: list_nat,X: nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) )
        = Ys2 )
      = ( ( Ys2 != nil_nat )
        & ( ( butlast_nat @ Ys2 )
          = Xs3 )
        & ( ( last_nat @ Ys2 )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_243_snoc__eq__iff__butlast,axiom,
    ! [Xs3: list_list_nat,X: list_nat,Ys2: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) )
        = Ys2 )
      = ( ( Ys2 != nil_list_nat )
        & ( ( butlast_list_nat @ Ys2 )
          = Xs3 )
        & ( ( last_list_nat @ Ys2 )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_244_Suc__length__conv,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ( suc @ N )
        = ( size_s3023201423986296836st_nat @ Xs3 ) )
      = ( ? [Y4: list_nat,Ys3: list_list_nat] :
            ( ( Xs3
              = ( cons_list_nat @ Y4 @ Ys3 ) )
            & ( ( size_s3023201423986296836st_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_245_Suc__length__conv,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs3 ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs3
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_246_length__Suc__conv,axiom,
    ! [Xs3: list_list_nat,N: nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( suc @ N ) )
      = ( ? [Y4: list_nat,Ys3: list_list_nat] :
            ( ( Xs3
              = ( cons_list_nat @ Y4 @ Ys3 ) )
            & ( ( size_s3023201423986296836st_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_247_length__Suc__conv,axiom,
    ! [Xs3: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs3
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_248_last_Osimps,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( ( Xs3 = nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs3 ) )
          = X ) )
      & ( ( Xs3 != nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs3 ) )
          = ( last_nat @ Xs3 ) ) ) ) ).

% last.simps
thf(fact_249_last_Osimps,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( ( Xs3 = nil_list_nat )
       => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
          = X ) )
      & ( ( Xs3 != nil_list_nat )
       => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
          = ( last_list_nat @ Xs3 ) ) ) ) ).

% last.simps
thf(fact_250_last__ConsL,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( Xs3 = nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs3 ) )
        = X ) ) ).

% last_ConsL
thf(fact_251_last__ConsL,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( Xs3 = nil_list_nat )
     => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
        = X ) ) ).

% last_ConsL
thf(fact_252_last__ConsR,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( Xs3 != nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs3 ) )
        = ( last_nat @ Xs3 ) ) ) ).

% last_ConsR
thf(fact_253_last__ConsR,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
        = ( last_list_nat @ Xs3 ) ) ) ).

% last_ConsR
thf(fact_254_last__append,axiom,
    ! [Ys2: list_nat,Xs3: list_nat] :
      ( ( ( Ys2 = nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs3 @ Ys2 ) )
          = ( last_nat @ Xs3 ) ) )
      & ( ( Ys2 != nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs3 @ Ys2 ) )
          = ( last_nat @ Ys2 ) ) ) ) ).

% last_append
thf(fact_255_last__append,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_nat] :
      ( ( ( Ys2 = nil_list_nat )
       => ( ( last_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
          = ( last_list_nat @ Xs3 ) ) )
      & ( ( Ys2 != nil_list_nat )
       => ( ( last_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
          = ( last_list_nat @ Ys2 ) ) ) ) ).

% last_append
thf(fact_256_longest__common__suffix,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
    ? [Ss: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ( Xs3
        = ( append_nat @ Xs4 @ Ss ) )
      & ( Ys2
        = ( append_nat @ Ys5 @ Ss ) )
      & ( ( Xs4 = nil_nat )
        | ( Ys5 = nil_nat )
        | ( ( last_nat @ Xs4 )
         != ( last_nat @ Ys5 ) ) ) ) ).

% longest_common_suffix
thf(fact_257_longest__common__suffix,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
    ? [Ss: list_list_nat,Xs4: list_list_nat,Ys5: list_list_nat] :
      ( ( Xs3
        = ( append_list_nat @ Xs4 @ Ss ) )
      & ( Ys2
        = ( append_list_nat @ Ys5 @ Ss ) )
      & ( ( Xs4 = nil_list_nat )
        | ( Ys5 = nil_list_nat )
        | ( ( last_list_nat @ Xs4 )
         != ( last_list_nat @ Ys5 ) ) ) ) ).

% longest_common_suffix
thf(fact_258_butlast_Osimps_I2_J,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( ( Xs3 = nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs3 ) )
          = nil_nat ) )
      & ( ( Xs3 != nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs3 ) )
          = ( cons_nat @ X @ ( butlast_nat @ Xs3 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_259_butlast_Osimps_I2_J,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( ( Xs3 = nil_list_nat )
       => ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
          = nil_list_nat ) )
      & ( ( Xs3 != nil_list_nat )
       => ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
          = ( cons_list_nat @ X @ ( butlast_list_nat @ Xs3 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_260_butlast__append,axiom,
    ! [Ys2: list_nat,Xs3: list_nat] :
      ( ( ( Ys2 = nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs3 @ Ys2 ) )
          = ( butlast_nat @ Xs3 ) ) )
      & ( ( Ys2 != nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs3 @ Ys2 ) )
          = ( append_nat @ Xs3 @ ( butlast_nat @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_261_butlast__append,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_nat] :
      ( ( ( Ys2 = nil_list_nat )
       => ( ( butlast_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
          = ( butlast_list_nat @ Xs3 ) ) )
      & ( ( Ys2 != nil_list_nat )
       => ( ( butlast_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
          = ( append_list_nat @ Xs3 @ ( butlast_list_nat @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_262_maps__simps_I2_J,axiom,
    ! [F: nat > list_nat] :
      ( ( maps_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_263_maps__simps_I2_J,axiom,
    ! [F: nat > list_list_nat] :
      ( ( maps_nat_list_nat @ F @ nil_nat )
      = nil_list_nat ) ).

% maps_simps(2)
thf(fact_264_maps__simps_I2_J,axiom,
    ! [F: list_nat > list_nat] :
      ( ( maps_list_nat_nat @ F @ nil_list_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_265_maps__simps_I2_J,axiom,
    ! [F: list_nat > list_list_nat] :
      ( ( maps_l5785965478274863235st_nat @ F @ nil_list_nat )
      = nil_list_nat ) ).

% maps_simps(2)
thf(fact_266_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_267_length__append__singleton,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( suc @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ).

% length_append_singleton
thf(fact_268_length__append__singleton,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs3 ) ) ) ).

% length_append_singleton
thf(fact_269_length__Cons,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( cons_list_nat @ X @ Xs3 ) )
      = ( suc @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ).

% length_Cons
thf(fact_270_length__Cons,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs3 ) )
      = ( suc @ ( size_size_list_nat @ Xs3 ) ) ) ).

% length_Cons
thf(fact_271_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_272_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( suc @ X23 )
        = ( suc @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% nat.inject
thf(fact_273_prefixes__snoc,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( prefixes_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) )
      = ( append_list_nat @ ( prefixes_nat @ Xs3 ) @ ( cons_list_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).

% prefixes_snoc
thf(fact_274_prefixes__snoc,axiom,
    ! [Xs3: list_list_nat,X: list_nat] :
      ( ( prefixes_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( append_list_list_nat @ ( prefixes_list_nat @ Xs3 ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).

% prefixes_snoc
thf(fact_275_length__stirling__row,axiom,
    ! [N: nat] :
      ( ( size_size_list_nat @ ( stirling_row @ N ) )
      = ( suc @ N ) ) ).

% length_stirling_row
thf(fact_276_prefixes__eq__snoc,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_list_nat,X: list_list_nat] :
      ( ( ( prefixes_list_nat @ Ys2 )
        = ( append_list_list_nat @ Xs3 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
      = ( ( ( ( Ys2 = nil_list_nat )
            & ( Xs3 = nil_list_list_nat ) )
          | ? [Z3: list_nat,Zs3: list_list_nat] :
              ( ( Ys2
                = ( append_list_nat @ Zs3 @ ( cons_list_nat @ Z3 @ nil_list_nat ) ) )
              & ( Xs3
                = ( prefixes_list_nat @ Zs3 ) ) ) )
        & ( X = Ys2 ) ) ) ).

% prefixes_eq_snoc
thf(fact_277_prefixes__eq__snoc,axiom,
    ! [Ys2: list_nat,Xs3: list_list_nat,X: list_nat] :
      ( ( ( prefixes_nat @ Ys2 )
        = ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( ( ( ( Ys2 = nil_nat )
            & ( Xs3 = nil_list_nat ) )
          | ? [Z3: nat,Zs3: list_nat] :
              ( ( Ys2
                = ( append_nat @ Zs3 @ ( cons_nat @ Z3 @ nil_nat ) ) )
              & ( Xs3
                = ( prefixes_nat @ Zs3 ) ) ) )
        & ( X = Ys2 ) ) ) ).

% prefixes_eq_snoc
thf(fact_278_sublists_Osimps_I1_J,axiom,
    ( ( sublists_list_nat @ nil_list_nat )
    = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).

% sublists.simps(1)
thf(fact_279_sublists_Osimps_I1_J,axiom,
    ( ( sublists_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% sublists.simps(1)
thf(fact_280_product__lists_Osimps_I1_J,axiom,
    ( ( produc6783906451316923569st_nat @ nil_list_list_nat )
    = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_281_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_282_prefixes__not__Nil,axiom,
    ! [Xs3: list_nat] :
      ( ( prefixes_nat @ Xs3 )
     != nil_list_nat ) ).

% prefixes_not_Nil
thf(fact_283_stirling__row__nonempty,axiom,
    ! [N: nat] :
      ( ( stirling_row @ N )
     != nil_nat ) ).

% stirling_row_nonempty
thf(fact_284_Suc__inject,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y2 ) )
     => ( X = Y2 ) ) ).

% Suc_inject
thf(fact_285_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_286_size__neq__size__imp__neq,axiom,
    ! [X: list_list_nat,Y2: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ X )
       != ( size_s3023201423986296836st_nat @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_287_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_288_size__neq__size__imp__neq,axiom,
    ! [X: char,Y2: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_289_prefixes_Osimps_I1_J,axiom,
    ( ( prefixes_list_nat @ nil_list_nat )
    = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).

% prefixes.simps(1)
thf(fact_290_prefixes_Osimps_I1_J,axiom,
    ( ( prefixes_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% prefixes.simps(1)
thf(fact_291_SuccI,axiom,
    ! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
     => ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_292_SuccI,axiom,
    ! [Kl: list_list_nat,K: list_nat,Kl2: set_list_list_nat] :
      ( ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 )
     => ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_293_SuccD,axiom,
    ! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
      ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
     => ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_294_SuccD,axiom,
    ! [K: list_nat,Kl2: set_list_list_nat,Kl: list_list_nat] :
      ( ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) )
     => ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_295_suffixes__eq__snoc,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_list_nat,X: list_list_nat] :
      ( ( ( suffixes_list_nat @ Ys2 )
        = ( append_list_list_nat @ Xs3 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
      = ( ( ( ( Ys2 = nil_list_nat )
            & ( Xs3 = nil_list_list_nat ) )
          | ? [Z3: list_nat,Zs3: list_list_nat] :
              ( ( Ys2
                = ( cons_list_nat @ Z3 @ Zs3 ) )
              & ( Xs3
                = ( suffixes_list_nat @ Zs3 ) ) ) )
        & ( X = Ys2 ) ) ) ).

% suffixes_eq_snoc
thf(fact_296_suffixes__eq__snoc,axiom,
    ! [Ys2: list_nat,Xs3: list_list_nat,X: list_nat] :
      ( ( ( suffixes_nat @ Ys2 )
        = ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( ( ( ( Ys2 = nil_nat )
            & ( Xs3 = nil_list_nat ) )
          | ? [Z3: nat,Zs3: list_nat] :
              ( ( Ys2
                = ( cons_nat @ Z3 @ Zs3 ) )
              & ( Xs3
                = ( suffixes_nat @ Zs3 ) ) ) )
        & ( X = Ys2 ) ) ) ).

% suffixes_eq_snoc
thf(fact_297_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( ( concat_list_nat @ Xss2 )
        = ( append_list_nat @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_list_list_nat )
         => ( ( Ys2 = nil_list_nat )
            & ( Zs = nil_list_nat ) ) )
        & ( ( Xss2 != nil_list_list_nat )
         => ? [Xss1: list_list_list_nat,Xs2: list_list_nat,Xs5: list_list_nat,Xss22: list_list_list_nat] :
              ( ( Xss2
                = ( append_list_list_nat @ Xss1 @ ( cons_list_list_nat @ ( append_list_nat @ Xs2 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs2 ) )
              & ( Zs
                = ( append_list_nat @ Xs5 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_298_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_list_nat )
         => ( ( Ys2 = nil_nat )
            & ( Zs = nil_nat ) ) )
        & ( ( Xss2 != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
              ( ( Xss2
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
              & ( Zs
                = ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_299_length__prefixes,axiom,
    ! [Xs3: list_list_nat] :
      ( ( size_s6248950052170075156st_nat @ ( prefixes_list_nat @ Xs3 ) )
      = ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) ) ).

