TPTP Problem File: SLH0891^1.p

%------------------------------------------------------------------------------
% 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    : Knights_Tour/0000_KnightsTour/prob_01040_040194__5896242_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1834 ( 573 unt; 553 typ;   0 def)
%            Number of atoms       : 3571 (1699 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 13355 ( 625   ~;  93   |; 270   &;10692   @)
%                                         (   0 <=>;1675  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   7 avg)
%            Number of types       :  110 ( 109 usr)
%            Number of type conns  : 1446 (1446   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  447 ( 444 usr;  26 con; 0-3 aty)
%            Number of variables   : 4317 ( 140   ^;3951   !; 226   ?;4317   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:00:55.797
%------------------------------------------------------------------------------
% Could-be-implicit typings (109)
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    product_prod_nat_int: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    list_int_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
    list_list_int: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    list_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

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

thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

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

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Num__Onum,type,
    num: $tType ).

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

thf(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (444)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Int__Oint,type,
    bNF_Gr1870224194279859149ft_int: set_list_int > int > set_list_int ).

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_OShift_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_Gr9175161896313899224nt_int: set_li3855193629254082847nt_int > product_prod_int_int > set_li3855193629254082847nt_int ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Int__Oint,type,
    bNF_Gr6350390219475566417cc_int: set_list_int > list_int > set_int ).

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

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_Gr2185362013987207252nt_int: set_li3855193629254082847nt_int > list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
    abs_abs_int: int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

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__Int__Oint,type,
    times_times_int: int > int > int ).

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_KnightsTour_Omirror1,type,
    mirror1: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_KnightsTour_Omirror1__aux,type,
    mirror1_aux: int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_KnightsTour_Omirror1__aux__rel,type,
    mirror1_aux_rel: produc661532565036771336nt_int > produc661532565036771336nt_int > $o ).

thf(sy_c_KnightsTour_Omirror1__square,type,
    mirror1_square: int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_KnightsTour_Omirror2,type,
    mirror2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_KnightsTour_Omirror2__aux,type,
    mirror2_aux: int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_KnightsTour_Omirror2__aux__rel,type,
    mirror2_aux_rel: produc661532565036771336nt_int > produc661532565036771336nt_int > $o ).

thf(sy_c_KnightsTour_Omirror2__square,type,
    mirror2_square: int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: list_int > list_int > list_int ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Int__Oint_J,type,
    append_list_int: list_list_int > list_list_int > list_list_int ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    append1237393811882493146nt_int: list_l1670014477004246597nt_int > list_l1670014477004246597nt_int > list_l1670014477004246597nt_int ).

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__Int__Oint_Mt__Int__Oint_J,type,
    append7030698103840186580nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    append6084095472372872615nt_int: list_P8440714079264627474nt_int > list_P8440714079264627474nt_int > list_P8440714079264627474nt_int ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
    append2855626057535414585nt_int: list_P3467955165070007716nt_int > list_P3467955165070007716nt_int > list_P3467955165070007716nt_int ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    append9111546825139864322nt_int: list_P2336717926344734829nt_int > list_P2336717926344734829nt_int > list_P2336717926344734829nt_int ).

thf(sy_c_List_Obind_001t__Int__Oint_001t__Int__Oint,type,
    bind_int_int: list_int > ( int > list_int ) > list_int ).

thf(sy_c_List_Obind_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bind_i1738094907376718532nt_int: list_int > ( int > list_P5707943133018811711nt_int ) > list_P5707943133018811711nt_int ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    bind_P1074787401898827934nt_int: list_P5707943133018811711nt_int > ( product_prod_int_int > list_int ) > list_int ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bind_P8056059132564497095nt_int: list_P5707943133018811711nt_int > ( product_prod_int_int > list_P5707943133018811711nt_int ) > list_P5707943133018811711nt_int ).

thf(sy_c_List_Obutlast_001t__Int__Oint,type,
    butlast_int: list_int > list_int ).

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

thf(sy_c_List_Obutlast_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    butlas2390654596765537291nt_int: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Ocan__select_001t__Int__Oint,type,
    can_select_int: ( int > $o ) > set_int > $o ).

thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
    can_select_nat: ( nat > $o ) > set_nat > $o ).

thf(sy_c_List_Oconcat_001t__Int__Oint,type,
    concat_int: list_list_int > list_int ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    concat4512918505337516154nt_int: list_l1670014477004246597nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Ocoset_001t__Int__Oint,type,
    coset_int: list_int > set_int ).

thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
    coset_nat: list_nat > set_nat ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    coset_3631019413092825772nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
    gen_length_int: nat > list_int > nat ).

thf(sy_c_List_Ogen__length_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    gen_le8428774395332151372nt_int: nat > list_P5707943133018811711nt_int > nat ).

thf(sy_c_List_Oinsert_001t__Int__Oint,type,
    insert_int: int > list_int > list_int ).

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

thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    insert5765537519290168021nt_int: product_prod_int_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Olast_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    last_i813233280922151042e_term: list_i8448526496819171953e_term > int > option6357759511663192854e_term ).

thf(sy_c_List_Olast_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    last_P8974135382477003148e_term: list_P1743416141875011707e_term > produc8551481072490612790e_term > option6357759511663192854e_term ).

thf(sy_c_List_Olast_001t__Int__Oint,type,
    last_int: list_int > int ).

thf(sy_c_List_Olast_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    last_l5818330359162608606nt_int: list_l1670014477004246597nt_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
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thf(sy_c_List_Olast_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Olistrel_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_List_Omap__filter_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Omap__tailrec__rev_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
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thf(sy_c_List_Omap__tailrec__rev__rel_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Omap__tailrec__rev__rel_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Omap__tailrec__rev__rel_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
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thf(sy_c_List_Omap__tailrec__rev__rel_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Omaps_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Omaps_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
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thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Omeasures_001t__Int__Oint,type,
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thf(sy_c_List_Omember_001t__Int__Oint,type,
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thf(sy_c_List_Omember_001t__Nat__Onat,type,
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thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_On__lists_001t__Int__Oint,type,
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thf(sy_c_List_On__lists_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Onth_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_List_Onull_001t__Int__Oint,type,
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thf(sy_c_List_Onull_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Orotate1_001t__Int__Oint,type,
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thf(sy_c_List_Orotate1_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oshuffles__rel_001t__Int__Oint,type,
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thf(sy_c_List_Oshuffles__rel_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Osplice_001t__Int__Oint,type,
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thf(sy_c_List_Osplice__rel_001t__Int__Oint,type,
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thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
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thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Otake_001t__Int__Oint,type,
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thf(sy_c_List_Otake_001t__Nat__Onat,type,
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thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Otranspose_001t__Int__Oint,type,
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thf(sy_c_List_Otranspose_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Otranspose__rel_001t__Int__Oint,type,
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thf(sy_c_List_Otranspose__rel_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Oupto,type,
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thf(sy_c_List_Oupto__aux,type,
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thf(sy_c_List_Oupto__rel,type,
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thf(sy_c_List_Ozip_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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    member4957952664650131097nt_int: produc759720530913461378nt_int > set_Pr5025984505184017080nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member6768764179952475034nt_int: produc6969430300573543555nt_int > set_Pr8667594812344408889nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member1869596255364795664nt_int: produc8664907604286745209nt_int > set_Pr2692125258033744815nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    member4298863958545693205nt_int: produc6531732081377312382nt_int > set_Pr6887017113042850356nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member3512312383758291599nt_int: produc1694388182729252728nt_int > set_Pr7582088413506099118nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member4311088582351766364et_int: produc9097395316738159027et_int > set_Pr2163364003801676051et_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Int__Oint_J,type,
    member2436464452258586801nt_int: produc9058040638841796122nt_int > set_Pr8964431684210982608nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member1281411469525323969st_int: produc8686859305351060906st_int > set_Pr6368618098895854816st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member1390679175989562640nt_int: produc1089560213143673063nt_int > set_Pr6304946757569631943nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member1920110580185997301nt_int: produc6945250483304103390nt_int > set_Pr6123067485828015636nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member1292241183792264892et_int: produc9133624956312949779et_int > set_Pr4810089274464741491et_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member4085533954029916580nt_int: produc2501202720802129403nt_int > set_Pr8634505666381077339nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    member6582457606847315088nt_int: produc2432570611225516007nt_int > set_Pr4708930517165415495nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    member3093737243930835296e_term: produc3832618445565238089e_term > set_Pr9020418513037773311e_term > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    member7154775377335690986e_term: produc1821397396195235155e_term > set_Pr7309386651551657097e_term > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
    member1729483249812673067nt_int: produc6754623152930779156nt_int > set_Pr53225590989397322nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J,type,
    member5907334269321869775nt_nat: produc1709102135585200056nt_nat > set_Pr2543308735921124078nt_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J,type,
    member211307870107589146nt_int: produc2223382658075010929nt_int > set_Pr5463530407906606289nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    member5126324565730479632et_int: produc212874708166070503et_int > set_Pr7486745082216227783et_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
    member1837137653922122684nt_nat: produc455149389588031763nt_nat > set_Pr6819878462438571379nt_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

thf(sy_v_i,type,
    i: int ).

thf(sy_v_j,type,
    j: int ).

thf(sy_v_m,type,
    m: int ).

thf(sy_v_ps,type,
    ps: list_P5707943133018811711nt_int ).

% Relevant facts (1273)
thf(fact_0_assms_I2_J,axiom,
    ( ( last_P3305686521732843992nt_int @ ps )
    = ( product_Pair_int_int @ i @ j ) ) ).

% assms(2)
thf(fact_1_assms_I1_J,axiom,
    ps != nil_Pr2300489316682597567nt_int ).

% assms(1)
thf(fact_2_prod_Oinject,axiom,
    ! [X1: nat,X2: set_int,Y1: nat,Y2: set_int] :
      ( ( ( produc29655638201817675et_int @ X1 @ X2 )
        = ( produc29655638201817675et_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3_prod_Oinject,axiom,
    ! [X1: int,X2: list_P5707943133018811711nt_int,Y1: int,Y2: list_P5707943133018811711nt_int] :
      ( ( ( produc8814303788642274490nt_int @ X1 @ X2 )
        = ( produc8814303788642274490nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_4_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( ( product_Pair_int_int @ X1 @ X2 )
        = ( product_Pair_int_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_5_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_6_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_7_old_Oprod_Oinject,axiom,
    ! [A: nat,B: set_int,A2: nat,B2: set_int] :
      ( ( ( produc29655638201817675et_int @ A @ B )
        = ( produc29655638201817675et_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
    ! [A: int,B: list_P5707943133018811711nt_int,A2: int,B2: list_P5707943133018811711nt_int] :
      ( ( ( produc8814303788642274490nt_int @ A @ B )
        = ( produc8814303788642274490nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_12_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D: int,I: int] :
          ( X
         != ( product_Pair_int_int @ D @ I ) ) ).