% length_prefixes
thf(fact_300_length__prefixes,axiom,
    ! [Xs3: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs3 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) ) ).

% length_prefixes
thf(fact_301_suffixes_Osimps_I2_J,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( suffixes_nat @ ( cons_nat @ X @ Xs3 ) )
      = ( append_list_nat @ ( suffixes_nat @ Xs3 ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs3 ) @ nil_list_nat ) ) ) ).

% suffixes.simps(2)
thf(fact_302_suffixes_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( suffixes_list_nat @ ( cons_list_nat @ X @ Xs3 ) )
      = ( append_list_list_nat @ ( suffixes_list_nat @ Xs3 ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ nil_list_list_nat ) ) ) ).

% suffixes.simps(2)
thf(fact_303_stirling__row__code_I1_J,axiom,
    ( ( stirling_row @ zero_zero_nat )
    = ( cons_nat @ one_one_nat @ nil_nat ) ) ).

% stirling_row_code(1)
thf(fact_304_last__list__update,axiom,
    ! [Xs3: list_list_nat,K: nat,X: list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( ( K
            = ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) )
         => ( ( last_list_nat @ ( list_update_list_nat @ Xs3 @ K @ X ) )
            = X ) )
        & ( ( K
           != ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) )
         => ( ( last_list_nat @ ( list_update_list_nat @ Xs3 @ K @ X ) )
            = ( last_list_nat @ Xs3 ) ) ) ) ) ).

% last_list_update
thf(fact_305_last__list__update,axiom,
    ! [Xs3: list_nat,K: nat,X: nat] :
      ( ( Xs3 != nil_nat )
     => ( ( ( K
            = ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) )
         => ( ( last_nat @ ( list_update_nat @ Xs3 @ K @ X ) )
            = X ) )
        & ( ( K
           != ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) )
         => ( ( last_nat @ ( list_update_nat @ Xs3 @ K @ X ) )
            = ( last_nat @ Xs3 ) ) ) ) ) ).

% last_list_update
thf(fact_306_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_307_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_308_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_309_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_310_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_311_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_312_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_313_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ( plus_plus_nat @ X @ Y2 )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_314_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y2: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y2 ) )
      = ( ( X = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_315_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_316_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_317_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_318_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_319_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_320_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_321_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_322_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_323_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_324_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_325_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_326_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_327_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_328_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_329_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_330_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_331_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_332_length__0__conv,axiom,
    ! [Xs3: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = zero_zero_nat )
      = ( Xs3 = nil_list_nat ) ) ).

% length_0_conv
thf(fact_333_length__0__conv,axiom,
    ! [Xs3: list_nat] :
      ( ( ( size_size_list_nat @ Xs3 )
        = zero_zero_nat )
      = ( Xs3 = nil_nat ) ) ).

% length_0_conv
thf(fact_334_length__append,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).

% length_append
thf(fact_335_length__append,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs3 @ Ys2 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_append
thf(fact_336_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_337_count__list__append,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,X: list_nat] :
      ( ( count_list_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ X )
      = ( plus_plus_nat @ ( count_list_list_nat @ Xs3 @ X ) @ ( count_list_list_nat @ Ys2 @ X ) ) ) ).

% count_list_append
thf(fact_338_count__list__append,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,X: nat] :
      ( ( count_list_nat @ ( append_nat @ Xs3 @ Ys2 ) @ X )
      = ( plus_plus_nat @ ( count_list_nat @ Xs3 @ X ) @ ( count_list_nat @ Ys2 @ X ) ) ) ).

% count_list_append
thf(fact_339_concat__append,axiom,
    ! [Xs3: list_list_list_nat,Ys2: list_list_list_nat] :
      ( ( concat_list_nat @ ( append_list_list_nat @ Xs3 @ Ys2 ) )
      = ( append_list_nat @ ( concat_list_nat @ Xs3 ) @ ( concat_list_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_340_concat__append,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( append_nat @ ( concat_nat @ Xs3 ) @ ( concat_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_341_length__butlast,axiom,
    ! [Xs3: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( butlast_list_nat @ Xs3 ) )
      = ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_342_length__butlast,axiom,
    ! [Xs3: list_nat] :
      ( ( size_size_list_nat @ ( butlast_nat @ Xs3 ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_343_length__suffixes,axiom,
    ! [Xs3: list_list_nat] :
      ( ( size_s6248950052170075156st_nat @ ( suffixes_list_nat @ Xs3 ) )
      = ( suc @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ).

% length_suffixes
thf(fact_344_length__suffixes,axiom,
    ! [Xs3: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs3 ) )
      = ( suc @ ( size_size_list_nat @ Xs3 ) ) ) ).

% length_suffixes
thf(fact_345_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_346_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_347_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_348_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_349_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_350_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_351_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_352_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_353_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_354_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_355_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_356_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_357_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_358_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_359_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_360_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_361_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_362_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_363_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_364_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_365_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_366_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_367_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_368_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_369_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_370_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_371_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_372_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_373_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_374_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_375_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_376_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_377_enum__rgfs_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X
           != ( suc @ N3 ) ) ) ).

% enum_rgfs.cases
thf(fact_378_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_379_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_380_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_381_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_382_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_383_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X2: nat,Y3: nat] :
              ( ( P @ X2 @ Y3 )
             => ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_384_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_385_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y2
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_386_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat
        = ( suc @ X23 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_387_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_388_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_389_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( zero_zero_nat
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_390_list_Osize_I4_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( cons_list_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_391_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_392_suffixes__not__Nil,axiom,
    ! [Xs3: list_nat] :
      ( ( suffixes_nat @ Xs3 )
     != nil_list_nat ) ).

% suffixes_not_Nil
thf(fact_393_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_394_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_395_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_396_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_397_list_Osize_I3_J,axiom,
    ( ( size_s3023201423986296836st_nat @ nil_list_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_398_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_399_concat_Osimps_I2_J,axiom,
    ! [X: list_list_nat,Xs3: list_list_list_nat] :
      ( ( concat_list_nat @ ( cons_list_list_nat @ X @ Xs3 ) )
      = ( append_list_nat @ X @ ( concat_list_nat @ Xs3 ) ) ) ).

% concat.simps(2)
thf(fact_400_concat_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( concat_nat @ ( cons_list_nat @ X @ Xs3 ) )
      = ( append_nat @ X @ ( concat_nat @ Xs3 ) ) ) ).

% concat.simps(2)
thf(fact_401_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_402_concat_Osimps_I1_J,axiom,
    ( ( concat_list_nat @ nil_list_list_nat )
    = nil_list_nat ) ).

% concat.simps(1)
thf(fact_403_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_404_list__update__code_I2_J,axiom,
    ! [X: nat,Xs3: list_nat,Y2: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs3 ) @ zero_zero_nat @ Y2 )
      = ( cons_nat @ Y2 @ Xs3 ) ) ).

% list_update_code(2)
thf(fact_405_list__update__code_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat,Y2: list_nat] :
      ( ( list_update_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ zero_zero_nat @ Y2 )
      = ( cons_list_nat @ Y2 @ Xs3 ) ) ).

% list_update_code(2)
thf(fact_406_count__list_Osimps_I1_J,axiom,
    ! [Y2: list_nat] :
      ( ( count_list_list_nat @ nil_list_nat @ Y2 )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_407_count__list_Osimps_I1_J,axiom,
    ! [Y2: nat] :
      ( ( count_list_nat @ nil_nat @ Y2 )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_408_stirling__row__code_I2_J,axiom,
    ! [N: nat] :
      ( ( stirling_row @ ( suc @ N ) )
      = ( stirling_row_aux_nat @ N @ zero_zero_nat @ ( stirling_row @ N ) ) ) ).

% stirling_row_code(2)
thf(fact_409_enum__rgfs_Osimps_I1_J,axiom,
    ( ( equiva7426478223624825838m_rgfs @ zero_zero_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% enum_rgfs.simps(1)
thf(fact_410_count__list_Osimps_I2_J,axiom,
    ! [X: list_nat,Y2: list_nat,Xs3: list_list_nat] :
      ( ( ( X = Y2 )
       => ( ( count_list_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ Y2 )
          = ( plus_plus_nat @ ( count_list_list_nat @ Xs3 @ Y2 ) @ one_one_nat ) ) )
      & ( ( X != Y2 )
       => ( ( count_list_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ Y2 )
          = ( count_list_list_nat @ Xs3 @ Y2 ) ) ) ) ).

% count_list.simps(2)
thf(fact_411_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Y2: nat,Xs3: list_nat] :
      ( ( ( X = Y2 )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs3 ) @ Y2 )
          = ( plus_plus_nat @ ( count_list_nat @ Xs3 @ Y2 ) @ one_one_nat ) ) )
      & ( ( X != Y2 )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs3 ) @ Y2 )
          = ( count_list_nat @ Xs3 @ Y2 ) ) ) ) ).

% count_list.simps(2)
thf(fact_412_suffixes_Osimps_I1_J,axiom,
    ( ( suffixes_list_nat @ nil_list_nat )
    = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).

% suffixes.simps(1)
thf(fact_413_suffixes_Osimps_I1_J,axiom,
    ( ( suffixes_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% suffixes.simps(1)
thf(fact_414_butlast__list__update,axiom,
    ! [K: nat,Xs3: list_list_nat,X: list_nat] :
      ( ( ( K
          = ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) )
       => ( ( butlast_list_nat @ ( list_update_list_nat @ Xs3 @ K @ X ) )
          = ( butlast_list_nat @ Xs3 ) ) )
      & ( ( K
         != ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) )
       => ( ( butlast_list_nat @ ( list_update_list_nat @ Xs3 @ K @ X ) )
          = ( list_update_list_nat @ ( butlast_list_nat @ Xs3 ) @ K @ X ) ) ) ) ).

% butlast_list_update
thf(fact_415_butlast__list__update,axiom,
    ! [K: nat,Xs3: list_nat,X: nat] :
      ( ( ( K
          = ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs3 @ K @ X ) )
          = ( butlast_nat @ Xs3 ) ) )
      & ( ( K
         != ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) )
       => ( ( butlast_nat @ ( list_update_nat @ Xs3 @ K @ X ) )
          = ( list_update_nat @ ( butlast_nat @ Xs3 ) @ K @ X ) ) ) ) ).

% butlast_list_update
thf(fact_416_concat__eq__appendD,axiom,
    ! [Xss2: list_list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( ( concat_list_nat @ Xss2 )
        = ( append_list_nat @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_list_list_nat )
       => ? [Xss12: list_list_list_nat,Xs: list_list_nat,Xs4: list_list_nat,Xss23: list_list_list_nat] :
            ( ( Xss2
              = ( append_list_list_nat @ Xss12 @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ Xs4 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append_list_nat @ ( concat_list_nat @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append_list_nat @ Xs4 @ ( concat_list_nat @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_417_concat__eq__appendD,axiom,
    ! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_list_nat )
       => ? [Xss12: list_list_nat,Xs: list_nat,Xs4: list_nat,Xss23: list_list_nat] :
            ( ( Xss2
              = ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs @ Xs4 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append_nat @ ( concat_nat @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append_nat @ Xs4 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_418_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_list_nat @ N @ nil_list_nat )
          = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_list_nat @ N @ nil_list_nat )
          = nil_list_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_419_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_420_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C2: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_421_n__lists_Osimps_I1_J,axiom,
    ! [Xs3: list_list_nat] :
      ( ( n_lists_list_nat @ zero_zero_nat @ Xs3 )
      = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_422_n__lists_Osimps_I1_J,axiom,
    ! [Xs3: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs3 )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_423_empty__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat] :
      ( ( member_list_nat @ nil_nat @ Kl2 )
     => ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_424_empty__Shift,axiom,
    ! [Kl2: set_list_list_nat,K: list_nat] :
      ( ( member_list_list_nat @ nil_list_nat @ Kl2 )
     => ( ( member_list_nat @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ nil_list_nat ) )
       => ( member_list_list_nat @ nil_list_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_425_Succ__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
      ( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
      = ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).

% Succ_Shift
thf(fact_426_Succ__Shift,axiom,
    ! [Kl2: set_list_list_nat,K: list_nat,Kl: list_list_nat] :
      ( ( bNF_Gr3053708287304744325st_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) @ Kl )
      = ( bNF_Gr3053708287304744325st_nat @ Kl2 @ ( cons_list_nat @ K @ Kl ) ) ) ).

% Succ_Shift
thf(fact_427_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( P @ A4 @ B4 )
          = ( P @ B4 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ A4 @ B4 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_428_ShiftD,axiom,
    ! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
      ( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
     => ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_429_ShiftD,axiom,
    ! [Kl: list_list_nat,Kl2: set_list_list_nat,K: list_nat] :
      ( ( member_list_list_nat @ Kl @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) )
     => ( member_list_list_nat @ ( cons_list_nat @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_430_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C2: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_431_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_432_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_433_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_434_Stirling__1,axiom,
    ! [N: nat] :
      ( ( stirling @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% Stirling_1
thf(fact_435_gen__length__def,axiom,
    ( gen_length_list_nat
    = ( ^ [N2: nat,Xs2: list_list_nat] : ( plus_plus_nat @ N2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_436_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N2: nat,Xs2: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_437_length__code,axiom,
    ( size_s3023201423986296836st_nat
    = ( gen_length_list_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_438_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_439__092_060open_062x_A_060_Argf__limit_Axs_A_L_A1_092_060close_062,axiom,
    ord_less_nat @ xa @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ xs ) @ one_one_nat ) ).

% \<open>x < rgf_limit xs + 1\<close>
thf(fact_440_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_441_gen__length__code_I2_J,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs3 ) )
      = ( gen_length_nat @ ( suc @ N ) @ Xs3 ) ) ).

% gen_length_code(2)
thf(fact_442_gen__length__code_I2_J,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( gen_length_list_nat @ N @ ( cons_list_nat @ X @ Xs3 ) )
      = ( gen_length_list_nat @ ( suc @ N ) @ Xs3 ) ) ).

% gen_length_code(2)
thf(fact_443_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_list_nat @ nil_list_nat )
    = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_444_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_445_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_446_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_447_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_448_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_449_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_450_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_451_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_452_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_453_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_454_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_455_Stirling__same,axiom,
    ! [N: nat] :
      ( ( stirling @ N @ N )
      = one_one_nat ) ).

% Stirling_same
thf(fact_456_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_457_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_458_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_459_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_460_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_461_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_462_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_463_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_464_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_465_Stirling__less,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( stirling @ N @ K )
        = zero_zero_nat ) ) ).

% Stirling_less
thf(fact_466_length__greater__0__conv,axiom,
    ! [Xs3: list_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) )
      = ( Xs3 != nil_list_nat ) ) ).

% length_greater_0_conv
thf(fact_467_length__greater__0__conv,axiom,
    ! [Xs3: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs3 ) )
      = ( Xs3 != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_468_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_469_linorder__neqE__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
     => ( ~ ( ord_less_nat @ X @ Y2 )
       => ( ord_less_nat @ Y2 @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_470_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
              & ~ ( P @ M4 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_471_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
             => ( P @ M4 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_472_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_473_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_474_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_475_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_476_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_477_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_478_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_479_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_480_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_481_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_482_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_483_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_484_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_485_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_486_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_487_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_488_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_489_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_490_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_491_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_492_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_493_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_494_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_495_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_496_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_497_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_498_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_499_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_500_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_501_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_502_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ I2 ) ) ) ) ).

% All_less_Suc
thf(fact_503_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_504_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_505_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_506_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_507_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_508_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_509_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_510_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N3 )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_511_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_512_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_513_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_514_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_515_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_516_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_517_length__induct,axiom,
    ! [P: list_list_nat > $o,Xs3: list_list_nat] :
      ( ! [Xs: list_list_nat] :
          ( ! [Ys6: list_list_nat] :
              ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys6 ) @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( P @ Ys6 ) )
         => ( P @ Xs ) )
     => ( P @ Xs3 ) ) ).

% length_induct
thf(fact_518_length__induct,axiom,
    ! [P: list_nat > $o,Xs3: list_nat] :
      ( ! [Xs: list_nat] :
          ( ! [Ys6: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs ) )
             => ( P @ Ys6 ) )
         => ( P @ Xs ) )
     => ( P @ Xs3 ) ) ).

% length_induct
thf(fact_519_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_520_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_521_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_522_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_523_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_524_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_525_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_526_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_527_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_528_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_529_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_530_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_531_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_532_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_533_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_534_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_535_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_536_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_537_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_538_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_539_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_540_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M2: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_541_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_542_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_543_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_544_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).

% less_natE
thf(fact_545_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_546_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_547_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_548_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_549_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_550_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_551_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_552_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_553_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_554_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_555_Stirling_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( stirling @ ( suc @ N ) @ zero_zero_nat )
      = zero_zero_nat ) ).

% Stirling.simps(3)
thf(fact_556_Stirling_Osimps_I2_J,axiom,
    ! [K: nat] :
      ( ( stirling @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% Stirling.simps(2)
thf(fact_557_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_558_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_list_nat @ N @ nil_list_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_559_Stirling_Osimps_I1_J,axiom,
    ( ( stirling @ zero_zero_nat @ zero_zero_nat )
    = one_one_nat ) ).

% Stirling.simps(1)
thf(fact_560_rgf__limit_Osimps_I1_J,axiom,
    ( ( equiva5889994315859557365_limit @ nil_nat )
    = zero_zero_nat ) ).

% rgf_limit.simps(1)
thf(fact_561_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_562_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_563_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_564_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_565_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_566_list__update__append1,axiom,
    ! [I: nat,Xs3: list_list_nat,Ys2: list_list_nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( list_update_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ I @ X )
        = ( append_list_nat @ ( list_update_list_nat @ Xs3 @ I @ X ) @ Ys2 ) ) ) ).

% list_update_append1
thf(fact_567_list__update__append1,axiom,
    ! [I: nat,Xs3: list_nat,Ys2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( list_update_nat @ ( append_nat @ Xs3 @ Ys2 ) @ I @ X )
        = ( append_nat @ ( list_update_nat @ Xs3 @ I @ X ) @ Ys2 ) ) ) ).

% list_update_append1
thf(fact_568_rgf__snoc,axiom,
    ! [Xs3: list_nat,X: nat] :
      ( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs3 @ ( cons_nat @ X @ nil_nat ) ) )
      = ( ( equiva3371634703666331078on_rgf @ Xs3 )
        & ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs3 ) @ one_one_nat ) ) ) ) ).

% rgf_snoc
thf(fact_569_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_570_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_571_list__update__append,axiom,
    ! [N: nat,Xs3: list_list_nat,Ys2: list_list_nat,X: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
       => ( ( list_update_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ N @ X )
          = ( append_list_nat @ ( list_update_list_nat @ Xs3 @ N @ X ) @ Ys2 ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
       => ( ( list_update_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ N @ X )
          = ( append_list_nat @ Xs3 @ ( list_update_list_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_572_list__update__append,axiom,
    ! [N: nat,Xs3: list_nat,Ys2: list_nat,X: nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
       => ( ( list_update_nat @ ( append_nat @ Xs3 @ Ys2 ) @ N @ X )
          = ( append_nat @ ( list_update_nat @ Xs3 @ N @ X ) @ Ys2 ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
       => ( ( list_update_nat @ ( append_nat @ Xs3 @ Ys2 ) @ N @ X )
          = ( append_nat @ Xs3 @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs3 ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_573_rgf__limit__snoc,axiom,
    ! [X: list_nat,Y2: nat] :
      ( ( equiva5889994315859557365_limit @ ( append_nat @ X @ ( cons_nat @ Y2 @ nil_nat ) ) )
      = ( ord_max_nat @ ( plus_plus_nat @ Y2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ X ) ) ) ).

% rgf_limit_snoc
thf(fact_574_rgf__limit_Oelims,axiom,
    ! [X: list_nat,Y2: nat] :
      ( ( ( equiva5889994315859557365_limit @ X )
        = Y2 )
     => ( ( ( X = nil_nat )
         => ( Y2 != zero_zero_nat ) )
       => ~ ! [X2: nat,Xs: list_nat] :
              ( ( X
                = ( cons_nat @ X2 @ Xs ) )
             => ( Y2
               != ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ) ) ).

% rgf_limit.elims
thf(fact_575_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs3 ) @ N )
        = ( nth_nat @ Xs3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_576_nth__Cons__pos,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ N )
        = ( nth_list_nat @ Xs3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_577_Stirling_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: nat] :
      ( ( ( stirling @ X @ Xa )
        = Y2 )
     => ( ( ( X = zero_zero_nat )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ( X = zero_zero_nat )
           => ( ? [K2: nat] :
                  ( Xa
                  = ( suc @ K2 ) )
             => ( Y2 != zero_zero_nat ) ) )
         => ( ( ? [N3: nat] :
                  ( X
                  = ( suc @ N3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( Y2 != zero_zero_nat ) ) )
           => ~ ! [N3: nat] :
                  ( ( X
                    = ( suc @ N3 ) )
                 => ! [K2: nat] :
                      ( ( Xa
                        = ( suc @ K2 ) )
                     => ( Y2
                       != ( plus_plus_nat @ ( times_times_nat @ ( suc @ K2 ) @ ( stirling @ N3 @ ( suc @ K2 ) ) ) @ ( stirling @ N3 @ K2 ) ) ) ) ) ) ) ) ) ).

% Stirling.elims
thf(fact_578_max_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ A )
      = A ) ).

% max.idem
thf(fact_579_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
      = ( ord_max_nat @ A @ B ) ) ).

% max.left_idem
thf(fact_580_max_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
      = ( ord_max_nat @ A @ B ) ) ).

% max.right_idem
thf(fact_581_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_582_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_583_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_584_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_585_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_586_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_587_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_588_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_589_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_590_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_591_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_592_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_593_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_594_max__less__iff__conj,axiom,
    ! [X: nat,Y2: nat,Z4: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y2 ) @ Z4 )
      = ( ( ord_less_nat @ X @ Z4 )
        & ( ord_less_nat @ Y2 @ Z4 ) ) ) ).

% max_less_iff_conj
thf(fact_595_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_596_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_597_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_598_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_599_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_600_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_601_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_602_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_603_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_604_list__update__id,axiom,
    ! [Xs3: list_nat,I: nat] :
      ( ( list_update_nat @ Xs3 @ I @ ( nth_nat @ Xs3 @ I ) )
      = Xs3 ) ).

% list_update_id
thf(fact_605_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs3: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs3 @ I @ X ) @ J )
        = ( nth_nat @ Xs3 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_606_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_607_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_608_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_609_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_610_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_611_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_612_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_613_nth__Cons__Suc,axiom,
    ! [X: nat,Xs3: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs3 ) @ ( suc @ N ) )
      = ( nth_nat @ Xs3 @ N ) ) ).

% nth_Cons_Suc
thf(fact_614_nth__Cons__Suc,axiom,
    ! [X: list_nat,Xs3: list_list_nat,N: nat] :
      ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ ( suc @ N ) )
      = ( nth_list_nat @ Xs3 @ N ) ) ).

% nth_Cons_Suc
thf(fact_615_nth__Cons__0,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs3 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_616_nth__Cons__0,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_617_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_618_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_619_nth__append__length,axiom,
    ! [Xs3: list_list_nat,X: list_nat,Ys2: list_list_nat] :
      ( ( nth_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs3 ) )
      = X ) ).

% nth_append_length
thf(fact_620_nth__append__length,axiom,
    ! [Xs3: list_nat,X: nat,Ys2: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs3 @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs3 ) )
      = X ) ).

% nth_append_length
thf(fact_621_nth__append__length__plus,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,N: nat] :
      ( ( nth_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) )
      = ( nth_list_nat @ Ys2 @ N ) ) ).

% nth_append_length_plus
thf(fact_622_nth__append__length__plus,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs3 @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs3 ) @ N ) )
      = ( nth_nat @ Ys2 @ N ) ) ).

% nth_append_length_plus
thf(fact_623_nth__list__update__eq,axiom,
    ! [I: nat,Xs3: list_list_nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( nth_list_nat @ ( list_update_list_nat @ Xs3 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_624_nth__list__update__eq,axiom,
    ! [I: nat,Xs3: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs3 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_625_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.assoc
thf(fact_626_max_Ocommute,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_max_nat @ B3 @ A3 ) ) ) ).

% max.commute
thf(fact_627_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
      = ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).

% max.left_commute
thf(fact_628_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_629_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_630_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_631_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_632_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_633_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_634_less__max__iff__disj,axiom,
    ! [Z4: nat,X: nat,Y2: nat] :
      ( ( ord_less_nat @ Z4 @ ( ord_max_nat @ X @ Y2 ) )
      = ( ( ord_less_nat @ Z4 @ X )
        | ( ord_less_nat @ Z4 @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_635_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_636_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( ord_max_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_637_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_638_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_639_max__add__distrib__left,axiom,
    ! [X: nat,Y2: nat,Z4: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y2 ) @ Z4 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z4 ) @ ( plus_plus_nat @ Y2 @ Z4 ) ) ) ).

% max_add_distrib_left
thf(fact_640_max__add__distrib__right,axiom,
    ! [X: nat,Y2: nat,Z4: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y2 @ Z4 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y2 ) @ ( plus_plus_nat @ X @ Z4 ) ) ) ).