% small_lazy'.cases
thf(fact_13_diff__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ A @ C ) )
     => ( B = C ) ) ).

% diff_left_imp_eq
thf(fact_14_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_15_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_16_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9133624956312949779et_int] :
      ~ ! [A3: nat,B3: set_int] :
          ( Y
         != ( produc29655638201817675et_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_17_old_Oprod_Oexhaust,axiom,
    ! [Y: produc661532565036771336nt_int] :
      ~ ! [A3: int,B3: list_P5707943133018811711nt_int] :
          ( Y
         != ( produc8814303788642274490nt_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_18_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_19_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_20_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_21_surj__pair,axiom,
    ! [P: produc9133624956312949779et_int] :
    ? [X3: nat,Y3: set_int] :
      ( P
      = ( produc29655638201817675et_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_22_surj__pair,axiom,
    ! [P: produc661532565036771336nt_int] :
    ? [X3: int,Y3: list_P5707943133018811711nt_int] :
      ( P
      = ( produc8814303788642274490nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_23_surj__pair,axiom,
    ! [P: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_24_surj__pair,axiom,
    ! [P: produc2285326912895808259nt_int] :
    ? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P
      = ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_25_surj__pair,axiom,
    ! [P: produc7773217078559923341nt_int] :
    ? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
      ( P
      = ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_26_prod__cases,axiom,
    ! [P2: produc9133624956312949779et_int > $o,P: produc9133624956312949779et_int] :
      ( ! [A3: nat,B3: set_int] : ( P2 @ ( produc29655638201817675et_int @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_27_prod__cases,axiom,
    ! [P2: produc661532565036771336nt_int > $o,P: produc661532565036771336nt_int] :
      ( ! [A3: int,B3: list_P5707943133018811711nt_int] : ( P2 @ ( produc8814303788642274490nt_int @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_28_prod__cases,axiom,
    ! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_29_prod__cases,axiom,
    ! [P2: produc2285326912895808259nt_int > $o,P: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P2 @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_30_prod__cases,axiom,
    ! [P2: produc7773217078559923341nt_int > $o,P: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P2 @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_31_step__checker_Ocases,axiom,
    ! [X: produc1219242969750017639nt_int] :
      ~ ! [I: int,J: int,I2: int,J2: int] :
          ( X
         != ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ I @ J ) @ ( product_Pair_int_int @ I2 @ J2 ) ) ) ).

% step_checker.cases
thf(fact_32_mirror2__aux__nil,axiom,
    ! [Ps: list_P5707943133018811711nt_int,M: int] :
      ( ( Ps = nil_Pr2300489316682597567nt_int )
      = ( ( mirror2_aux @ M @ Ps )
        = nil_Pr2300489316682597567nt_int ) ) ).

% mirror2_aux_nil
thf(fact_33_mirror2__aux_Osimps_I1_J,axiom,
    ! [M: int] :
      ( ( mirror2_aux @ M @ nil_Pr2300489316682597567nt_int )
      = nil_Pr2300489316682597567nt_int ) ).

% mirror2_aux.simps(1)
thf(fact_34_prod__induct3,axiom,
    ! [P2: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] : ( P2 @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_35_prod__induct3,axiom,
    ! [P2: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] : ( P2 @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_36_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_37_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_38_Pair__inject,axiom,
    ! [A: nat,B: set_int,A2: nat,B2: set_int] :
      ( ( ( produc29655638201817675et_int @ A @ B )
        = ( produc29655638201817675et_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_39_Pair__inject,axiom,
    ! [A: int,B: list_P5707943133018811711nt_int,A2: int,B2: list_P5707943133018811711nt_int] :
      ( ( ( produc8814303788642274490nt_int @ A @ B )
        = ( produc8814303788642274490nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_40_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_41_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_42_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_43_last__mirror1__aux,axiom,
    ! [Ps: list_P5707943133018811711nt_int,I3: int,J3: int,N: int] :
      ( ( Ps != nil_Pr2300489316682597567nt_int )
     => ( ( ( last_P3305686521732843992nt_int @ Ps )
          = ( product_Pair_int_int @ I3 @ J3 ) )
       => ( ( last_P3305686521732843992nt_int @ ( mirror1_aux @ N @ Ps ) )
          = ( product_Pair_int_int @ ( minus_minus_int @ N @ I3 ) @ J3 ) ) ) ) ).

% last_mirror1_aux
thf(fact_44_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F @ ( product_Pair_int_int @ D @ I ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_45_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F: int > option6357759511663192854e_term,D: int,I: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F @ ( product_Pair_int_int @ D @ I ) ) ) ).

% exhaustive_int'.cases
thf(fact_46_mirror2__nil,axiom,
    ! [Ps: list_P5707943133018811711nt_int] :
      ( ( Ps = nil_Pr2300489316682597567nt_int )
      = ( ( mirror2 @ Ps )
        = nil_Pr2300489316682597567nt_int ) ) ).

% mirror2_nil
thf(fact_47_mirror1__nil,axiom,
    ! [Ps: list_P5707943133018811711nt_int] :
      ( ( Ps = nil_Pr2300489316682597567nt_int )
      = ( ( mirror1 @ Ps )
        = nil_Pr2300489316682597567nt_int ) ) ).

% mirror1_nil
thf(fact_48_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: set_int,R2: set_Pr4810089274464741491et_int,S2: set_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_49_ssubst__Pair__rhs,axiom,
    ! [R: int,S: list_P5707943133018811711nt_int,R2: set_Pr1843331182504696254nt_int,S2: list_P5707943133018811711nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_50_ssubst__Pair__rhs,axiom,
    ! [R: int,S: int,R2: set_Pr958786334691620121nt_int,S2: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_51_ssubst__Pair__rhs,axiom,
    ! [R: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R2: set_Pr9222295170931077689nt_int,S2: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_52_ssubst__Pair__rhs,axiom,
    ! [R: int > option6357759511663192854e_term,S: product_prod_int_int,R2: set_Pr1872883991513573699nt_int,S2: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_53_curryI,axiom,
    ! [F2: produc9133624956312949779et_int > $o,A: nat,B: set_int] :
      ( ( F2 @ ( produc29655638201817675et_int @ A @ B ) )
     => ( produc703246036942720985_int_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_54_curryI,axiom,
    ! [F2: produc661532565036771336nt_int > $o,A: int,B: list_P5707943133018811711nt_int] :
      ( ( F2 @ ( produc8814303788642274490nt_int @ A @ B ) )
     => ( produc5448946096399320464_int_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_55_curryI,axiom,
    ! [F2: product_prod_int_int > $o,A: int,B: int] :
      ( ( F2 @ ( product_Pair_int_int @ A @ B ) )
     => ( produc175634133007206835_int_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_56_curryI,axiom,
    ! [F2: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F2 @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( produc730925184835016917_int_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_57_curryI,axiom,
    ! [F2: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F2 @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( produc9098658269643458507_int_o @ F2 @ A @ B ) ) ).

% curryI
thf(fact_58_swap__simp,axiom,
    ! [X: set_int,Y: nat] :
      ( ( produc3309827273037374443nt_nat @ ( produc2828793327748666955nt_nat @ X @ Y ) )
      = ( produc29655638201817675et_int @ Y @ X ) ) ).

% swap_simp
thf(fact_59_swap__simp,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: int] :
      ( ( produc7218770527457448564nt_int @ ( produc8426000078211274772nt_int @ X @ Y ) )
      = ( produc8814303788642274490nt_int @ Y @ X ) ) ).

% swap_simp
thf(fact_60_swap__simp,axiom,
    ! [X: product_prod_int_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( produc1360913580328244269e_term @ ( produc5219906170222623181e_term @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ Y @ X ) ) ).

% swap_simp
thf(fact_61_swap__simp,axiom,
    ! [X: product_prod_int_int,Y: int > option6357759511663192854e_term] :
      ( ( produc2386993194027891363e_term @ ( produc7516973260301520963e_term @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ Y @ X ) ) ).

% swap_simp
thf(fact_62_swap__simp,axiom,
    ! [X: nat,Y: set_int] :
      ( ( produc510689583490525163et_int @ ( produc29655638201817675et_int @ X @ Y ) )
      = ( produc2828793327748666955nt_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_63_swap__simp,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int] :
      ( ( produc7607074237888448282nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) )
      = ( produc8426000078211274772nt_int @ Y @ X ) ) ).

% swap_simp
thf(fact_64_swap__simp,axiom,
    ! [X: int,Y: int] :
      ( ( product_swap_int_int @ ( product_Pair_int_int @ X @ Y ) )
      = ( product_Pair_int_int @ Y @ X ) ) ).

% swap_simp
thf(fact_65_swap__simp,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc1841954058824580629nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5219906170222623181e_term @ Y @ X ) ) ).

% swap_simp
thf(fact_66_swap__simp,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc8399074013560602399nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc7516973260301520963e_term @ Y @ X ) ) ).

% swap_simp
thf(fact_67_curryD,axiom,
    ! [F2: produc9133624956312949779et_int > $o,A: nat,B: set_int] :
      ( ( produc703246036942720985_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc29655638201817675et_int @ A @ B ) ) ) ).

% curryD
thf(fact_68_curryD,axiom,
    ! [F2: produc661532565036771336nt_int > $o,A: int,B: list_P5707943133018811711nt_int] :
      ( ( produc5448946096399320464_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc8814303788642274490nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_69_curryD,axiom,
    ! [F2: product_prod_int_int > $o,A: int,B: int] :
      ( ( produc175634133007206835_int_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_int_int @ A @ B ) ) ) ).

% curryD
thf(fact_70_curryD,axiom,
    ! [F2: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_71_curryD,axiom,
    ! [F2: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_72_curryE,axiom,
    ! [F2: produc9133624956312949779et_int > $o,A: nat,B: set_int] :
      ( ( produc703246036942720985_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc29655638201817675et_int @ A @ B ) ) ) ).

% curryE
thf(fact_73_curryE,axiom,
    ! [F2: produc661532565036771336nt_int > $o,A: int,B: list_P5707943133018811711nt_int] :
      ( ( produc5448946096399320464_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc8814303788642274490nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_74_curryE,axiom,
    ! [F2: product_prod_int_int > $o,A: int,B: int] :
      ( ( produc175634133007206835_int_o @ F2 @ A @ B )
     => ( F2 @ ( product_Pair_int_int @ A @ B ) ) ) ).