% max_add_distrib_right
thf(fact_641_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_642_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_643_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_644_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_645_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_646_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_647_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_648_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_649_crossproduct__eq,axiom,
    ! [W2: nat,Y2: nat,X: nat,Z4: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y2 ) @ ( times_times_nat @ X @ Z4 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z4 ) @ ( times_times_nat @ X @ Y2 ) ) )
      = ( ( W2 = X )
        | ( Y2 = Z4 ) ) ) ).

% crossproduct_eq
thf(fact_650_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_651_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_652_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_653_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_654_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_655_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_656_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_657_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_658_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_659_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_660_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_661_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_662_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_663_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_664_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_665_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_666_nth__equalityI,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs3 )
        = ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
           => ( ( nth_list_nat @ Xs3 @ I3 )
              = ( nth_list_nat @ Ys2 @ I3 ) ) )
       => ( Xs3 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_667_nth__equalityI,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs3 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
           => ( ( nth_nat @ Xs3 @ I3 )
              = ( nth_nat @ Ys2 @ I3 ) ) )
       => ( Xs3 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_668_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X4: list_nat] : ( P @ I2 @ X4 ) ) )
      = ( ? [Xs2: list_list_nat] :
            ( ( ( size_s3023201423986296836st_nat @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_list_nat @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_669_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X4: nat] : ( P @ I2 @ X4 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_670_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_list_nat,Z: list_list_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_list_nat,Ys3: list_list_nat] :
          ( ( ( size_s3023201423986296836st_nat @ Xs2 )
            = ( size_s3023201423986296836st_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
             => ( ( nth_list_nat @ Xs2 @ I2 )
                = ( nth_list_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_671_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I2 )
                = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_672_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_673_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_674_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_675_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_676_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_677_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_678_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_679_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_680_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_681_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_682_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_683_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_684_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_685_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_686_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_687_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_688_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_689_nth__list__update,axiom,
    ! [I: nat,Xs3: list_list_nat,J: nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( ( I = J )
         => ( ( nth_list_nat @ ( list_update_list_nat @ Xs3 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_list_nat @ ( list_update_list_nat @ Xs3 @ I @ X ) @ J )
            = ( nth_list_nat @ Xs3 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_690_nth__list__update,axiom,
    ! [I: nat,Xs3: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs3 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs3 @ I @ X ) @ J )
            = ( nth_nat @ Xs3 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_691_list__update__same__conv,axiom,
    ! [I: nat,Xs3: list_list_nat,X: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( ( list_update_list_nat @ Xs3 @ I @ X )
          = Xs3 )
        = ( ( nth_list_nat @ Xs3 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_692_list__update__same__conv,axiom,
    ! [I: nat,Xs3: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( ( list_update_nat @ Xs3 @ I @ X )
          = Xs3 )
        = ( ( nth_nat @ Xs3 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_693_nth__butlast,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ ( butlast_list_nat @ Xs3 ) ) )
     => ( ( nth_list_nat @ ( butlast_list_nat @ Xs3 ) @ N )
        = ( nth_list_nat @ Xs3 @ N ) ) ) ).

% nth_butlast
thf(fact_694_nth__butlast,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs3 ) ) )
     => ( ( nth_nat @ ( butlast_nat @ Xs3 ) @ N )
        = ( nth_nat @ Xs3 @ N ) ) ) ).

% nth_butlast
thf(fact_695_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_696_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_697_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_698_Stirling_Osimps_I4_J,axiom,
    ! [N: nat,K: nat] :
      ( ( stirling @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( suc @ K ) @ ( stirling @ N @ ( suc @ K ) ) ) @ ( stirling @ N @ K ) ) ) ).

% Stirling.simps(4)
thf(fact_699_stirling__row__aux_Osimps_I2_J,axiom,
    ! [N: nat,Y2: nat,X: nat,Xs3: list_nat] :
      ( ( stirling_row_aux_nat @ N @ Y2 @ ( cons_nat @ X @ Xs3 ) )
      = ( cons_nat @ ( plus_plus_nat @ Y2 @ ( times_times_nat @ N @ X ) ) @ ( stirling_row_aux_nat @ N @ X @ Xs3 ) ) ) ).

% stirling_row_aux.simps(2)
thf(fact_700_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs3 ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs3 ) @ N )
          = ( nth_nat @ Xs3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_701_nth__Cons_H,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ N )
          = ( nth_list_nat @ Xs3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_702_nth__append,axiom,
    ! [N: nat,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
       => ( ( nth_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ N )
          = ( nth_list_nat @ Xs3 @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
       => ( ( nth_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ N )
          = ( nth_list_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ) ) ) ).

% nth_append
thf(fact_703_nth__append,axiom,
    ! [N: nat,Xs3: list_nat,Ys2: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
       => ( ( nth_nat @ ( append_nat @ Xs3 @ Ys2 ) @ N )
          = ( nth_nat @ Xs3 @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
       => ( ( nth_nat @ ( append_nat @ Xs3 @ Ys2 ) @ N )
          = ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).

% nth_append
thf(fact_704_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_705_rgf__limit_Osimps_I2_J,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( equiva5889994315859557365_limit @ ( cons_nat @ X @ Xs3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs3 ) ) ) ).

% rgf_limit.simps(2)
thf(fact_706_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y2: nat,Xs3: list_nat,N: nat] :
      ( ( X != Y2 )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs3 ) @ N )
          = Y2 )
        = ( ( ( nth_nat @ Xs3 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y2 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_707_nth__non__equal__first__eq,axiom,
    ! [X: list_nat,Y2: list_nat,Xs3: list_list_nat,N: nat] :
      ( ( X != Y2 )
     => ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ N )
          = Y2 )
        = ( ( ( nth_list_nat @ Xs3 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y2 )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_708_last__conv__nth,axiom,
    ! [Xs3: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( last_list_nat @ Xs3 )
        = ( nth_list_nat @ Xs3 @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_709_last__conv__nth,axiom,
    ! [Xs3: list_nat] :
      ( ( Xs3 != nil_nat )
     => ( ( last_nat @ Xs3 )
        = ( nth_nat @ Xs3 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_710_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_711_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_712_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_713_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_714_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_715_stirling_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: nat] :
      ( ( ( stirling2 @ X @ Xa )
        = Y2 )
     => ( ( ( X = zero_zero_nat )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ( X = zero_zero_nat )
           => ( ? [K2: nat] :
                  ( Xa
                  = ( suc @ K2 ) )
             => ( Y2 != zero_zero_nat ) ) )
         => ( ( ? [N3: nat] :
                  ( X
                  = ( suc @ N3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( Y2 != zero_zero_nat ) ) )
           => ~ ! [N3: nat] :
                  ( ( X
                    = ( suc @ N3 ) )
                 => ! [K2: nat] :
                      ( ( Xa
                        = ( suc @ K2 ) )
                     => ( Y2
                       != ( plus_plus_nat @ ( times_times_nat @ N3 @ ( stirling2 @ N3 @ ( suc @ K2 ) ) ) @ ( stirling2 @ N3 @ K2 ) ) ) ) ) ) ) ) ) ).

% stirling.elims
thf(fact_716_take__Suc__conv__app__nth,axiom,
    ! [I: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( take_list_nat @ ( suc @ I ) @ Xs3 )
        = ( append_list_nat @ ( take_list_nat @ I @ Xs3 ) @ ( cons_list_nat @ ( nth_list_nat @ Xs3 @ I ) @ nil_list_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_717_take__Suc__conv__app__nth,axiom,
    ! [I: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( take_nat @ ( suc @ I ) @ Xs3 )
        = ( append_nat @ ( take_nat @ I @ Xs3 ) @ ( cons_nat @ ( nth_nat @ Xs3 @ I ) @ nil_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_718_stirling__same,axiom,
    ! [N: nat] :
      ( ( stirling2 @ N @ N )
      = one_one_nat ) ).

% stirling_same
thf(fact_719_take__Suc__Cons,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs3 ) )
      = ( cons_nat @ X @ ( take_nat @ N @ Xs3 ) ) ) ).

% take_Suc_Cons
thf(fact_720_take__Suc__Cons,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( take_list_nat @ ( suc @ N ) @ ( cons_list_nat @ X @ Xs3 ) )
      = ( cons_list_nat @ X @ ( take_list_nat @ N @ Xs3 ) ) ) ).

% take_Suc_Cons
thf(fact_721_take__eq__Nil2,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( nil_nat
        = ( take_nat @ N @ Xs3 ) )
      = ( ( N = zero_zero_nat )
        | ( Xs3 = nil_nat ) ) ) ).

% take_eq_Nil2
thf(fact_722_take__eq__Nil2,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( nil_list_nat
        = ( take_list_nat @ N @ Xs3 ) )
      = ( ( N = zero_zero_nat )
        | ( Xs3 = nil_list_nat ) ) ) ).

% take_eq_Nil2
thf(fact_723_take__eq__Nil,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ( take_nat @ N @ Xs3 )
        = nil_nat )
      = ( ( N = zero_zero_nat )
        | ( Xs3 = nil_nat ) ) ) ).

% take_eq_Nil
thf(fact_724_take__eq__Nil,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ( take_list_nat @ N @ Xs3 )
        = nil_list_nat )
      = ( ( N = zero_zero_nat )
        | ( Xs3 = nil_list_nat ) ) ) ).

% take_eq_Nil
thf(fact_725_take0,axiom,
    ( ( take_nat @ zero_zero_nat )
    = ( ^ [Xs2: list_nat] : nil_nat ) ) ).

% take0
thf(fact_726_take0,axiom,
    ( ( take_list_nat @ zero_zero_nat )
    = ( ^ [Xs2: list_list_nat] : nil_list_nat ) ) ).

% take0
thf(fact_727_nth__take,axiom,
    ! [I: nat,N: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( take_nat @ N @ Xs3 ) @ I )
        = ( nth_nat @ Xs3 @ I ) ) ) ).

% nth_take
thf(fact_728_stirling__0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( stirling2 @ N @ zero_zero_nat )
        = zero_zero_nat ) ) ).

% stirling_0
thf(fact_729_stirling__less,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( stirling2 @ N @ K )
        = zero_zero_nat ) ) ).

% stirling_less
thf(fact_730_take__append,axiom,
    ! [N: nat,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( take_list_nat @ N @ ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( append_list_nat @ ( take_list_nat @ N @ Xs3 ) @ ( take_list_nat @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) ) @ Ys2 ) ) ) ).

% take_append
thf(fact_731_take__append,axiom,
    ! [N: nat,Xs3: list_nat,Ys2: list_nat] :
      ( ( take_nat @ N @ ( append_nat @ Xs3 @ Ys2 ) )
      = ( append_nat @ ( take_nat @ N @ Xs3 ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs3 ) ) @ Ys2 ) ) ) ).

% take_append
thf(fact_732_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_nat @ N @ nil_nat )
      = nil_nat ) ).

% take_Nil
thf(fact_733_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_list_nat @ N @ nil_list_nat )
      = nil_list_nat ) ).

% take_Nil
thf(fact_734_take__0,axiom,
    ! [Xs3: list_nat] :
      ( ( take_nat @ zero_zero_nat @ Xs3 )
      = nil_nat ) ).

% take_0
thf(fact_735_take__0,axiom,
    ! [Xs3: list_list_nat] :
      ( ( take_list_nat @ zero_zero_nat @ Xs3 )
      = nil_list_nat ) ).

% take_0
thf(fact_736_stirling_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( stirling2 @ ( suc @ N ) @ zero_zero_nat )
      = zero_zero_nat ) ).

% stirling.simps(3)
thf(fact_737_stirling_Osimps_I2_J,axiom,
    ! [K: nat] :
      ( ( stirling2 @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% stirling.simps(2)
thf(fact_738_stirling_Osimps_I1_J,axiom,
    ( ( stirling2 @ zero_zero_nat @ zero_zero_nat )
    = one_one_nat ) ).

% stirling.simps(1)
thf(fact_739_take__butlast,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( take_list_nat @ N @ ( butlast_list_nat @ Xs3 ) )
        = ( take_list_nat @ N @ Xs3 ) ) ) ).

% take_butlast
thf(fact_740_take__butlast,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
     => ( ( take_nat @ N @ ( butlast_nat @ Xs3 ) )
        = ( take_nat @ N @ Xs3 ) ) ) ).

% take_butlast
thf(fact_741_butlast__conv__take,axiom,
    ( butlast_list_nat
    = ( ^ [Xs2: list_list_nat] : ( take_list_nat @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) @ Xs2 ) ) ) ).

% butlast_conv_take
thf(fact_742_butlast__conv__take,axiom,
    ( butlast_nat
    = ( ^ [Xs2: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) @ Xs2 ) ) ) ).

% butlast_conv_take
thf(fact_743_stirling_Osimps_I4_J,axiom,
    ! [N: nat,K: nat] :
      ( ( stirling2 @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( stirling2 @ N @ ( suc @ K ) ) ) @ ( stirling2 @ N @ K ) ) ) ).

% stirling.simps(4)
thf(fact_744_take__Cons_H,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( take_nat @ N @ ( cons_nat @ X @ Xs3 ) )
          = nil_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( take_nat @ N @ ( cons_nat @ X @ Xs3 ) )
          = ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs3 ) ) ) ) ) ).

% take_Cons'
thf(fact_745_take__Cons_H,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( take_list_nat @ N @ ( cons_list_nat @ X @ Xs3 ) )
          = nil_list_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( take_list_nat @ N @ ( cons_list_nat @ X @ Xs3 ) )
          = ( cons_list_nat @ X @ ( take_list_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs3 ) ) ) ) ) ).

% take_Cons'
thf(fact_746_stirling__code,axiom,
    ( stirling2
    = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( K3 = zero_zero_nat ) @ ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( K3 = N2 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N2 ) @ K3 ) ) ) ) ) ) ).

% stirling_code
thf(fact_747_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs3: list_list_nat,A: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( list_update_list_nat @ Xs3 @ I @ A )
        = ( append_list_nat @ ( take_list_nat @ I @ Xs3 ) @ ( cons_list_nat @ A @ ( drop_list_nat @ ( suc @ I ) @ Xs3 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_748_upd__conv__take__nth__drop,axiom,
    ! [I: nat,Xs3: list_nat,A: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( list_update_nat @ Xs3 @ I @ A )
        = ( append_nat @ ( take_nat @ I @ Xs3 ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs3 ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_749_id__take__nth__drop,axiom,
    ! [I: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( Xs3
        = ( append_list_nat @ ( take_list_nat @ I @ Xs3 ) @ ( cons_list_nat @ ( nth_list_nat @ Xs3 @ I ) @ ( drop_list_nat @ ( suc @ I ) @ Xs3 ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_750_id__take__nth__drop,axiom,
    ! [I: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( Xs3
        = ( append_nat @ ( take_nat @ I @ Xs3 ) @ ( cons_nat @ ( nth_nat @ Xs3 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs3 ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_751_rgf__limit_Opelims,axiom,
    ! [X: list_nat,Y2: nat] :
      ( ( ( equiva5889994315859557365_limit @ X )
        = Y2 )
     => ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y2 = zero_zero_nat )
             => ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
         => ~ ! [X2: nat,Xs: list_nat] :
                ( ( X
                  = ( cons_nat @ X2 @ Xs ) )
               => ( ( Y2
                    = ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) )
                 => ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X2 @ Xs ) ) ) ) ) ) ) ).

% rgf_limit.pelims
thf(fact_752_drop__Suc__Cons,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs3 ) )
      = ( drop_nat @ N @ Xs3 ) ) ).

% drop_Suc_Cons
thf(fact_753_drop__Suc__Cons,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( drop_list_nat @ ( suc @ N ) @ ( cons_list_nat @ X @ Xs3 ) )
      = ( drop_list_nat @ N @ Xs3 ) ) ).

% drop_Suc_Cons
thf(fact_754_length__drop,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( drop_list_nat @ N @ Xs3 ) )
      = ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ).

% length_drop
thf(fact_755_length__drop,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( size_size_list_nat @ ( drop_nat @ N @ Xs3 ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ).

% length_drop
thf(fact_756_append__take__drop__id,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( append_nat @ ( take_nat @ N @ Xs3 ) @ ( drop_nat @ N @ Xs3 ) )
      = Xs3 ) ).

% append_take_drop_id
thf(fact_757_append__take__drop__id,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( append_list_nat @ ( take_list_nat @ N @ Xs3 ) @ ( drop_list_nat @ N @ Xs3 ) )
      = Xs3 ) ).

% append_take_drop_id
thf(fact_758_drop__append,axiom,
    ! [N: nat,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( drop_list_nat @ N @ ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( append_list_nat @ ( drop_list_nat @ N @ Xs3 ) @ ( drop_list_nat @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) ) @ Ys2 ) ) ) ).

% drop_append
thf(fact_759_drop__append,axiom,
    ! [N: nat,Xs3: list_nat,Ys2: list_nat] :
      ( ( drop_nat @ N @ ( append_nat @ Xs3 @ Ys2 ) )
      = ( append_nat @ ( drop_nat @ N @ Xs3 ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs3 ) ) @ Ys2 ) ) ) ).

% drop_append
thf(fact_760_last__drop,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( last_list_nat @ ( drop_list_nat @ N @ Xs3 ) )
        = ( last_list_nat @ Xs3 ) ) ) ).

% last_drop
thf(fact_761_last__drop,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
     => ( ( last_nat @ ( drop_nat @ N @ Xs3 ) )
        = ( last_nat @ Xs3 ) ) ) ).

% last_drop
thf(fact_762_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_763_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_list_nat @ N @ nil_list_nat )
      = nil_list_nat ) ).

% drop_Nil
thf(fact_764_nth__via__drop,axiom,
    ! [N: nat,Xs3: list_nat,Y2: nat,Ys2: list_nat] :
      ( ( ( drop_nat @ N @ Xs3 )
        = ( cons_nat @ Y2 @ Ys2 ) )
     => ( ( nth_nat @ Xs3 @ N )
        = Y2 ) ) ).

% nth_via_drop
thf(fact_765_nth__via__drop,axiom,
    ! [N: nat,Xs3: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
      ( ( ( drop_list_nat @ N @ Xs3 )
        = ( cons_list_nat @ Y2 @ Ys2 ) )
     => ( ( nth_list_nat @ Xs3 @ N )
        = Y2 ) ) ).

% nth_via_drop
thf(fact_766_append__eq__conv__conj,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( ( append_list_nat @ Xs3 @ Ys2 )
        = Zs )
      = ( ( Xs3
          = ( take_list_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ Zs ) )
        & ( Ys2
          = ( drop_list_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_767_append__eq__conv__conj,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs3 @ Ys2 )
        = Zs )
      = ( ( Xs3
          = ( take_nat @ ( size_size_list_nat @ Xs3 ) @ Zs ) )
        & ( Ys2
          = ( drop_nat @ ( size_size_list_nat @ Xs3 ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_768_take__add,axiom,
    ! [I: nat,J: nat,Xs3: list_nat] :
      ( ( take_nat @ ( plus_plus_nat @ I @ J ) @ Xs3 )
      = ( append_nat @ ( take_nat @ I @ Xs3 ) @ ( take_nat @ J @ ( drop_nat @ I @ Xs3 ) ) ) ) ).

% take_add
thf(fact_769_take__add,axiom,
    ! [I: nat,J: nat,Xs3: list_list_nat] :
      ( ( take_list_nat @ ( plus_plus_nat @ I @ J ) @ Xs3 )
      = ( append_list_nat @ ( take_list_nat @ I @ Xs3 ) @ ( take_list_nat @ J @ ( drop_list_nat @ I @ Xs3 ) ) ) ) ).

% take_add
thf(fact_770_drop__Cons_H,axiom,
    ! [N: nat,X: nat,Xs3: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( drop_nat @ N @ ( cons_nat @ X @ Xs3 ) )
          = ( cons_nat @ X @ Xs3 ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( drop_nat @ N @ ( cons_nat @ X @ Xs3 ) )
          = ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs3 ) ) ) ) ).

% drop_Cons'
thf(fact_771_drop__Cons_H,axiom,
    ! [N: nat,X: list_nat,Xs3: list_list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( drop_list_nat @ N @ ( cons_list_nat @ X @ Xs3 ) )
          = ( cons_list_nat @ X @ Xs3 ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( drop_list_nat @ N @ ( cons_list_nat @ X @ Xs3 ) )
          = ( drop_list_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs3 ) ) ) ) ).

% drop_Cons'
thf(fact_772_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( cons_list_nat @ ( nth_list_nat @ Xs3 @ I ) @ ( drop_list_nat @ ( suc @ I ) @ Xs3 ) )
        = ( drop_list_nat @ I @ Xs3 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_773_Cons__nth__drop__Suc,axiom,
    ! [I: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs3 ) )
     => ( ( cons_nat @ ( nth_nat @ Xs3 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs3 ) )
        = ( drop_nat @ I @ Xs3 ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_774_take__hd__drop,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( append_list_nat @ ( take_list_nat @ N @ Xs3 ) @ ( cons_list_nat @ ( hd_list_nat @ ( drop_list_nat @ N @ Xs3 ) ) @ nil_list_nat ) )
        = ( take_list_nat @ ( suc @ N ) @ Xs3 ) ) ) ).

% take_hd_drop
thf(fact_775_take__hd__drop,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
     => ( ( append_nat @ ( take_nat @ N @ Xs3 ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs3 ) ) @ nil_nat ) )
        = ( take_nat @ ( suc @ N ) @ Xs3 ) ) ) ).

% take_hd_drop
thf(fact_776_append__one__prefix,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ Ys2 )
     => ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) )
       => ( prefix_list_nat @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ ( nth_list_nat @ Ys2 @ ( size_s3023201423986296836st_nat @ Xs3 ) ) @ nil_list_nat ) ) @ Ys2 ) ) ) ).

% append_one_prefix
thf(fact_777_append__one__prefix,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( prefix_nat @ Xs3 @ Ys2 )
     => ( ( ord_less_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) )
       => ( prefix_nat @ ( append_nat @ Xs3 @ ( cons_nat @ ( nth_nat @ Ys2 @ ( size_size_list_nat @ Xs3 ) ) @ nil_nat ) ) @ Ys2 ) ) ) ).

% append_one_prefix
thf(fact_778_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_779_prefix__order_Odual__order_Orefl,axiom,
    ! [A: list_nat] : ( prefix_nat @ A @ A ) ).

% prefix_order.dual_order.refl
thf(fact_780_prefix__order_Oorder__refl,axiom,
    ! [X: list_nat] : ( prefix_nat @ X @ X ) ).

% prefix_order.order_refl
thf(fact_781_Cons__prefix__Cons,axiom,
    ! [X: list_nat,Xs3: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
      ( ( prefix_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ ( cons_list_nat @ Y2 @ Ys2 ) )
      = ( ( X = Y2 )
        & ( prefix_list_nat @ Xs3 @ Ys2 ) ) ) ).

% Cons_prefix_Cons
thf(fact_782_Cons__prefix__Cons,axiom,
    ! [X: nat,Xs3: list_nat,Y2: nat,Ys2: list_nat] :
      ( ( prefix_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y2 @ Ys2 ) )
      = ( ( X = Y2 )
        & ( prefix_nat @ Xs3 @ Ys2 ) ) ) ).

% Cons_prefix_Cons
thf(fact_783_prefix__code_I1_J,axiom,
    ! [Xs3: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs3 ) ).

% prefix_code(1)
thf(fact_784_prefix__code_I1_J,axiom,
    ! [Xs3: list_nat] : ( prefix_nat @ nil_nat @ Xs3 ) ).

% prefix_code(1)
thf(fact_785_prefix__bot_Oextremum__unique,axiom,
    ! [A: list_list_nat] :
      ( ( prefix_list_nat @ A @ nil_list_nat )
      = ( A = nil_list_nat ) ) ).

% prefix_bot.extremum_unique
thf(fact_786_prefix__bot_Oextremum__unique,axiom,
    ! [A: list_nat] :
      ( ( prefix_nat @ A @ nil_nat )
      = ( A = nil_nat ) ) ).

% prefix_bot.extremum_unique
thf(fact_787_prefix__Nil,axiom,
    ! [Xs3: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ nil_list_nat )
      = ( Xs3 = nil_list_nat ) ) ).

% prefix_Nil
thf(fact_788_prefix__Nil,axiom,
    ! [Xs3: list_nat] :
      ( ( prefix_nat @ Xs3 @ nil_nat )
      = ( Xs3 = nil_nat ) ) ).

% prefix_Nil
thf(fact_789_same__prefix__prefix,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( prefix_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ ( append_list_nat @ Xs3 @ Zs ) )
      = ( prefix_list_nat @ Ys2 @ Zs ) ) ).

% same_prefix_prefix
thf(fact_790_same__prefix__prefix,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( prefix_nat @ ( append_nat @ Xs3 @ Ys2 ) @ ( append_nat @ Xs3 @ Zs ) )
      = ( prefix_nat @ Ys2 @ Zs ) ) ).

% same_prefix_prefix
thf(fact_791_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_792_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_793_hd__prefixes,axiom,
    ! [Xs3: list_nat] :
      ( ( hd_list_nat @ ( prefixes_nat @ Xs3 ) )
      = nil_nat ) ).

% hd_prefixes
thf(fact_794_hd__prefixes,axiom,
    ! [Xs3: list_list_nat] :
      ( ( hd_list_list_nat @ ( prefixes_list_nat @ Xs3 ) )
      = nil_list_nat ) ).

% hd_prefixes
thf(fact_795_hd__suffixes,axiom,
    ! [Xs3: list_nat] :
      ( ( hd_list_nat @ ( suffixes_nat @ Xs3 ) )
      = nil_nat ) ).