% curryE
thf(fact_75_curryE,axiom,
    ! [F2: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_76_curryE,axiom,
    ! [F2: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F2 @ A @ B )
     => ( F2 @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_77_mirror1__aux__nil,axiom,
    ! [Ps: list_P5707943133018811711nt_int,M: int] :
      ( ( Ps = nil_Pr2300489316682597567nt_int )
      = ( ( mirror1_aux @ M @ Ps )
        = nil_Pr2300489316682597567nt_int ) ) ).

% mirror1_aux_nil
thf(fact_78_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_79_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat2 @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_80_mirror1__aux_Osimps_I1_J,axiom,
    ! [N: int] :
      ( ( mirror1_aux @ N @ nil_Pr2300489316682597567nt_int )
      = nil_Pr2300489316682597567nt_int ) ).

% mirror1_aux.simps(1)
thf(fact_81_list__ex1__simps_I1_J,axiom,
    ! [P2: product_prod_int_int > $o] :
      ~ ( list_e5465588451548443778nt_int @ P2 @ nil_Pr2300489316682597567nt_int ) ).

% list_ex1_simps(1)
thf(fact_82_list__ex1__simps_I1_J,axiom,
    ! [P2: int > $o] :
      ~ ( list_ex1_int @ P2 @ nil_int ) ).

% list_ex1_simps(1)
thf(fact_83_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y4: set_int,Y: set_int,R2: nat > set_Pr2522554150109002629et_int] :
      ( ( P2 @ X )
     => ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ Y4 @ Y ) @ ( R2 @ X ) )
       => ( member5126324565730479632et_int @ ( produc985091676681408599et_int @ ( produc29655638201817675et_int @ X @ Y4 ) @ ( produc29655638201817675et_int @ X @ Y ) ) @ ( same_fst_nat_set_int @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_84_same__fstI,axiom,
    ! [P2: int > $o,X: int,Y4: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R2: int > set_Pr6304946757569631943nt_int] :
      ( ( P2 @ X )
     => ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Y4 @ Y ) @ ( R2 @ X ) )
       => ( member211307870107589146nt_int @ ( produc1418755309762820265nt_int @ ( produc8814303788642274490nt_int @ X @ Y4 ) @ ( produc8814303788642274490nt_int @ X @ Y ) ) @ ( same_f3003577036519843749nt_int @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_85_same__fstI,axiom,
    ! [P2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int,Y: product_prod_int_int,R2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P2 @ X )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y4 @ Y ) @ ( R2 @ X ) )
       => ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ X @ Y4 ) @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( same_f5472592420709775776nt_int @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_86_same__fstI,axiom,
    ! [P2: ( int > option6357759511663192854e_term ) > $o,X: int > option6357759511663192854e_term,Y4: product_prod_int_int,Y: product_prod_int_int,R2: ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P2 @ X )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y4 @ Y ) @ ( R2 @ X ) )
       => ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ X @ Y4 ) @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( same_f6662725367016992042nt_int @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_87_same__fstI,axiom,
    ! [P2: int > $o,X: int,Y4: int,Y: int,R2: int > set_Pr958786334691620121nt_int] :
      ( ( P2 @ X )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y4 @ Y ) @ ( R2 @ X ) )
       => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ ( product_Pair_int_int @ X @ Y ) ) @ ( same_fst_int_int @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_88_product_Osimps_I1_J,axiom,
    ! [Uu: list_int] :
      ( ( product_int_int @ nil_int @ Uu )
      = nil_Pr2300489316682597567nt_int ) ).

% product.simps(1)
thf(fact_89_bind__simps_I1_J,axiom,
    ! [F2: product_prod_int_int > list_P5707943133018811711nt_int] :
      ( ( bind_P8056059132564497095nt_int @ nil_Pr2300489316682597567nt_int @ F2 )
      = nil_Pr2300489316682597567nt_int ) ).

% bind_simps(1)
thf(fact_90_bind__simps_I1_J,axiom,
    ! [F2: product_prod_int_int > list_int] :
      ( ( bind_P1074787401898827934nt_int @ nil_Pr2300489316682597567nt_int @ F2 )
      = nil_int ) ).

% bind_simps(1)
thf(fact_91_bind__simps_I1_J,axiom,
    ! [F2: int > list_P5707943133018811711nt_int] :
      ( ( bind_i1738094907376718532nt_int @ nil_int @ F2 )
      = nil_Pr2300489316682597567nt_int ) ).

% bind_simps(1)
thf(fact_92_bind__simps_I1_J,axiom,
    ! [F2: int > list_int] :
      ( ( bind_int_int @ nil_int @ F2 )
      = nil_int ) ).

% bind_simps(1)
thf(fact_93_member__rec_I2_J,axiom,
    ! [Y: product_prod_int_int] :
      ~ ( member2925713097805433300nt_int @ nil_Pr2300489316682597567nt_int @ Y ) ).

% member_rec(2)
thf(fact_94_member__rec_I2_J,axiom,
    ! [Y: int] :
      ~ ( member_int @ nil_int @ Y ) ).

% member_rec(2)
thf(fact_95_in__measures_I1_J,axiom,
    ! [X: int,Y: int] :
      ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ nil_int_nat ) ) ).

% in_measures(1)
thf(fact_96_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le8428774395332151372nt_int @ N @ nil_Pr2300489316682597567nt_int )
      = N ) ).

% gen_length_code(1)
thf(fact_97_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_int @ N @ nil_int )
      = N ) ).

% gen_length_code(1)
thf(fact_98_maps__simps_I2_J,axiom,
    ! [F2: product_prod_int_int > list_P5707943133018811711nt_int] :
      ( ( maps_P133354487822995585nt_int @ F2 @ nil_Pr2300489316682597567nt_int )
      = nil_Pr2300489316682597567nt_int ) ).

% maps_simps(2)
thf(fact_99_maps__simps_I2_J,axiom,
    ! [F2: product_prod_int_int > list_int] :
      ( ( maps_P493294916530037604nt_int @ F2 @ nil_Pr2300489316682597567nt_int )
      = nil_int ) ).

% maps_simps(2)
thf(fact_100_maps__simps_I2_J,axiom,
    ! [F2: int > list_P5707943133018811711nt_int] :
      ( ( maps_i1156602422007928202nt_int @ F2 @ nil_int )
      = nil_Pr2300489316682597567nt_int ) ).

% maps_simps(2)
thf(fact_101_maps__simps_I2_J,axiom,
    ! [F2: int > list_int] :
      ( ( maps_int_int @ F2 @ nil_int )
      = nil_int ) ).

% maps_simps(2)
thf(fact_102_pair__in__swap__image,axiom,
    ! [Y: set_int,X: nat,A4: set_Pr4810089274464741491et_int] :
      ( ( member1837137653922122684nt_nat @ ( produc2828793327748666955nt_nat @ Y @ X ) @ ( image_2292489492271515589nt_nat @ produc510689583490525163et_int @ A4 ) )
      = ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_103_pair__in__swap__image,axiom,
    ! [Y: list_P5707943133018811711nt_int,X: int,A4: set_Pr1843331182504696254nt_int] :
      ( ( member2436464452258586801nt_int @ ( produc8426000078211274772nt_int @ Y @ X ) @ ( image_9042332981003017769nt_int @ produc7607074237888448282nt_int @ A4 ) )
      = ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_104_pair__in__swap__image,axiom,
    ! [Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,A4: set_Pr9222295170931077689nt_int] :
      ( ( member7154775377335690986e_term @ ( produc5219906170222623181e_term @ Y @ X ) @ ( image_3490000813295636061e_term @ produc1841954058824580629nt_int @ A4 ) )
      = ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_105_pair__in__swap__image,axiom,
    ! [Y: product_prod_int_int,X: int > option6357759511663192854e_term,A4: set_Pr1872883991513573699nt_int] :
      ( ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ Y @ X ) @ ( image_463623704579739357e_term @ produc8399074013560602399nt_int @ A4 ) )
      = ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_106_pair__in__swap__image,axiom,
    ! [Y: nat,X: set_int,A4: set_Pr6819878462438571379nt_nat] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ Y @ X ) @ ( image_6524025564092292549et_int @ produc3309827273037374443nt_nat @ A4 ) )
      = ( member1837137653922122684nt_nat @ ( produc2828793327748666955nt_nat @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_107_pair__in__swap__image,axiom,
    ! [Y: int,X: list_P5707943133018811711nt_int,A4: set_Pr8964431684210982608nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ Y @ X ) @ ( image_1973658408154710569nt_int @ produc7218770527457448564nt_int @ A4 ) )
      = ( member2436464452258586801nt_int @ ( produc8426000078211274772nt_int @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_108_pair__in__swap__image,axiom,
    ! [Y: int,X: int,A4: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ ( image_2653370878348428101nt_int @ product_swap_int_int @ A4 ) )
      = ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_109_pair__in__swap__image,axiom,
    ! [Y: produc8551481072490612790e_term > option6357759511663192854e_term,X: product_prod_int_int,A4: set_Pr7309386651551657097e_term] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y @ X ) @ ( image_1640129903684976221nt_int @ produc1360913580328244269e_term @ A4 ) )
      = ( member7154775377335690986e_term @ ( produc5219906170222623181e_term @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_110_pair__in__swap__image,axiom,
    ! [Y: int > option6357759511663192854e_term,X: product_prod_int_int,A4: set_Pr9020418513037773311e_term] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y @ X ) @ ( image_1292712855239066845nt_int @ produc2386993194027891363e_term @ A4 ) )
      = ( member3093737243930835296e_term @ ( produc7516973260301520963e_term @ X @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_111_eq__Nil__null,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( Xs = nil_Pr2300489316682597567nt_int )
      = ( null_P4123483736518020807nt_int @ Xs ) ) ).

% eq_Nil_null
thf(fact_112_eq__Nil__null,axiom,
    ! [Xs: list_int] :
      ( ( Xs = nil_int )
      = ( null_int @ Xs ) ) ).

% eq_Nil_null
thf(fact_113_null__rec_I2_J,axiom,
    null_P4123483736518020807nt_int @ nil_Pr2300489316682597567nt_int ).

% null_rec(2)
thf(fact_114_null__rec_I2_J,axiom,
    null_int @ nil_int ).

% null_rec(2)
thf(fact_115_image__eqI,axiom,
    ! [B: nat,F2: nat > nat,X: nat,A4: set_nat] :
      ( ( B
        = ( F2 @ X ) )
     => ( ( member_nat2 @ X @ A4 )
       => ( member_nat2 @ B @ ( image_nat_nat @ F2 @ A4 ) ) ) ) ).

% image_eqI
thf(fact_116_Inf_OINF__cong,axiom,
    ! [A4: set_nat,B4: set_nat,C3: nat > nat,D3: nat > nat,Inf: set_nat > nat] :
      ( ( A4 = B4 )
     => ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ B4 )
           => ( ( C3 @ X3 )
              = ( D3 @ X3 ) ) )
       => ( ( Inf @ ( image_nat_nat @ C3 @ A4 ) )
          = ( Inf @ ( image_nat_nat @ D3 @ B4 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_117_Sup_OSUP__cong,axiom,
    ! [A4: set_nat,B4: set_nat,C3: nat > nat,D3: nat > nat,Sup: set_nat > nat] :
      ( ( A4 = B4 )
     => ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ B4 )
           => ( ( C3 @ X3 )
              = ( D3 @ X3 ) ) )
       => ( ( Sup @ ( image_nat_nat @ C3 @ A4 ) )
          = ( Sup @ ( image_nat_nat @ D3 @ B4 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_118_imageI,axiom,
    ! [X: nat,A4: set_nat,F2: nat > nat] :
      ( ( member_nat2 @ X @ A4 )
     => ( member_nat2 @ ( F2 @ X ) @ ( image_nat_nat @ F2 @ A4 ) ) ) ).

% imageI
thf(fact_119_image__iff,axiom,
    ! [Z: nat,F2: nat > nat,A4: set_nat] :
      ( ( member_nat2 @ Z @ ( image_nat_nat @ F2 @ A4 ) )
      = ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ A4 )
            & ( Z
              = ( F2 @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_120_bex__imageD,axiom,
    ! [F2: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat2 @ X5 @ ( image_nat_nat @ F2 @ A4 ) )
          & ( P2 @ X5 ) )
     => ? [X3: nat] :
          ( ( member_nat2 @ X3 @ A4 )
          & ( P2 @ ( F2 @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_121_image__cong,axiom,
    ! [M2: set_nat,N2: set_nat,F2: nat > nat,G: nat > nat] :
      ( ( M2 = N2 )
     => ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ N2 )
           => ( ( F2 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_nat @ F2 @ M2 )
          = ( image_nat_nat @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_122_ball__imageD,axiom,
    ! [F2: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat2 @ X3 @ ( image_nat_nat @ F2 @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X5: nat] :
          ( ( member_nat2 @ X5 @ A4 )
         => ( P2 @ ( F2 @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_123_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: nat,F2: nat > nat] :
      ( ( member_nat2 @ X @ A4 )
     => ( ( B
          = ( F2 @ X ) )
       => ( member_nat2 @ B @ ( image_nat_nat @ F2 @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_124_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F2: int > int,G: int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_125_map__prod__imageI,axiom,
    ! [A: nat,B: set_int,R2: set_Pr4810089274464741491et_int,F2: nat > int,G: set_int > int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ A @ B ) @ R2 )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_922801827940673131nt_int @ ( produc1603449573289796100nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_126_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F2: int > nat,G: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8001630411029145503et_int @ ( produc626993783231305220et_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_127_map__prod__imageI,axiom,
    ! [A: nat,B: set_int,R2: set_Pr4810089274464741491et_int,F2: nat > nat,G: set_int > set_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ A @ B ) @ R2 )
     => ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_1747593022141657797et_int @ ( produc767733989823756638et_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_128_map__prod__imageI,axiom,
    ! [A: int,B: list_P5707943133018811711nt_int,R2: set_Pr1843331182504696254nt_int,F2: int > int,G: list_P5707943133018811711nt_int > int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ A @ B ) @ R2 )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_1604825640560108266nt_int @ ( produc7810239482201755035nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_129_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F2: int > int,G: int > list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8353976552941304700nt_int @ ( produc8198543192632754753nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_130_map__prod__imageI,axiom,
    ! [A: nat,B: set_int,R2: set_Pr4810089274464741491et_int,F2: nat > int,G: set_int > list_P5707943133018811711nt_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ A @ B ) @ R2 )
     => ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_6253543406118951894nt_int @ ( produc5750686221896385383nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_131_map__prod__imageI,axiom,
    ! [A: int,B: list_P5707943133018811711nt_int,R2: set_Pr1843331182504696254nt_int,F2: int > nat,G: list_P5707943133018811711nt_int > set_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ A @ B ) @ R2 )
     => ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_8576745712708285252et_int @ ( produc7200407630492273325et_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_132_map__prod__imageI,axiom,
    ! [A: int,B: list_P5707943133018811711nt_int,R2: set_Pr1843331182504696254nt_int,F2: int > int,G: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ A @ B ) @ R2 )
     => ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_645824907197992983nt_int @ ( produc5369303456246265488nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_133_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F2: int > int > option6357759511663192854e_term,G: int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 )
     => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_7679467227446020609nt_int @ ( produc995849431746665926nt_int @ F2 @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_134_UNIV__I,axiom,
    ! [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).

% UNIV_I
thf(fact_135_map__prod__simp,axiom,
    ! [F2: int > int,G: int > int,A: int,B: int] :
      ( ( produc6036585564866528938nt_int @ F2 @ G @ ( product_Pair_int_int @ A @ B ) )
      = ( product_Pair_int_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_136_map__prod__simp,axiom,
    ! [F2: nat > int,G: set_int > int,A: nat,B: set_int] :
      ( ( produc1603449573289796100nt_int @ F2 @ G @ ( produc29655638201817675et_int @ A @ B ) )
      = ( product_Pair_int_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_137_map__prod__simp,axiom,
    ! [F2: int > nat,G: int > set_int,A: int,B: int] :
      ( ( produc626993783231305220et_int @ F2 @ G @ ( product_Pair_int_int @ A @ B ) )
      = ( produc29655638201817675et_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_138_map__prod__simp,axiom,
    ! [F2: nat > nat,G: set_int > set_int,A: nat,B: set_int] :
      ( ( produc767733989823756638et_int @ F2 @ G @ ( produc29655638201817675et_int @ A @ B ) )
      = ( produc29655638201817675et_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_139_map__prod__simp,axiom,
    ! [F2: int > int,G: list_P5707943133018811711nt_int > int,A: int,B: list_P5707943133018811711nt_int] :
      ( ( produc7810239482201755035nt_int @ F2 @ G @ ( produc8814303788642274490nt_int @ A @ B ) )
      = ( product_Pair_int_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_140_map__prod__simp,axiom,
    ! [F2: int > int,G: int > list_P5707943133018811711nt_int,A: int,B: int] :
      ( ( produc8198543192632754753nt_int @ F2 @ G @ ( product_Pair_int_int @ A @ B ) )
      = ( produc8814303788642274490nt_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_141_map__prod__simp,axiom,
    ! [F2: nat > int,G: set_int > list_P5707943133018811711nt_int,A: nat,B: set_int] :
      ( ( produc5750686221896385383nt_int @ F2 @ G @ ( produc29655638201817675et_int @ A @ B ) )
      = ( produc8814303788642274490nt_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_142_map__prod__simp,axiom,
    ! [F2: int > nat,G: list_P5707943133018811711nt_int > set_int,A: int,B: list_P5707943133018811711nt_int] :
      ( ( produc7200407630492273325et_int @ F2 @ G @ ( produc8814303788642274490nt_int @ A @ B ) )
      = ( produc29655638201817675et_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_143_map__prod__simp,axiom,
    ! [F2: int > int,G: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int,A: int,B: list_P5707943133018811711nt_int] :
      ( ( produc5369303456246265488nt_int @ F2 @ G @ ( produc8814303788642274490nt_int @ A @ B ) )
      = ( produc8814303788642274490nt_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_144_map__prod__simp,axiom,
    ! [F2: int > int > option6357759511663192854e_term,G: int > product_prod_int_int,A: int,B: int] :
      ( ( produc995849431746665926nt_int @ F2 @ G @ ( product_Pair_int_int @ A @ B ) )
      = ( produc4305682042979456191nt_int @ ( F2 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_145_surj__diff,axiom,
    ! [A: int] :
      ( ( image_int_int @ ( minus_minus_int @ A ) @ top_top_set_int )
      = top_top_set_int ) ).

% surj_diff
thf(fact_146_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_147_UNIV__eq__I,axiom,
    ! [A4: set_nat] :
      ( ! [X3: nat] : ( member_nat2 @ X3 @ A4 )
     => ( top_top_set_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_148_range__eqI,axiom,
    ! [B: nat,F2: nat > nat,X: nat] :
      ( ( B
        = ( F2 @ X ) )
     => ( member_nat2 @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_149_rangeI,axiom,
    ! [F2: nat > nat,X: nat] : ( member_nat2 @ ( F2 @ X ) @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).

% rangeI
thf(fact_150_map__prod__surj,axiom,
    ! [F2: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F2 @ G ) @ top_to4669805908274784177at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_151_surjD,axiom,
    ! [F2: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X3: nat] :
          ( Y
          = ( F2 @ X3 ) ) ) ).

% surjD
thf(fact_152_surjE,axiom,
    ! [F2: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X3: nat] :
            ( Y
           != ( F2 @ X3 ) ) ) ).

% surjE
thf(fact_153_surjI,axiom,
    ! [G: nat > nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( G @ ( F2 @ X3 ) )
          = X3 )
     => ( ( image_nat_nat @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_154_surj__def,axiom,
    ! [F2: nat > nat] :
      ( ( ( image_nat_nat @ F2 @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y5: nat] :
          ? [X4: nat] :
            ( Y5
            = ( F2 @ X4 ) ) ) ) ).