% hd_suffixes
thf(fact_796_hd__suffixes,axiom,
    ! [Xs3: list_list_nat] :
      ( ( hd_list_list_nat @ ( suffixes_list_nat @ Xs3 ) )
      = nil_list_nat ) ).

% hd_suffixes
thf(fact_797_same__prefix__nil,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( prefix_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ Xs3 )
      = ( Ys2 = nil_list_nat ) ) ).

% same_prefix_nil
thf(fact_798_same__prefix__nil,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( prefix_nat @ ( append_nat @ Xs3 @ Ys2 ) @ Xs3 )
      = ( Ys2 = nil_nat ) ) ).

% same_prefix_nil
thf(fact_799_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_800_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_801_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_802_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_803_hd__append2,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( Xs3 != nil_nat )
     => ( ( hd_nat @ ( append_nat @ Xs3 @ Ys2 ) )
        = ( hd_nat @ Xs3 ) ) ) ).

% hd_append2
thf(fact_804_hd__append2,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( hd_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
        = ( hd_list_nat @ Xs3 ) ) ) ).

% hd_append2
thf(fact_805_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_806_prefix__snoc,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Y2: list_nat] :
      ( ( prefix_list_nat @ Xs3 @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
      = ( ( Xs3
          = ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
        | ( prefix_list_nat @ Xs3 @ Ys2 ) ) ) ).

% prefix_snoc
thf(fact_807_prefix__snoc,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Y2: nat] :
      ( ( prefix_nat @ Xs3 @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
      = ( ( Xs3
          = ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
        | ( prefix_nat @ Xs3 @ Ys2 ) ) ) ).

% prefix_snoc
thf(fact_808_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_809_prefix__bot_Obot__least,axiom,
    ! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).

% prefix_bot.bot_least
thf(fact_810_prefix__bot_Obot__least,axiom,
    ! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).

% prefix_bot.bot_least
thf(fact_811_prefix__bot_Oextremum__uniqueI,axiom,
    ! [A: list_list_nat] :
      ( ( prefix_list_nat @ A @ nil_list_nat )
     => ( A = nil_list_nat ) ) ).

% prefix_bot.extremum_uniqueI
thf(fact_812_prefix__bot_Oextremum__uniqueI,axiom,
    ! [A: list_nat] :
      ( ( prefix_nat @ A @ nil_nat )
     => ( A = nil_nat ) ) ).

% prefix_bot.extremum_uniqueI
thf(fact_813_Nil__prefix,axiom,
    ! [Xs3: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs3 ) ).

% Nil_prefix
thf(fact_814_Nil__prefix,axiom,
    ! [Xs3: list_nat] : ( prefix_nat @ nil_nat @ Xs3 ) ).

% Nil_prefix
thf(fact_815_prefixE,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ Ys2 )
     => ~ ! [Zs2: list_list_nat] :
            ( Ys2
           != ( append_list_nat @ Xs3 @ Zs2 ) ) ) ).

% prefixE
thf(fact_816_prefixE,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( prefix_nat @ Xs3 @ Ys2 )
     => ~ ! [Zs2: list_nat] :
            ( Ys2
           != ( append_nat @ Xs3 @ Zs2 ) ) ) ).

% prefixE
thf(fact_817_prefixI,axiom,
    ! [Ys2: list_list_nat,Xs3: list_list_nat,Zs: list_list_nat] :
      ( ( Ys2
        = ( append_list_nat @ Xs3 @ Zs ) )
     => ( prefix_list_nat @ Xs3 @ Ys2 ) ) ).

% prefixI
thf(fact_818_prefixI,axiom,
    ! [Ys2: list_nat,Xs3: list_nat,Zs: list_nat] :
      ( ( Ys2
        = ( append_nat @ Xs3 @ Zs ) )
     => ( prefix_nat @ Xs3 @ Ys2 ) ) ).

% prefixI
thf(fact_819_prefix__def,axiom,
    ( prefix_list_nat
    = ( ^ [Xs2: list_list_nat,Ys3: list_list_nat] :
        ? [Zs3: list_list_nat] :
          ( Ys3
          = ( append_list_nat @ Xs2 @ Zs3 ) ) ) ) ).

% prefix_def
thf(fact_820_prefix__def,axiom,
    ( prefix_nat
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
        ? [Zs3: list_nat] :
          ( Ys3
          = ( append_nat @ Xs2 @ Zs3 ) ) ) ) ).

% prefix_def
thf(fact_821_prefix__append,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ ( append_list_nat @ Ys2 @ Zs ) )
      = ( ( prefix_list_nat @ Xs3 @ Ys2 )
        | ? [Us2: list_list_nat] :
            ( ( Xs3
              = ( append_list_nat @ Ys2 @ Us2 ) )
            & ( prefix_list_nat @ Us2 @ Zs ) ) ) ) ).

% prefix_append
thf(fact_822_prefix__append,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( prefix_nat @ Xs3 @ ( append_nat @ Ys2 @ Zs ) )
      = ( ( prefix_nat @ Xs3 @ Ys2 )
        | ? [Us2: list_nat] :
            ( ( Xs3
              = ( append_nat @ Ys2 @ Us2 ) )
            & ( prefix_nat @ Us2 @ Zs ) ) ) ) ).

% prefix_append
thf(fact_823_prefix__prefix,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ Ys2 )
     => ( prefix_list_nat @ Xs3 @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).

% prefix_prefix
thf(fact_824_prefix__prefix,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( prefix_nat @ Xs3 @ Ys2 )
     => ( prefix_nat @ Xs3 @ ( append_nat @ Ys2 @ Zs ) ) ) ).

% prefix_prefix
thf(fact_825_append__prefixD,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
      ( ( prefix_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) @ Zs )
     => ( prefix_list_nat @ Xs3 @ Zs ) ) ).

% append_prefixD
thf(fact_826_append__prefixD,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( prefix_nat @ ( append_nat @ Xs3 @ Ys2 ) @ Zs )
     => ( prefix_nat @ Xs3 @ Zs ) ) ).

% append_prefixD
thf(fact_827_list_Osel_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_828_list_Osel_I1_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] :
      ( ( hd_list_nat @ ( cons_list_nat @ X21 @ X22 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_829_filter__mono__prefix,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,P: list_nat > $o] :
      ( ( prefix_list_nat @ Xs3 @ Ys2 )
     => ( prefix_list_nat @ ( filter_list_nat @ P @ Xs3 ) @ ( filter_list_nat @ P @ Ys2 ) ) ) ).

% filter_mono_prefix
thf(fact_830_filter__mono__prefix,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,P: nat > $o] :
      ( ( prefix_nat @ Xs3 @ Ys2 )
     => ( prefix_nat @ ( filter_nat @ P @ Xs3 ) @ ( filter_nat @ P @ Ys2 ) ) ) ).

% filter_mono_prefix
thf(fact_831_prefix__same__cases,axiom,
    ! [Xs_1: list_nat,Ys2: list_nat,Xs_2: list_nat] :
      ( ( prefix_nat @ Xs_1 @ Ys2 )
     => ( ( prefix_nat @ Xs_2 @ Ys2 )
       => ( ( prefix_nat @ Xs_1 @ Xs_2 )
          | ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).

% prefix_same_cases
thf(fact_832_prefix__order_Odual__order_Oantisym,axiom,
    ! [B: list_nat,A: list_nat] :
      ( ( prefix_nat @ B @ A )
     => ( ( prefix_nat @ A @ B )
       => ( A = B ) ) ) ).

% prefix_order.dual_order.antisym
thf(fact_833_prefix__order_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [A3: list_nat,B3: list_nat] :
          ( ( prefix_nat @ B3 @ A3 )
          & ( prefix_nat @ A3 @ B3 ) ) ) ) ).

% prefix_order.dual_order.eq_iff
thf(fact_834_prefix__order_Odual__order_Otrans,axiom,
    ! [B: list_nat,A: list_nat,C: list_nat] :
      ( ( prefix_nat @ B @ A )
     => ( ( prefix_nat @ C @ B )
       => ( prefix_nat @ C @ A ) ) ) ).

% prefix_order.dual_order.trans
thf(fact_835_prefix__order_Oord__le__eq__trans,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( prefix_nat @ A @ B )
     => ( ( B = C )
       => ( prefix_nat @ A @ C ) ) ) ).

% prefix_order.ord_le_eq_trans
thf(fact_836_prefix__order_Oord__eq__le__trans,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( A = B )
     => ( ( prefix_nat @ B @ C )
       => ( prefix_nat @ A @ C ) ) ) ).

% prefix_order.ord_eq_le_trans
thf(fact_837_prefix__order_Oorder__antisym,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( prefix_nat @ X @ Y2 )
     => ( ( prefix_nat @ Y2 @ X )
       => ( X = Y2 ) ) ) ).

% prefix_order.order_antisym
thf(fact_838_prefix__order_Oorder__eq__iff,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [X3: list_nat,Y4: list_nat] :
          ( ( prefix_nat @ X3 @ Y4 )
          & ( prefix_nat @ Y4 @ X3 ) ) ) ) ).

% prefix_order.order_eq_iff
thf(fact_839_prefix__order_Oantisym__conv,axiom,
    ! [Y2: list_nat,X: list_nat] :
      ( ( prefix_nat @ Y2 @ X )
     => ( ( prefix_nat @ X @ Y2 )
        = ( X = Y2 ) ) ) ).

% prefix_order.antisym_conv
thf(fact_840_prefix__order_Oorder__trans,axiom,
    ! [X: list_nat,Y2: list_nat,Z4: list_nat] :
      ( ( prefix_nat @ X @ Y2 )
     => ( ( prefix_nat @ Y2 @ Z4 )
       => ( prefix_nat @ X @ Z4 ) ) ) ).

% prefix_order.order_trans
thf(fact_841_prefix__order_Oeq__refl,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( X = Y2 )
     => ( prefix_nat @ X @ Y2 ) ) ).

% prefix_order.eq_refl
thf(fact_842_prefix__order_Oantisym,axiom,
    ! [A: list_nat,B: list_nat] :
      ( ( prefix_nat @ A @ B )
     => ( ( prefix_nat @ B @ A )
       => ( A = B ) ) ) ).

% prefix_order.antisym
thf(fact_843_prefix__order_Oeq__iff,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [A3: list_nat,B3: list_nat] :
          ( ( prefix_nat @ A3 @ B3 )
          & ( prefix_nat @ B3 @ A3 ) ) ) ) ).

% prefix_order.eq_iff
thf(fact_844_prefix__order_Otrans,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( prefix_nat @ A @ B )
     => ( ( prefix_nat @ B @ C )
       => ( prefix_nat @ A @ C ) ) ) ).

% prefix_order.trans
thf(fact_845_hd__concat,axiom,
    ! [Xs3: list_list_list_nat] :
      ( ( Xs3 != nil_list_list_nat )
     => ( ( ( hd_list_list_nat @ Xs3 )
         != nil_list_nat )
       => ( ( hd_list_nat @ ( concat_list_nat @ Xs3 ) )
          = ( hd_list_nat @ ( hd_list_list_nat @ Xs3 ) ) ) ) ) ).

% hd_concat
thf(fact_846_hd__concat,axiom,
    ! [Xs3: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( ( hd_list_nat @ Xs3 )
         != nil_nat )
       => ( ( hd_nat @ ( concat_nat @ Xs3 ) )
          = ( hd_nat @ ( hd_list_nat @ Xs3 ) ) ) ) ) ).

% hd_concat
thf(fact_847_prefixeq__butlast,axiom,
    ! [Xs3: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs3 ) @ Xs3 ) ).

% prefixeq_butlast
thf(fact_848_take__is__prefix,axiom,
    ! [N: nat,Xs3: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs3 ) @ Xs3 ) ).

% take_is_prefix
thf(fact_849_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_850_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_851_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_852_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_853_not__prefix__induct,axiom,
    ! [Ps: list_list_nat,Ls: list_list_nat,P: list_list_nat > list_list_nat > $o] :
      ( ~ ( prefix_list_nat @ Ps @ Ls )
     => ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
       => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat] :
              ( ( X2 != Y3 )
             => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) )
         => ( ! [X2: list_nat,Xs: list_list_nat,Y3: list_nat,Ys: list_list_nat] :
                ( ( X2 = Y3 )
               => ( ~ ( prefix_list_nat @ Xs @ Ys )
                 => ( ( P @ Xs @ Ys )
                   => ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y3 @ Ys ) ) ) ) )
           => ( P @ Ps @ Ls ) ) ) ) ) ).

% not_prefix_induct
thf(fact_854_not__prefix__induct,axiom,
    ! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
      ( ~ ( prefix_nat @ Ps @ Ls )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
       => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
              ( ( X2 != Y3 )
             => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) )
         => ( ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
                ( ( X2 = Y3 )
               => ( ~ ( prefix_nat @ Xs @ Ys )
                 => ( ( P @ Xs @ Ys )
                   => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) )
           => ( P @ Ps @ Ls ) ) ) ) ) ).