% surj_def
thf(fact_155_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F2: int > int,G: int > int,R2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( image_2653370878348428101nt_int @ ( produc6036585564866528938nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( C
              = ( product_Pair_int_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_156_prod__fun__imageE,axiom,
    ! [C: produc9133624956312949779et_int,F2: int > nat,G: int > set_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member1292241183792264892et_int @ C @ ( image_8001630411029145503et_int @ ( produc626993783231305220et_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( C
              = ( produc29655638201817675et_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_157_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F2: nat > int,G: set_int > int,R2: set_Pr4810089274464741491et_int] :
      ( ( member5262025264175285858nt_int @ C @ ( image_922801827940673131nt_int @ ( produc1603449573289796100nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: set_int] :
            ( ( C
              = ( product_Pair_int_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_158_prod__fun__imageE,axiom,
    ! [C: produc9133624956312949779et_int,F2: nat > nat,G: set_int > set_int,R2: set_Pr4810089274464741491et_int] :
      ( ( member1292241183792264892et_int @ C @ ( image_1747593022141657797et_int @ ( produc767733989823756638et_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: set_int] :
            ( ( C
              = ( produc29655638201817675et_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_159_prod__fun__imageE,axiom,
    ! [C: produc661532565036771336nt_int,F2: int > int,G: int > list_P5707943133018811711nt_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member3263328415308337823nt_int @ C @ ( image_8353976552941304700nt_int @ ( produc8198543192632754753nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( C
              = ( produc8814303788642274490nt_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_160_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F2: int > int,G: list_P5707943133018811711nt_int > int,R2: set_Pr1843331182504696254nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( image_1604825640560108266nt_int @ ( produc7810239482201755035nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int,Y3: list_P5707943133018811711nt_int] :
            ( ( C
              = ( product_Pair_int_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_161_prod__fun__imageE,axiom,
    ! [C: produc9133624956312949779et_int,F2: int > nat,G: list_P5707943133018811711nt_int > set_int,R2: set_Pr1843331182504696254nt_int] :
      ( ( member1292241183792264892et_int @ C @ ( image_8576745712708285252et_int @ ( produc7200407630492273325et_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int,Y3: list_P5707943133018811711nt_int] :
            ( ( C
              = ( produc29655638201817675et_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_162_prod__fun__imageE,axiom,
    ! [C: produc661532565036771336nt_int,F2: nat > int,G: set_int > list_P5707943133018811711nt_int,R2: set_Pr4810089274464741491et_int] :
      ( ( member3263328415308337823nt_int @ C @ ( image_6253543406118951894nt_int @ ( produc5750686221896385383nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: set_int] :
            ( ( C
              = ( produc8814303788642274490nt_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_163_prod__fun__imageE,axiom,
    ! [C: produc661532565036771336nt_int,F2: int > int,G: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int,R2: set_Pr1843331182504696254nt_int] :
      ( ( member3263328415308337823nt_int @ C @ ( image_645824907197992983nt_int @ ( produc5369303456246265488nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int,Y3: list_P5707943133018811711nt_int] :
            ( ( C
              = ( produc8814303788642274490nt_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_164_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F2: ( int > option6357759511663192854e_term ) > int,G: product_prod_int_int > int,R2: set_Pr1872883991513573699nt_int] :
      ( ( member5262025264175285858nt_int @ C @ ( image_3108077152341812517nt_int @ ( produc7152297802020950944nt_int @ F2 @ G ) @ R2 ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( ( C
              = ( product_Pair_int_int @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_165_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_166_UNIV__coset,axiom,
    ( top_to4366644338036079209nt_int
    = ( coset_3631019413092825772nt_int @ nil_Pr2300489316682597567nt_int ) ) ).

% UNIV_coset
thf(fact_167_UNIV__coset,axiom,
    ( top_top_set_int
    = ( coset_int @ nil_int ) ) ).

% UNIV_coset
thf(fact_168_in__lex__prod,axiom,
    ! [A: nat,B: set_int,A2: nat,B2: set_int,R: set_Pr1261947904930325089at_nat,S: set_Pr2522554150109002629et_int] :
      ( ( member5126324565730479632et_int @ ( produc985091676681408599et_int @ ( produc29655638201817675et_int @ A @ B ) @ ( produc29655638201817675et_int @ A2 @ B2 ) ) @ ( lex_prod_nat_set_int @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_169_in__lex__prod,axiom,
    ! [A: int,B: list_P5707943133018811711nt_int,A2: int,B2: list_P5707943133018811711nt_int,R: set_Pr958786334691620121nt_int,S: set_Pr6304946757569631943nt_int] :
      ( ( member211307870107589146nt_int @ ( produc1418755309762820265nt_int @ ( produc8814303788642274490nt_int @ A @ B ) @ ( produc8814303788642274490nt_int @ A2 @ B2 ) ) @ ( lex_pr1845011153532943010nt_int @ R @ S ) )
      = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_170_in__lex__prod,axiom,
    ! [A: int,B: int,A2: int,B2: int,R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ A2 @ B2 ) ) @ ( lex_prod_int_int @ R @ S ) )
      = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_171_in__lex__prod,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,R: set_Pr4675205556198584599e_term,S: set_Pr2560585780119916871nt_int] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5700946648718959541nt_int @ A2 @ B2 ) ) @ ( lex_pr4548132154224949661nt_int @ R @ S ) )
      = ( ( member8796044733138646368e_term @ ( produc8124735558134971759e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_172_in__lex__prod,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: int > option6357759511663192854e_term,B2: product_prod_int_int,R: set_Pr7140365069852192003e_term,S: set_Pr2560585780119916871nt_int] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc4305682042979456191nt_int @ A2 @ B2 ) ) @ ( lex_pr7234932505866811047nt_int @ R @ S ) )
      = ( ( member6538663773838357324e_term @ ( produc4510790414113558107e_term @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_173_last__appendR,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( Ys != nil_Pr2300489316682597567nt_int )
     => ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
        = ( last_P3305686521732843992nt_int @ Ys ) ) ) ).

% last_appendR
thf(fact_174_last__appendR,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys != nil_int )
     => ( ( last_int @ ( append_int @ Xs @ Ys ) )
        = ( last_int @ Ys ) ) ) ).

% last_appendR
thf(fact_175_last__appendL,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( Ys = nil_Pr2300489316682597567nt_int )
     => ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
        = ( last_P3305686521732843992nt_int @ Xs ) ) ) ).

% last_appendL
thf(fact_176_last__appendL,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys = nil_int )
     => ( ( last_int @ ( append_int @ Xs @ Ys ) )
        = ( last_int @ Xs ) ) ) ).

% last_appendL
thf(fact_177_map__filter__simps_I2_J,axiom,
    ! [F2: product_prod_int_int > option4624381673175914239nt_int] :
      ( ( map_fi1866282469256664714nt_int @ F2 @ nil_Pr2300489316682597567nt_int )
      = nil_Pr2300489316682597567nt_int ) ).

% map_filter_simps(2)
thf(fact_178_map__filter__simps_I2_J,axiom,
    ! [F2: product_prod_int_int > option_int] :
      ( ( map_fi5298100290067045531nt_int @ F2 @ nil_Pr2300489316682597567nt_int )
      = nil_int ) ).

% map_filter_simps(2)
thf(fact_179_map__filter__simps_I2_J,axiom,
    ! [F2: int > option4624381673175914239nt_int] :
      ( ( map_fi5961407795544936129nt_int @ F2 @ nil_int )
      = nil_Pr2300489316682597567nt_int ) ).

% map_filter_simps(2)
thf(fact_180_map__filter__simps_I2_J,axiom,
    ! [F2: int > option_int] :
      ( ( map_filter_int_int @ F2 @ nil_int )
      = nil_int ) ).

% map_filter_simps(2)
thf(fact_181_same__append__eq,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = ( append_int @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_182_append__same__eq,axiom,
    ! [Ys: list_int,Xs: list_int,Zs: list_int] :
      ( ( ( append_int @ Ys @ Xs )
        = ( append_int @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_183_append__assoc,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs )
      = ( append_int @ Xs @ ( append_int @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_184_append_Oassoc,axiom,
    ! [A: list_int,B: list_int,C: list_int] :
      ( ( append_int @ ( append_int @ A @ B ) @ C )
      = ( append_int @ A @ ( append_int @ B @ C ) ) ) ).

% append.assoc
thf(fact_185_append_Oright__neutral,axiom,
    ! [A: list_P5707943133018811711nt_int] :
      ( ( append7030698103840186580nt_int @ A @ nil_Pr2300489316682597567nt_int )
      = A ) ).

% append.right_neutral
thf(fact_186_append_Oright__neutral,axiom,
    ! [A: list_int] :
      ( ( append_int @ A @ nil_int )
      = A ) ).

% append.right_neutral
thf(fact_187_append__Nil2,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( append7030698103840186580nt_int @ Xs @ nil_Pr2300489316682597567nt_int )
      = Xs ) ).

% append_Nil2
thf(fact_188_append__Nil2,axiom,
    ! [Xs: list_int] :
      ( ( append_int @ Xs @ nil_int )
      = Xs ) ).

% append_Nil2
thf(fact_189_append__self__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( append7030698103840186580nt_int @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr2300489316682597567nt_int ) ) ).

% append_self_conv
thf(fact_190_append__self__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_int ) ) ).

% append_self_conv
thf(fact_191_self__append__conv,axiom,
    ! [Y: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( Y
        = ( append7030698103840186580nt_int @ Y @ Ys ) )
      = ( Ys = nil_Pr2300489316682597567nt_int ) ) ).

% self_append_conv
thf(fact_192_self__append__conv,axiom,
    ! [Y: list_int,Ys: list_int] :
      ( ( Y
        = ( append_int @ Y @ Ys ) )
      = ( Ys = nil_int ) ) ).

% self_append_conv
thf(fact_193_append__self__conv2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( append7030698103840186580nt_int @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% append_self_conv2
thf(fact_194_append__self__conv2,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_int ) ) ).

% append_self_conv2
thf(fact_195_self__append__conv2,axiom,
    ! [Y: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( Y
        = ( append7030698103840186580nt_int @ Xs @ Y ) )
      = ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% self_append_conv2
thf(fact_196_self__append__conv2,axiom,
    ! [Y: list_int,Xs: list_int] :
      ( ( Y
        = ( append_int @ Xs @ Y ) )
      = ( Xs = nil_int ) ) ).

% self_append_conv2
thf(fact_197_Nil__is__append__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( nil_Pr2300489316682597567nt_int
        = ( append7030698103840186580nt_int @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        & ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% Nil_is_append_conv
thf(fact_198_Nil__is__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_int
        = ( append_int @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% Nil_is_append_conv
thf(fact_199_append__is__Nil__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( append7030698103840186580nt_int @ Xs @ Ys )
        = nil_Pr2300489316682597567nt_int )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        & ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% append_is_Nil_conv
thf(fact_200_append__is__Nil__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = nil_int )
      = ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% append_is_Nil_conv
thf(fact_201_prod_Ocollapse,axiom,
    ! [Prod: produc9133624956312949779et_int] :
      ( ( produc29655638201817675et_int @ ( produc9078099356025976375et_int @ Prod ) @ ( produc1962771869185418873et_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_202_prod_Ocollapse,axiom,
    ! [Prod: produc661532565036771336nt_int] :
      ( ( produc8814303788642274490nt_int @ ( produc3131058726980325326nt_int @ Prod ) @ ( produc669174308195927564nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_203_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] :
      ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_204_prod_Ocollapse,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_205_prod_Ocollapse,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_206_append__eq__append__conv2,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = ( append_int @ Zs @ Ts ) )
      = ( ? [Us: list_int] :
            ( ( ( Xs
                = ( append_int @ Zs @ Us ) )
              & ( ( append_int @ Us @ Ys )
                = Ts ) )
            | ( ( ( append_int @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append_int @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_207_append__eq__appendI,axiom,
    ! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us2: list_int] :
      ( ( ( append_int @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_int @ Xs1 @ Us2 ) )
       => ( ( append_int @ Xs @ Ys )
          = ( append_int @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_208_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9133624956312949779et_int] :
      ( Prod
      = ( produc29655638201817675et_int @ ( produc9078099356025976375et_int @ Prod ) @ ( produc1962771869185418873et_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_209_prod_Oexhaust__sel,axiom,
    ! [Prod: produc661532565036771336nt_int] :
      ( Prod
      = ( produc8814303788642274490nt_int @ ( produc3131058726980325326nt_int @ Prod ) @ ( produc669174308195927564nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_210_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] :
      ( Prod
      = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_211_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( Prod
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_212_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( Prod
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_213_surjective__pairing,axiom,
    ! [T: produc9133624956312949779et_int] :
      ( T
      = ( produc29655638201817675et_int @ ( produc9078099356025976375et_int @ T ) @ ( produc1962771869185418873et_int @ T ) ) ) ).

% surjective_pairing
thf(fact_214_surjective__pairing,axiom,
    ! [T: produc661532565036771336nt_int] :
      ( T
      = ( produc8814303788642274490nt_int @ ( produc3131058726980325326nt_int @ T ) @ ( produc669174308195927564nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_215_surjective__pairing,axiom,
    ! [T: product_prod_int_int] :
      ( T
      = ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).

% surjective_pairing
thf(fact_216_surjective__pairing,axiom,
    ! [T: produc2285326912895808259nt_int] :
      ( T
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ T ) @ ( produc7328097813583171335nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_217_surjective__pairing,axiom,
    ! [T: produc7773217078559923341nt_int] :
      ( T
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ T ) @ ( produc3162348030201620241nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_218_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X4: nat] : ( member_nat2 @ X4 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_219_prod_Oswap__def,axiom,
    ( produc3309827273037374443nt_nat
    = ( ^ [P3: produc455149389588031763nt_nat] : ( produc29655638201817675et_int @ ( produc4761909558732268153nt_nat @ P3 ) @ ( produc2653865008718049847nt_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_220_prod_Oswap__def,axiom,
    ( produc7218770527457448564nt_int
    = ( ^ [P3: produc9058040638841796122nt_int] : ( produc8814303788642274490nt_int @ ( produc280870597764927846nt_int @ P3 ) @ ( produc2742755016549325608nt_int @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_221_prod_Oswap__def,axiom,
    ( product_swap_int_int
    = ( ^ [P3: product_prod_int_int] : ( product_Pair_int_int @ ( product_snd_int_int @ P3 ) @ ( product_fst_int_int @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_222_prod_Oswap__def,axiom,
    ( produc1360913580328244269e_term
    = ( ^ [P3: produc1821397396195235155e_term] : ( produc5700946648718959541nt_int @ ( produc6847057335086834975e_term @ P3 ) @ ( produc6335124012134732001e_term @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_223_prod_Oswap__def,axiom,
    ( produc2386993194027891363e_term
    = ( ^ [P3: produc3832618445565238089e_term] : ( produc4305682042979456191nt_int @ ( produc6373639247523685013e_term @ P3 ) @ ( produc217921407547260247e_term @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_224_snd__conv,axiom,
    ! [X1: nat,X2: set_int] :
      ( ( produc1962771869185418873et_int @ ( produc29655638201817675et_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_225_snd__conv,axiom,
    ! [X1: int,X2: list_P5707943133018811711nt_int] :
      ( ( produc669174308195927564nt_int @ ( produc8814303788642274490nt_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_226_snd__conv,axiom,
    ! [X1: int,X2: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_227_snd__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_228_snd__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_229_snd__eqD,axiom,
    ! [X: nat,Y: set_int,A: set_int] :
      ( ( ( produc1962771869185418873et_int @ ( produc29655638201817675et_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_230_snd__eqD,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int,A: list_P5707943133018811711nt_int] :
      ( ( ( produc669174308195927564nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_231_snd__eqD,axiom,
    ! [X: int,Y: int,A: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_232_snd__eqD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_233_snd__eqD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_234_fst__conv,axiom,
    ! [X1: nat,X2: set_int] :
      ( ( produc9078099356025976375et_int @ ( produc29655638201817675et_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_235_fst__conv,axiom,
    ! [X1: int,X2: list_P5707943133018811711nt_int] :
      ( ( produc3131058726980325326nt_int @ ( produc8814303788642274490nt_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_236_fst__conv,axiom,
    ! [X1: int,X2: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_237_fst__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_238_fst__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_239_fst__eqD,axiom,
    ! [X: nat,Y: set_int,A: nat] :
      ( ( ( produc9078099356025976375et_int @ ( produc29655638201817675et_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_240_fst__eqD,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int,A: int] :
      ( ( ( produc3131058726980325326nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_241_fst__eqD,axiom,
    ! [X: int,Y: int,A: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_242_fst__eqD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_243_fst__eqD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: int > option6357759511663192854e_term] :
      ( ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_244_append__Nil,axiom,
    ! [Ys: list_P5707943133018811711nt_int] :
      ( ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_245_append__Nil,axiom,
    ! [Ys: list_int] :
      ( ( append_int @ nil_int @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_246_append_Oleft__neutral,axiom,
    ! [A: list_P5707943133018811711nt_int] :
      ( ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ A )
      = A ) ).

% append.left_neutral
thf(fact_247_append_Oleft__neutral,axiom,
    ! [A: list_int] :
      ( ( append_int @ nil_int @ A )
      = A ) ).

% append.left_neutral
thf(fact_248_eq__Nil__appendI,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append7030698103840186580nt_int @ nil_Pr2300489316682597567nt_int @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_249_eq__Nil__appendI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_int @ nil_int @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_250_last__append,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( ( Ys = nil_Pr2300489316682597567nt_int )
       => ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
          = ( last_P3305686521732843992nt_int @ Xs ) ) )
      & ( ( Ys != nil_Pr2300489316682597567nt_int )
       => ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
          = ( last_P3305686521732843992nt_int @ Ys ) ) ) ) ).

% last_append
thf(fact_251_last__append,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( ( Ys = nil_int )
       => ( ( last_int @ ( append_int @ Xs @ Ys ) )
          = ( last_int @ Xs ) ) )
      & ( ( Ys != nil_int )
       => ( ( last_int @ ( append_int @ Xs @ Ys ) )
          = ( last_int @ Ys ) ) ) ) ).

% last_append
thf(fact_252_longest__common__suffix,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
    ? [Ss: list_P5707943133018811711nt_int,Xs2: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
      ( ( Xs
        = ( append7030698103840186580nt_int @ Xs2 @ Ss ) )
      & ( Ys
        = ( append7030698103840186580nt_int @ Ys2 @ Ss ) )
      & ( ( Xs2 = nil_Pr2300489316682597567nt_int )
        | ( Ys2 = nil_Pr2300489316682597567nt_int )
        | ( ( last_P3305686521732843992nt_int @ Xs2 )
         != ( last_P3305686521732843992nt_int @ Ys2 ) ) ) ) ).

% longest_common_suffix
thf(fact_253_longest__common__suffix,axiom,
    ! [Xs: list_int,Ys: list_int] :
    ? [Ss: list_int,Xs2: list_int,Ys2: list_int] :
      ( ( Xs
        = ( append_int @ Xs2 @ Ss ) )
      & ( Ys
        = ( append_int @ Ys2 @ Ss ) )
      & ( ( Xs2 = nil_int )
        | ( Ys2 = nil_int )
        | ( ( last_int @ Xs2 )
         != ( last_int @ Ys2 ) ) ) ) ).

% longest_common_suffix
thf(fact_254_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > set_int > $o,X: nat,Y: set_int,A: produc9133624956312949779et_int] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc29655638201817675et_int @ X @ Y ) )
       => ( P2 @ ( produc9078099356025976375et_int @ A ) @ ( produc1962771869185418873et_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_255_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: int > list_P5707943133018811711nt_int > $o,X: int,Y: list_P5707943133018811711nt_int,A: produc661532565036771336nt_int] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc8814303788642274490nt_int @ X @ Y ) )
       => ( P2 @ ( produc3131058726980325326nt_int @ A ) @ ( produc669174308195927564nt_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_256_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: int > int > $o,X: int,Y: int,A: product_prod_int_int] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( product_Pair_int_int @ X @ Y ) )
       => ( P2 @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_257_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: produc2285326912895808259nt_int] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc5700946648718959541nt_int @ X @ Y ) )
       => ( P2 @ ( produc6816164490631068361nt_int @ A ) @ ( produc7328097813583171335nt_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_258_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: produc7773217078559923341nt_int] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc4305682042979456191nt_int @ X @ Y ) )
       => ( P2 @ ( produc6230002227079971283nt_int @ A ) @ ( produc3162348030201620241nt_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_259_conjI__realizer,axiom,
    ! [P2: nat > $o,P: nat,Q: set_int > $o,Q2: set_int] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc9078099356025976375et_int @ ( produc29655638201817675et_int @ P @ Q2 ) ) )
          & ( Q @ ( produc1962771869185418873et_int @ ( produc29655638201817675et_int @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_260_conjI__realizer,axiom,
    ! [P2: int > $o,P: int,Q: list_P5707943133018811711nt_int > $o,Q2: list_P5707943133018811711nt_int] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc3131058726980325326nt_int @ ( produc8814303788642274490nt_int @ P @ Q2 ) ) )
          & ( Q @ ( produc669174308195927564nt_int @ ( produc8814303788642274490nt_int @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_261_conjI__realizer,axiom,
    ! [P2: int > $o,P: int,Q: int > $o,Q2: int] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_int_int @ ( product_Pair_int_int @ P @ Q2 ) ) )
          & ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_262_conjI__realizer,axiom,
    ! [P2: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P: produc8551481072490612790e_term > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q2: product_prod_int_int] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ P @ Q2 ) ) )
          & ( Q @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_263_conjI__realizer,axiom,
    ! [P2: ( int > option6357759511663192854e_term ) > $o,P: int > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q2: product_prod_int_int] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ P @ Q2 ) ) )
          & ( Q @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_264_exI__realizer,axiom,
    ! [P2: set_int > nat > $o,Y: set_int,X: nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc1962771869185418873et_int @ ( produc29655638201817675et_int @ X @ Y ) ) @ ( produc9078099356025976375et_int @ ( produc29655638201817675et_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_265_exI__realizer,axiom,
    ! [P2: list_P5707943133018811711nt_int > int > $o,Y: list_P5707943133018811711nt_int,X: int] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc669174308195927564nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) ) @ ( produc3131058726980325326nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_266_exI__realizer,axiom,
    ! [P2: int > int > $o,Y: int,X: int] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_267_exI__realizer,axiom,
    ! [P2: product_prod_int_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_268_exI__realizer,axiom,
    ! [P2: product_prod_int_int > ( int > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: int > option6357759511663192854e_term] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_269_eq__snd__iff,axiom,
    ! [B: set_int,P: produc9133624956312949779et_int] :
      ( ( B
        = ( produc1962771869185418873et_int @ P ) )
      = ( ? [A5: nat] :
            ( P
            = ( produc29655638201817675et_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_270_eq__snd__iff,axiom,
    ! [B: list_P5707943133018811711nt_int,P: produc661532565036771336nt_int] :
      ( ( B
        = ( produc669174308195927564nt_int @ P ) )
      = ( ? [A5: int] :
            ( P
            = ( produc8814303788642274490nt_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_271_eq__snd__iff,axiom,
    ! [B: int,P: product_prod_int_int] :
      ( ( B
        = ( product_snd_int_int @ P ) )
      = ( ? [A5: int] :
            ( P
            = ( product_Pair_int_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_272_eq__snd__iff,axiom,
    ! [B: product_prod_int_int,P: produc2285326912895808259nt_int] :
      ( ( B
        = ( produc7328097813583171335nt_int @ P ) )
      = ( ? [A5: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( P
            = ( produc5700946648718959541nt_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_273_eq__snd__iff,axiom,
    ! [B: product_prod_int_int,P: produc7773217078559923341nt_int] :
      ( ( B
        = ( produc3162348030201620241nt_int @ P ) )
      = ( ? [A5: int > option6357759511663192854e_term] :
            ( P
            = ( produc4305682042979456191nt_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_274_sndI,axiom,
    ! [X: produc9133624956312949779et_int,Y: nat,Z: set_int] :
      ( ( X
        = ( produc29655638201817675et_int @ Y @ Z ) )
     => ( ( produc1962771869185418873et_int @ X )
        = Z ) ) ).