% not_prefix_induct
thf(fact_855_not__prefix__cases,axiom,
    ! [Ps: list_list_nat,Ls: list_list_nat] :
      ( ~ ( prefix_list_nat @ Ps @ Ls )
     => ( ( ( Ps != nil_list_nat )
         => ( Ls != nil_list_nat ) )
       => ( ! [A4: list_nat,As: list_list_nat] :
              ( ( Ps
                = ( cons_list_nat @ A4 @ As ) )
             => ! [X2: list_nat,Xs: list_list_nat] :
                  ( ( Ls
                    = ( cons_list_nat @ X2 @ Xs ) )
                 => ( ( X2 = A4 )
                   => ( prefix_list_nat @ As @ Xs ) ) ) )
         => ~ ! [A4: list_nat] :
                ( ? [As: list_list_nat] :
                    ( Ps
                    = ( cons_list_nat @ A4 @ As ) )
               => ! [X2: list_nat] :
                    ( ? [Xs: list_list_nat] :
                        ( Ls
                        = ( cons_list_nat @ X2 @ Xs ) )
                   => ( X2 = A4 ) ) ) ) ) ) ).

% not_prefix_cases
thf(fact_856_not__prefix__cases,axiom,
    ! [Ps: list_nat,Ls: list_nat] :
      ( ~ ( prefix_nat @ Ps @ Ls )
     => ( ( ( Ps != nil_nat )
         => ( Ls != nil_nat ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Ps
                = ( cons_nat @ A4 @ As ) )
             => ! [X2: nat,Xs: list_nat] :
                  ( ( Ls
                    = ( cons_nat @ X2 @ Xs ) )
                 => ( ( X2 = A4 )
                   => ( prefix_nat @ As @ Xs ) ) ) )
         => ~ ! [A4: nat] :
                ( ? [As: list_nat] :
                    ( Ps
                    = ( cons_nat @ A4 @ As ) )
               => ! [X2: nat] :
                    ( ? [Xs: list_nat] :
                        ( Ls
                        = ( cons_nat @ X2 @ Xs ) )
                   => ( X2 = A4 ) ) ) ) ) ) ).

% not_prefix_cases
thf(fact_857_prefix__Cons,axiom,
    ! [Xs3: list_list_nat,Y2: list_nat,Ys2: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ ( cons_list_nat @ Y2 @ Ys2 ) )
      = ( ( Xs3 = nil_list_nat )
        | ? [Zs3: list_list_nat] :
            ( ( Xs3
              = ( cons_list_nat @ Y2 @ Zs3 ) )
            & ( prefix_list_nat @ Zs3 @ Ys2 ) ) ) ) ).

% prefix_Cons
thf(fact_858_prefix__Cons,axiom,
    ! [Xs3: list_nat,Y2: nat,Ys2: list_nat] :
      ( ( prefix_nat @ Xs3 @ ( cons_nat @ Y2 @ Ys2 ) )
      = ( ( Xs3 = nil_nat )
        | ? [Zs3: list_nat] :
            ( ( Xs3
              = ( cons_nat @ Y2 @ Zs3 ) )
            & ( prefix_nat @ Zs3 @ Ys2 ) ) ) ) ).

% prefix_Cons
thf(fact_859_prefix__code_I2_J,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ~ ( prefix_list_nat @ ( cons_list_nat @ X @ Xs3 ) @ nil_list_nat ) ).

% prefix_code(2)
thf(fact_860_prefix__code_I2_J,axiom,
    ! [X: nat,Xs3: list_nat] :
      ~ ( prefix_nat @ ( cons_nat @ X @ Xs3 ) @ nil_nat ) ).

% prefix_code(2)
thf(fact_861_longest__common__prefix,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
    ? [Ps2: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ( Xs3
        = ( append_nat @ Ps2 @ Xs4 ) )
      & ( Ys2
        = ( append_nat @ Ps2 @ Ys5 ) )
      & ( ( Xs4 = nil_nat )
        | ( Ys5 = nil_nat )
        | ( ( hd_nat @ Xs4 )
         != ( hd_nat @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_862_longest__common__prefix,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
    ? [Ps2: list_list_nat,Xs4: list_list_nat,Ys5: list_list_nat] :
      ( ( Xs3
        = ( append_list_nat @ Ps2 @ Xs4 ) )
      & ( Ys2
        = ( append_list_nat @ Ps2 @ Ys5 ) )
      & ( ( Xs4 = nil_list_nat )
        | ( Ys5 = nil_list_nat )
        | ( ( hd_list_nat @ Xs4 )
         != ( hd_list_nat @ Ys5 ) ) ) ) ).

% longest_common_prefix
thf(fact_863_hd__append,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( ( Xs3 = nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs3 @ Ys2 ) )
          = ( hd_nat @ Ys2 ) ) )
      & ( ( Xs3 != nil_nat )
       => ( ( hd_nat @ ( append_nat @ Xs3 @ Ys2 ) )
          = ( hd_nat @ Xs3 ) ) ) ) ).

% hd_append
thf(fact_864_hd__append,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ( Xs3 = nil_list_nat )
       => ( ( hd_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
          = ( hd_list_nat @ Ys2 ) ) )
      & ( ( Xs3 != nil_list_nat )
       => ( ( hd_list_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
          = ( hd_list_nat @ Xs3 ) ) ) ) ).

% hd_append
thf(fact_865_hd__Nil__eq__last,axiom,
    ( ( hd_nat @ nil_nat )
    = ( last_nat @ nil_nat ) ) ).

% hd_Nil_eq_last
thf(fact_866_hd__Nil__eq__last,axiom,
    ( ( hd_list_nat @ nil_list_nat )
    = ( last_list_nat @ nil_list_nat ) ) ).

% hd_Nil_eq_last
thf(fact_867_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_868_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_869_hd__conv__nth,axiom,
    ! [Xs3: list_nat] :
      ( ( Xs3 != nil_nat )
     => ( ( hd_nat @ Xs3 )
        = ( nth_nat @ Xs3 @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_870_hd__conv__nth,axiom,
    ! [Xs3: list_list_nat] :
      ( ( Xs3 != nil_list_nat )
     => ( ( hd_list_nat @ Xs3 )
        = ( nth_list_nat @ Xs3 @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_871_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_872_hd__drop__conv__nth,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( hd_list_nat @ ( drop_list_nat @ N @ Xs3 ) )
        = ( nth_list_nat @ Xs3 @ N ) ) ) ).

% hd_drop_conv_nth
thf(fact_873_hd__drop__conv__nth,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs3 ) )
     => ( ( hd_nat @ ( drop_nat @ N @ Xs3 ) )
        = ( nth_nat @ Xs3 @ N ) ) ) ).

% hd_drop_conv_nth
thf(fact_874_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_875_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_876_rgf__def,axiom,
    ( equiva3371634703666331078on_rgf
    = ( ^ [X3: list_nat] :
        ! [Ys3: list_nat,Y4: nat] :
          ( ( prefix_nat @ ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) @ X3 )
         => ( ord_less_eq_nat @ Y4 @ ( equiva5889994315859557365_limit @ Ys3 ) ) ) ) ) ).

% rgf_def
thf(fact_877_butlast__take,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( butlast_list_nat @ ( take_list_nat @ N @ Xs3 ) )
        = ( take_list_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs3 ) ) ) ).

% butlast_take
thf(fact_878_butlast__take,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs3 ) )
     => ( ( butlast_nat @ ( take_nat @ N @ Xs3 ) )
        = ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs3 ) ) ) ).

% butlast_take
thf(fact_879_length__product,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( size_s7664791237847770771st_nat @ ( produc2861266219255159431st_nat @ Xs3 @ Ys2 ) )
      = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_880_length__product,axiom,
    ! [Xs3: list_list_nat,Ys2: list_nat] :
      ( ( size_s6663376490332876291at_nat @ ( product_list_nat_nat @ Xs3 @ Ys2 ) )
      = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_881_length__product,axiom,
    ! [Xs3: list_nat,Ys2: list_list_nat] :
      ( ( size_s9035287501014481795st_nat @ ( product_nat_list_nat @ Xs3 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_882_length__product,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs3 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_883_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_884_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_885_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_886_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_887_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_888_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_889_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_890_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_891_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_892_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_893_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_894_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_895_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_896_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_897_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_898_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_899_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_900_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_901_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_902_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_903_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_904_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_905_take__all,axiom,
    ! [Xs3: list_list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N )
     => ( ( take_list_nat @ N @ Xs3 )
        = Xs3 ) ) ).

% take_all
thf(fact_906_take__all,axiom,
    ! [Xs3: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N )
     => ( ( take_nat @ N @ Xs3 )
        = Xs3 ) ) ).

% take_all
thf(fact_907_take__all__iff,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ( take_list_nat @ N @ Xs3 )
        = Xs3 )
      = ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ).

% take_all_iff
thf(fact_908_take__all__iff,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ( take_nat @ N @ Xs3 )
        = Xs3 )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ).

% take_all_iff
thf(fact_909_list__update__beyond,axiom,
    ! [Xs3: list_list_nat,I: nat,X: list_nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ I )
     => ( ( list_update_list_nat @ Xs3 @ I @ X )
        = Xs3 ) ) ).

% list_update_beyond
thf(fact_910_list__update__beyond,axiom,
    ! [Xs3: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ I )
     => ( ( list_update_nat @ Xs3 @ I @ X )
        = Xs3 ) ) ).

% list_update_beyond
thf(fact_911_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_912_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_913_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_914_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_915_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_916_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_917_drop__eq__Nil2,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( nil_list_nat
        = ( drop_list_nat @ N @ Xs3 ) )
      = ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_918_drop__eq__Nil2,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( nil_nat
        = ( drop_nat @ N @ Xs3 ) )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_919_drop__eq__Nil,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ( drop_list_nat @ N @ Xs3 )
        = nil_list_nat )
      = ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_920_drop__eq__Nil,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ( drop_nat @ N @ Xs3 )
        = nil_nat )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ).

% drop_eq_Nil
thf(fact_921_drop__all,axiom,
    ! [Xs3: list_list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N )
     => ( ( drop_list_nat @ N @ Xs3 )
        = nil_list_nat ) ) ).

% drop_all
thf(fact_922_drop__all,axiom,
    ! [Xs3: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N )
     => ( ( drop_nat @ N @ Xs3 )
        = nil_nat ) ) ).

% drop_all
thf(fact_923_rotate1__length01,axiom,
    ! [Xs3: list_list_nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ one_one_nat )
     => ( ( rotate1_list_nat @ Xs3 )
        = Xs3 ) ) ).

% rotate1_length01
thf(fact_924_rotate1__length01,axiom,
    ! [Xs3: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs3 )
        = Xs3 ) ) ).

% rotate1_length01
thf(fact_925_nth__drop,axiom,
    ! [N: nat,Xs3: list_list_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( nth_list_nat @ ( drop_list_nat @ N @ Xs3 ) @ I )
        = ( nth_list_nat @ Xs3 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_926_nth__drop,axiom,
    ! [N: nat,Xs3: list_nat,I: nat] :
      ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs3 ) )
     => ( ( nth_nat @ ( drop_nat @ N @ Xs3 ) @ I )
        = ( nth_nat @ Xs3 @ ( plus_plus_nat @ N @ I ) ) ) ) ).

% nth_drop
thf(fact_927_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_928_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_929_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_930_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_931_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_932_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_933_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_934_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_935_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_936_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_937_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_938_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_939_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_940_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_941_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_942_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_943_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_944_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_945_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_946_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_947_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_948_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_949_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_950_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_951_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_952_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_max_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% max.absorb_iff2
thf(fact_953_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_max_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_954_le__max__iff__disj,axiom,
    ! [Z4: nat,X: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ Z4 @ ( ord_max_nat @ X @ Y2 ) )
      = ( ( ord_less_eq_nat @ Z4 @ X )
        | ( ord_less_eq_nat @ Z4 @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_955_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_956_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_957_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( A3
          = ( ord_max_nat @ A3 @ B3 ) ) ) ) ).