% sndI
thf(fact_275_sndI,axiom,
    ! [X: produc661532565036771336nt_int,Y: int,Z: list_P5707943133018811711nt_int] :
      ( ( X
        = ( produc8814303788642274490nt_int @ Y @ Z ) )
     => ( ( produc669174308195927564nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_276_sndI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_snd_int_int @ X )
        = Z ) ) ).

% sndI
thf(fact_277_sndI,axiom,
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc7328097813583171335nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_278_sndI,axiom,
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc3162348030201620241nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_279_eq__fst__iff,axiom,
    ! [A: nat,P: produc9133624956312949779et_int] :
      ( ( A
        = ( produc9078099356025976375et_int @ P ) )
      = ( ? [B5: set_int] :
            ( P
            = ( produc29655638201817675et_int @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_280_eq__fst__iff,axiom,
    ! [A: int,P: produc661532565036771336nt_int] :
      ( ( A
        = ( produc3131058726980325326nt_int @ P ) )
      = ( ? [B5: list_P5707943133018811711nt_int] :
            ( P
            = ( produc8814303788642274490nt_int @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_281_eq__fst__iff,axiom,
    ! [A: int,P: product_prod_int_int] :
      ( ( A
        = ( product_fst_int_int @ P ) )
      = ( ? [B5: int] :
            ( P
            = ( product_Pair_int_int @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_282_eq__fst__iff,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,P: produc2285326912895808259nt_int] :
      ( ( A
        = ( produc6816164490631068361nt_int @ P ) )
      = ( ? [B5: product_prod_int_int] :
            ( P
            = ( produc5700946648718959541nt_int @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_283_eq__fst__iff,axiom,
    ! [A: int > option6357759511663192854e_term,P: produc7773217078559923341nt_int] :
      ( ( A
        = ( produc6230002227079971283nt_int @ P ) )
      = ( ? [B5: product_prod_int_int] :
            ( P
            = ( produc4305682042979456191nt_int @ A @ B5 ) ) ) ) ).

% eq_fst_iff
thf(fact_284_fstI,axiom,
    ! [X: produc9133624956312949779et_int,Y: nat,Z: set_int] :
      ( ( X
        = ( produc29655638201817675et_int @ Y @ Z ) )
     => ( ( produc9078099356025976375et_int @ X )
        = Y ) ) ).

% fstI
thf(fact_285_fstI,axiom,
    ! [X: produc661532565036771336nt_int,Y: int,Z: list_P5707943133018811711nt_int] :
      ( ( X
        = ( produc8814303788642274490nt_int @ Y @ Z ) )
     => ( ( produc3131058726980325326nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_286_fstI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_fst_int_int @ X )
        = Y ) ) ).

% fstI
thf(fact_287_fstI,axiom,
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc6816164490631068361nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_288_fstI,axiom,
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc6230002227079971283nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_289_last__snoc,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( last_P3305686521732843992nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
      = X ) ).

% last_snoc
thf(fact_290_last__snoc,axiom,
    ! [Xs: list_int,X: int] :
      ( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
      = X ) ).

% last_snoc
thf(fact_291_asymI,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
         => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X3 ) @ R ) )
     => ( asym_on_int @ top_top_set_int @ R ) ) ).

% asymI
thf(fact_292_list_Oinject,axiom,
    ! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int,Y21: product_prod_int_int,Y22: list_P5707943133018811711nt_int] :
      ( ( ( cons_P3334398858971670639nt_int @ X21 @ X22 )
        = ( cons_P3334398858971670639nt_int @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_293_list_Oinject,axiom,
    ! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
      ( ( ( cons_int @ X21 @ X22 )
        = ( cons_int @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_294_asym__onI,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat2 @ X3 @ A4 )
         => ( ( member_nat2 @ Y3 @ A4 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_on_nat @ A4 @ R ) ) ).

% asym_onI
thf(fact_295_asym__onI,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member_int2 @ X3 @ A4 )
         => ( ( member_int2 @ Y3 @ A4 )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
             => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_on_int @ A4 @ R ) ) ).

% asym_onI
thf(fact_296_append1__eq__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
        = ( append7030698103840186580nt_int @ Ys @ ( cons_P3334398858971670639nt_int @ Y @ nil_Pr2300489316682597567nt_int ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_297_append1__eq__conv,axiom,
    ! [Xs: list_int,X: int,Ys: list_int,Y: int] :
      ( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
        = ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_298_bind__simps_I2_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,F2: product_prod_int_int > list_int] :
      ( ( bind_P1074787401898827934nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ F2 )
      = ( append_int @ ( F2 @ X ) @ ( bind_P1074787401898827934nt_int @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_299_bind__simps_I2_J,axiom,
    ! [X: int,Xs: list_int,F2: int > list_int] :
      ( ( bind_int_int @ ( cons_int @ X @ Xs ) @ F2 )
      = ( append_int @ ( F2 @ X ) @ ( bind_int_int @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_300_List_Otranspose_Ocases,axiom,
    ! [X: list_l1670014477004246597nt_int] :
      ( ( X != nil_li8670148097206105925nt_int )
     => ( ! [Xss: list_l1670014477004246597nt_int] :
            ( X
           != ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss ) )
       => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Xss: list_l1670014477004246597nt_int] :
              ( X
             != ( cons_l7309679040211256053nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% List.transpose.cases
thf(fact_301_List_Otranspose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] :
            ( X
           != ( cons_list_int @ nil_int @ Xss ) )
       => ~ ! [X3: int,Xs3: list_int,Xss: list_list_int] :
              ( X
             != ( cons_list_int @ ( cons_int @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% List.transpose.cases
thf(fact_302_not__Cons__self2,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( cons_P3334398858971670639nt_int @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_303_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] :
      ( ( cons_int @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_304_list__nonempty__induct,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > $o] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ( ! [X3: product_prod_int_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) )
       => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
              ( ( Xs3 != nil_Pr2300489316682597567nt_int )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_305_list__nonempty__induct,axiom,
    ! [Xs: list_int,P2: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X3: int] : ( P2 @ ( cons_int @ X3 @ nil_int ) )
       => ( ! [X3: int,Xs3: list_int] :
              ( ( Xs3 != nil_int )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_int @ X3 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_306_list__induct2_H,axiom,
    ! [P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
     => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ nil_Pr2300489316682597567nt_int )
       => ( ! [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] : ( P2 @ nil_Pr2300489316682597567nt_int @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
                ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_307_list__induct2_H,axiom,
    ! [P2: list_P5707943133018811711nt_int > list_int > $o,Xs: list_P5707943133018811711nt_int,Ys: list_int] :
      ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_int )
     => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ nil_int )
       => ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_Pr2300489316682597567nt_int @ ( cons_int @ Y3 @ Ys3 ) )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: int,Ys3: list_int] :
                ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_308_list__induct2_H,axiom,
    ! [P2: list_int > list_P5707943133018811711nt_int > $o,Xs: list_int,Ys: list_P5707943133018811711nt_int] :
      ( ( P2 @ nil_int @ nil_Pr2300489316682597567nt_int )
     => ( ! [X3: int,Xs3: list_int] : ( P2 @ ( cons_int @ X3 @ Xs3 ) @ nil_Pr2300489316682597567nt_int )
       => ( ! [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] : ( P2 @ nil_int @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
         => ( ! [X3: int,Xs3: list_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
                ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_309_list__induct2_H,axiom,
    ! [P2: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
      ( ( P2 @ nil_int @ nil_int )
     => ( ! [X3: int,Xs3: list_int] : ( P2 @ ( cons_int @ X3 @ Xs3 ) @ nil_int )
       => ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_int @ ( cons_int @ Y3 @ Ys3 ) )
         => ( ! [X3: int,Xs3: list_int,Y3: int,Ys3: list_int] :
                ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_310_neq__Nil__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
      = ( ? [Y5: product_prod_int_int,Ys4: list_P5707943133018811711nt_int] :
            ( Xs
            = ( cons_P3334398858971670639nt_int @ Y5 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_311_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
      = ( ? [Y5: int,Ys4: list_int] :
            ( Xs
            = ( cons_int @ Y5 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_312_successively_Ocases,axiom,
    ! [X: produc1050408459402128056nt_int] :
      ( ! [P4: product_prod_int_int > product_prod_int_int > $o] :
          ( X
         != ( produc3328129369365053992nt_int @ P4 @ nil_Pr2300489316682597567nt_int ) )
     => ( ! [P4: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int] :
            ( X
           != ( produc3328129369365053992nt_int @ P4 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) ) )
       => ~ ! [P4: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,Y3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
              ( X
             != ( produc3328129369365053992nt_int @ P4 @ ( cons_P3334398858971670639nt_int @ X3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_313_successively_Ocases,axiom,
    ! [X: produc5834231552977413017st_int] :
      ( ! [P4: int > int > $o] :
          ( X
         != ( produc8618682346314911123st_int @ P4 @ nil_int ) )
     => ( ! [P4: int > int > $o,X3: int] :
            ( X
           != ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ nil_int ) ) )
       => ~ ! [P4: int > int > $o,X3: int,Y3: int,Xs3: list_int] :
              ( X
             != ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_314_remdups__adj_Ocases,axiom,
    ! [X: list_P5707943133018811711nt_int] :
      ( ( X != nil_Pr2300489316682597567nt_int )
     => ( ! [X3: product_prod_int_int] :
            ( X
           != ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) )
       => ~ ! [X3: product_prod_int_int,Y3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
              ( X
             != ( cons_P3334398858971670639nt_int @ X3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_315_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X3: int] :
            ( X
           != ( cons_int @ X3 @ nil_int ) )
       => ~ ! [X3: int,Y3: int,Xs3: list_int] :
              ( X
             != ( cons_int @ X3 @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_316_sorted__wrt_Ocases,axiom,
    ! [X: produc1050408459402128056nt_int] :
      ( ! [P4: product_prod_int_int > product_prod_int_int > $o] :
          ( X
         != ( produc3328129369365053992nt_int @ P4 @ nil_Pr2300489316682597567nt_int ) )
     => ~ ! [P4: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
            ( X
           != ( produc3328129369365053992nt_int @ P4 @ ( cons_P3334398858971670639nt_int @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_317_sorted__wrt_Ocases,axiom,
    ! [X: produc5834231552977413017st_int] :
      ( ! [P4: int > int > $o] :
          ( X
         != ( produc8618682346314911123st_int @ P4 @ nil_int ) )
     => ~ ! [P4: int > int > $o,X3: int,Ys3: list_int] :
            ( X
           != ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_318_shuffles_Ocases,axiom,
    ! [X: produc1089560213143673063nt_int] :
      ( ! [Ys3: list_P5707943133018811711nt_int] :
          ( X
         != ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys3 ) )
     => ( ! [Xs3: list_P5707943133018811711nt_int] :
            ( X
           != ( produc1932183703851549015nt_int @ Xs3 @ nil_Pr2300489316682597567nt_int ) )
       => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
              ( X
             != ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_319_shuffles_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [Ys3: list_int] :
          ( X
         != ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
     => ( ! [Xs3: list_int] :
            ( X
           != ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
       => ~ ! [X3: int,Xs3: list_int,Y3: int,Ys3: list_int] :
              ( X
             != ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_320_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X3: int,Xs3: list_int] :
          ( X
         != ( cons_int @ X3 @ Xs3 ) )
     => ( X = nil_int ) ) ).