% max.order_iff
thf(fact_958_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_959_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_960_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_961_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_962_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_963_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_964_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_965_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_966_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_967_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_968_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_969_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_970_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X2: nat] :
            ( ( P @ X2 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_971_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_972_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_973_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_974_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_nat @ M2 @ N2 )
          | ( M2 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_975_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_976_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M2 @ N2 )
          & ( M2 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_977_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_978_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_979_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_980_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_981_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_982_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_983_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_984_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M3: nat] :
          ( M6
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_985_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_986_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_987_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_988_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
             => ( P @ M4 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_989_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_990_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X2: nat] : ( R2 @ X2 @ X2 )
       => ( ! [X2: nat,Y3: nat,Z2: nat] :
              ( ( R2 @ X2 @ Y3 )
             => ( ( R2 @ Y3 @ Z2 )
               => ( R2 @ X2 @ Z2 ) ) )
         => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
           => ( R2 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_991_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_992_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_993_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_994_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C2: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_995_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_996_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_997_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_998_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_999_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1000_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1001_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1002_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1003_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1004_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_1005_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1006_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1007_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1008_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_1009_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1010_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1011_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1012_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1013_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1014_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1015_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1016_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1017_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1018_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1019_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1020_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ( ( plus_plus_nat @ X @ Y2 )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1021_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y2 )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1022_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1023_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1024_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_1025_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1026_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1027_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1028_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1029_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1030_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1031_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_1032_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1033_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1034_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1035_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1036_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1037_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1038_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1039_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1040_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1041_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1042_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1043_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1044_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1045_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1046_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1047_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1048_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1049_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1050_impossible__Cons,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat,X: list_nat] :
      ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) )
     => ( Xs3
       != ( cons_list_nat @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_1051_impossible__Cons,axiom,
    ! [Xs3: list_nat,Ys2: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) )
     => ( Xs3
       != ( cons_nat @ X @ Ys2 ) ) ) ).

% impossible_Cons
thf(fact_1052_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1053_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1054_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1055_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_1056_prefix__order_Olift__Suc__mono__le,axiom,
    ! [F: nat > list_nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( prefix_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( prefix_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% prefix_order.lift_Suc_mono_le
thf(fact_1057_prefix__order_Olift__Suc__antimono__le,axiom,
    ! [F: nat > list_nat,N: nat,N4: nat] :
      ( ! [N3: nat] : ( prefix_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( prefix_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% prefix_order.lift_Suc_antimono_le
thf(fact_1058_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1059_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1060_prefix__length__prefix,axiom,
    ! [Ps: list_list_nat,Xs3: list_list_nat,Qs: list_list_nat] :
      ( ( prefix_list_nat @ Ps @ Xs3 )
     => ( ( prefix_list_nat @ Qs @ Xs3 )
       => ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Ps ) @ ( size_s3023201423986296836st_nat @ Qs ) )
         => ( prefix_list_nat @ Ps @ Qs ) ) ) ) ).

% prefix_length_prefix
thf(fact_1061_prefix__length__prefix,axiom,
    ! [Ps: list_nat,Xs3: list_nat,Qs: list_nat] :
      ( ( prefix_nat @ Ps @ Xs3 )
     => ( ( prefix_nat @ Qs @ Xs3 )
       => ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
         => ( prefix_nat @ Ps @ Qs ) ) ) ) ).

% prefix_length_prefix
thf(fact_1062_prefix__length__le,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( prefix_list_nat @ Xs3 @ Ys2 )
     => ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).

% prefix_length_le
thf(fact_1063_prefix__length__le,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( prefix_nat @ Xs3 @ Ys2 )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% prefix_length_le
thf(fact_1064_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1065_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1066_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1067_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1068_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1069_length__filter__le,axiom,
    ! [P: list_nat > $o,Xs3: list_list_nat] : ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ ( filter_list_nat @ P @ Xs3 ) ) @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ).

% length_filter_le
thf(fact_1070_length__filter__le,axiom,
    ! [P: nat > $o,Xs3: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs3 ) ) @ ( size_size_list_nat @ Xs3 ) ) ).

% length_filter_le
thf(fact_1071_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_1072_count__le__length,axiom,
    ! [Xs3: list_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs3 @ X ) @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ).

% count_le_length
thf(fact_1073_count__le__length,axiom,
    ! [Xs3: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs3 @ X ) @ ( size_size_list_nat @ Xs3 ) ) ).

% count_le_length
thf(fact_1074_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_1075_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1076_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_1077_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1078_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_1079_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_1080_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1081_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1082_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_1083_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1084_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_1085_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1086_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1087_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1088_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_1089_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1090_Suc__le__length__iff,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s3023201423986296836st_nat @ Xs3 ) )
      = ( ? [X3: list_nat,Ys3: list_list_nat] :
            ( ( Xs3
              = ( cons_list_nat @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_1091_Suc__le__length__iff,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs3 ) )
      = ( ? [X3: nat,Ys3: list_nat] :
            ( ( Xs3
              = ( cons_nat @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_1092_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1093_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1094_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_1095_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_1096_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_1097_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_1098_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_1099_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_1100_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_1101_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_1102_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_1103_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_1104_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_1105_nth__take__lemma,axiom,
    ! [K: nat,Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_s3023201423986296836st_nat @ Xs3 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s3023201423986296836st_nat @ Ys2 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ K )
             => ( ( nth_list_nat @ Xs3 @ I3 )
                = ( nth_list_nat @ Ys2 @ I3 ) ) )
         => ( ( take_list_nat @ K @ Xs3 )
            = ( take_list_nat @ K @ Ys2 ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1106_nth__take__lemma,axiom,
    ! [K: nat,Xs3: list_nat,Ys2: list_nat] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs3 ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys2 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ K )
             => ( ( nth_nat @ Xs3 @ I3 )
                = ( nth_nat @ Ys2 @ I3 ) ) )
         => ( ( take_nat @ K @ Xs3 )
            = ( take_nat @ K @ Ys2 ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1107_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_list_nat,Xs_2: list_list_nat,Ys_1: list_list_nat,Ys_2: list_list_nat] :
      ( ( ( append_list_nat @ Xs_1 @ Xs_2 )
        = ( append_list_nat @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs_1 ) @ ( size_s3023201423986296836st_nat @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_list_nat @ ( size_s3023201423986296836st_nat @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append_list_nat @ ( drop_list_nat @ ( size_s3023201423986296836st_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs_1 ) @ ( size_s3023201423986296836st_nat @ Ys_1 ) )
         => ( ( ( take_list_nat @ ( size_s3023201423986296836st_nat @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append_list_nat @ ( drop_list_nat @ ( size_s3023201423986296836st_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_1108_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
      ( ( ( append_nat @ Xs_1 @ Xs_2 )
        = ( append_nat @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
         => ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_1109_nth__stirling__row,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( nth_nat @ ( stirling_row @ N ) @ K )
        = ( stirling2 @ N @ K ) ) ) ).

% nth_stirling_row
thf(fact_1110_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1111_distinct__adj__append__iff,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( distinct_adj_nat @ ( append_nat @ Xs3 @ Ys2 ) )
      = ( ( distinct_adj_nat @ Xs3 )
        & ( distinct_adj_nat @ Ys2 )
        & ( ( Xs3 = nil_nat )
          | ( Ys2 = nil_nat )
          | ( ( last_nat @ Xs3 )
           != ( hd_nat @ Ys2 ) ) ) ) ) ).

% distinct_adj_append_iff
thf(fact_1112_distinct__adj__append__iff,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( distin876741697294417026st_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
      = ( ( distin876741697294417026st_nat @ Xs3 )
        & ( distin876741697294417026st_nat @ Ys2 )
        & ( ( Xs3 = nil_list_nat )
          | ( Ys2 = nil_list_nat )
          | ( ( last_list_nat @ Xs3 )
           != ( hd_list_nat @ Ys2 ) ) ) ) ) ).

% distinct_adj_append_iff
thf(fact_1113_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1114_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1115_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_1116_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1117_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1118_distinct__adj__Cons__Cons,axiom,
    ! [X: nat,Y2: nat,Xs3: list_nat] :
      ( ( distinct_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs3 ) ) )
      = ( ( X != Y2 )
        & ( distinct_adj_nat @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ).

% distinct_adj_Cons_Cons
thf(fact_1119_distinct__adj__Cons__Cons,axiom,
    ! [X: list_nat,Y2: list_nat,Xs3: list_list_nat] :
      ( ( distin876741697294417026st_nat @ ( cons_list_nat @ X @ ( cons_list_nat @ Y2 @ Xs3 ) ) )
      = ( ( X != Y2 )
        & ( distin876741697294417026st_nat @ ( cons_list_nat @ Y2 @ Xs3 ) ) ) ) ).

% distinct_adj_Cons_Cons
thf(fact_1120_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_1121_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1122_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1123_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
        = ( ord_less_nat @ X @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_1124_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1125_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1126_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1127_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y2 ) )
        = ( ord_less_eq_nat @ X @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_1128_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_1129_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_1130_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_1131_distinct__adj__Nil,axiom,
    distinct_adj_nat @ nil_nat ).

% distinct_adj_Nil
thf(fact_1132_distinct__adj__Nil,axiom,
    distin876741697294417026st_nat @ nil_list_nat ).

% distinct_adj_Nil
thf(fact_1133_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_1134_distinct__adj__singleton,axiom,
    ! [X: nat] : ( distinct_adj_nat @ ( cons_nat @ X @ nil_nat ) ) ).

% distinct_adj_singleton
thf(fact_1135_distinct__adj__singleton,axiom,
    ! [X: list_nat] : ( distin876741697294417026st_nat @ ( cons_list_nat @ X @ nil_list_nat ) ) ).

% distinct_adj_singleton
thf(fact_1136_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_1137_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_1138_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_1139_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_1140_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1141_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1142_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_1143_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1144_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1145_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1146_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1147_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_1148_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_1149_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_1150_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_1151_distinct__adj__ConsD,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( distinct_adj_nat @ ( cons_nat @ X @ Xs3 ) )
     => ( distinct_adj_nat @ Xs3 ) ) ).

% distinct_adj_ConsD
thf(fact_1152_distinct__adj__ConsD,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( distin876741697294417026st_nat @ ( cons_list_nat @ X @ Xs3 ) )
     => ( distin876741697294417026st_nat @ Xs3 ) ) ).

% distinct_adj_ConsD
thf(fact_1153_distinct__adj__appendD2,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( distinct_adj_nat @ ( append_nat @ Xs3 @ Ys2 ) )
     => ( distinct_adj_nat @ Ys2 ) ) ).

% distinct_adj_appendD2
thf(fact_1154_distinct__adj__appendD2,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( distin876741697294417026st_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
     => ( distin876741697294417026st_nat @ Ys2 ) ) ).

% distinct_adj_appendD2
thf(fact_1155_distinct__adj__appendD1,axiom,
    ! [Xs3: list_nat,Ys2: list_nat] :
      ( ( distinct_adj_nat @ ( append_nat @ Xs3 @ Ys2 ) )
     => ( distinct_adj_nat @ Xs3 ) ) ).

% distinct_adj_appendD1
thf(fact_1156_distinct__adj__appendD1,axiom,
    ! [Xs3: list_list_nat,Ys2: list_list_nat] :
      ( ( distin876741697294417026st_nat @ ( append_list_nat @ Xs3 @ Ys2 ) )
     => ( distin876741697294417026st_nat @ Xs3 ) ) ).

% distinct_adj_appendD1
thf(fact_1157_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_1158_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_1159_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_1160_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_1161_left__right__inverse__power,axiom,
    ! [X: nat,Y2: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y2 )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y2 @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_1162_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_1163_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_1164_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_1165_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_1166_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_1167_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1168_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1169_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1170_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1171_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_1172_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_1173_distinct__adj__Cons,axiom,
    ! [X: nat,Xs3: list_nat] :
      ( ( distinct_adj_nat @ ( cons_nat @ X @ Xs3 ) )
      = ( ( Xs3 = nil_nat )
        | ( ( X
           != ( hd_nat @ Xs3 ) )
          & ( distinct_adj_nat @ Xs3 ) ) ) ) ).

% distinct_adj_Cons
thf(fact_1174_distinct__adj__Cons,axiom,
    ! [X: list_nat,Xs3: list_list_nat] :
      ( ( distin876741697294417026st_nat @ ( cons_list_nat @ X @ Xs3 ) )
      = ( ( Xs3 = nil_list_nat )
        | ( ( X
           != ( hd_list_nat @ Xs3 ) )
          & ( distin876741697294417026st_nat @ Xs3 ) ) ) ) ).

% distinct_adj_Cons
thf(fact_1175_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_1176_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P2: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1177_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_1178_length__n__lists,axiom,
    ! [N: nat,Xs3: list_list_nat] :
      ( ( size_s6248950052170075156st_nat @ ( n_lists_list_nat @ N @ Xs3 ) )
      = ( power_power_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ).

% length_n_lists
thf(fact_1179_length__n__lists,axiom,
    ! [N: nat,Xs3: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N @ Xs3 ) )
      = ( power_power_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ).

% length_n_lists
thf(fact_1180_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_1181_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_1182_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_1183_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_1184_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1185_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_1186_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_1187_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_1188_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_1189_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_1190_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_1191_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_1192_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_1193_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_1194_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_1195_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] :
      ( ( if_nat @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] :
      ( ( if_nat @ $true @ X @ Y2 )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( size_s3023201423986296836st_nat
      @ ( filter_list_nat
        @ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
          @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) )
        @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ) ) ) )
    = ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ) ) @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ) ) ).

%------------------------------------------------------------------------------