% min_list.cases
thf(fact_321_splice_Ocases,axiom,
    ! [X: produc1089560213143673063nt_int] :
      ( ! [Ys3: list_P5707943133018811711nt_int] :
          ( X
         != ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys3 ) )
     => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Ys3: list_P5707943133018811711nt_int] :
            ( X
           != ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ Ys3 ) ) ) ).

% splice.cases
thf(fact_322_splice_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [Ys3: list_int] :
          ( X
         != ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
     => ~ ! [X3: int,Xs3: list_int,Ys3: list_int] :
            ( X
           != ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs3 ) @ Ys3 ) ) ) ).

% splice.cases
thf(fact_323_list_Oexhaust,axiom,
    ! [Y: list_P5707943133018811711nt_int] :
      ( ( Y != nil_Pr2300489316682597567nt_int )
     => ~ ! [X212: product_prod_int_int,X222: list_P5707943133018811711nt_int] :
            ( Y
           != ( cons_P3334398858971670639nt_int @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_324_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X222: list_int] :
            ( Y
           != ( cons_int @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_325_list_OdiscI,axiom,
    ! [List: list_P5707943133018811711nt_int,X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
      ( ( List
        = ( cons_P3334398858971670639nt_int @ X21 @ X22 ) )
     => ( List != nil_Pr2300489316682597567nt_int ) ) ).

% list.discI
thf(fact_326_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X22: list_int] :
      ( ( List
        = ( cons_int @ X21 @ X22 ) )
     => ( List != nil_int ) ) ).

% list.discI
thf(fact_327_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] :
      ( nil_Pr2300489316682597567nt_int
     != ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_328_list_Odistinct_I1_J,axiom,
    ! [X21: int,X22: list_int] :
      ( nil_int
     != ( cons_int @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_329_append__Cons,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys )
      = ( cons_P3334398858971670639nt_int @ X @ ( append7030698103840186580nt_int @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_330_append__Cons,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] :
      ( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
      = ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_331_Cons__eq__appendI,axiom,
    ! [X: product_prod_int_int,Xs1: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
      ( ( ( cons_P3334398858971670639nt_int @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append7030698103840186580nt_int @ Xs1 @ Zs ) )
       => ( ( cons_P3334398858971670639nt_int @ X @ Xs )
          = ( append7030698103840186580nt_int @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_332_Cons__eq__appendI,axiom,
    ! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
      ( ( ( cons_int @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_int @ Xs1 @ Zs ) )
       => ( ( cons_int @ X @ Xs )
          = ( append_int @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_333_path__checker_Ocases,axiom,
    ! [X: produc2007852851243229709nt_int] :
      ( ! [B3: set_Pr958786334691620121nt_int] :
          ( X
         != ( produc2261658324281137661nt_int @ B3 @ nil_Pr2300489316682597567nt_int ) )
     => ( ! [B3: set_Pr958786334691620121nt_int,S_i: product_prod_int_int] :
            ( X
           != ( produc2261658324281137661nt_int @ B3 @ ( cons_P3334398858971670639nt_int @ S_i @ nil_Pr2300489316682597567nt_int ) ) )
       => ~ ! [B3: set_Pr958786334691620121nt_int,S_i: product_prod_int_int,S_j: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
              ( X
             != ( produc2261658324281137661nt_int @ B3 @ ( cons_P3334398858971670639nt_int @ S_i @ ( cons_P3334398858971670639nt_int @ S_j @ Ps2 ) ) ) ) ) ) ).

% path_checker.cases
thf(fact_334_mirror2__aux_Ocases,axiom,
    ! [X: produc661532565036771336nt_int] :
      ( ! [M3: int] :
          ( X
         != ( produc8814303788642274490nt_int @ M3 @ nil_Pr2300489316682597567nt_int ) )
     => ~ ! [M3: int,S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
            ( X
           != ( produc8814303788642274490nt_int @ M3 @ ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) ) ) ) ).

% mirror2_aux.cases
thf(fact_335_KnightsTour_Otranspose_Ocases,axiom,
    ! [X: list_P5707943133018811711nt_int] :
      ( ( X != nil_Pr2300489316682597567nt_int )
     => ~ ! [S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
            ( X
           != ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) ) ) ).

% KnightsTour.transpose.cases
thf(fact_336_asym__onD,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ A4 @ R )
     => ( ( member_nat2 @ X @ A4 )
       => ( ( member_nat2 @ Y @ A4 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_337_asym__onD,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ( asym_on_int @ A4 @ R )
     => ( ( member_int2 @ X @ A4 )
       => ( ( member_int2 @ Y @ A4 )
         => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_338_asym__on__def,axiom,
    ( asym_on_int
    = ( ^ [A6: set_int,R3: set_Pr958786334691620121nt_int] :
        ! [X4: int] :
          ( ( member_int2 @ X4 @ A6 )
         => ! [Y5: int] :
              ( ( member_int2 @ Y5 @ A6 )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R3 )
               => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_339_null__rec_I1_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ~ ( null_P4123483736518020807nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_340_null__rec_I1_J,axiom,
    ! [X: int,Xs: list_int] :
      ~ ( null_int @ ( cons_int @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_341_member__rec_I1_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( member2925713097805433300nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member2925713097805433300nt_int @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_342_member__rec_I1_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( member_int @ ( cons_int @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_int @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_343_rev__nonempty__induct,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > $o] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ( ! [X3: product_prod_int_int] : ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) )
       => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
              ( ( Xs3 != nil_Pr2300489316682597567nt_int )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( append7030698103840186580nt_int @ Xs3 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_344_rev__nonempty__induct,axiom,
    ! [Xs: list_int,P2: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X3: int] : ( P2 @ ( cons_int @ X3 @ nil_int ) )
       => ( ! [X3: int,Xs3: list_int] :
              ( ( Xs3 != nil_int )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( append_int @ Xs3 @ ( cons_int @ X3 @ nil_int ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_345_append__eq__Cons__conv,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( ( append7030698103840186580nt_int @ Ys @ Zs )
        = ( cons_P3334398858971670639nt_int @ X @ Xs ) )
      = ( ( ( Ys = nil_Pr2300489316682597567nt_int )
          & ( Zs
            = ( cons_P3334398858971670639nt_int @ X @ Xs ) ) )
        | ? [Ys5: list_P5707943133018811711nt_int] :
            ( ( Ys
              = ( cons_P3334398858971670639nt_int @ X @ Ys5 ) )
            & ( ( append7030698103840186580nt_int @ Ys5 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_346_append__eq__Cons__conv,axiom,
    ! [Ys: list_int,Zs: list_int,X: int,Xs: list_int] :
      ( ( ( append_int @ Ys @ Zs )
        = ( cons_int @ X @ Xs ) )
      = ( ( ( Ys = nil_int )
          & ( Zs
            = ( cons_int @ X @ Xs ) ) )
        | ? [Ys5: list_int] :
            ( ( Ys
              = ( cons_int @ X @ Ys5 ) )
            & ( ( append_int @ Ys5 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_347_Cons__eq__append__conv,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
      ( ( ( cons_P3334398858971670639nt_int @ X @ Xs )
        = ( append7030698103840186580nt_int @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Pr2300489316682597567nt_int )
          & ( ( cons_P3334398858971670639nt_int @ X @ Xs )
            = Zs ) )
        | ? [Ys5: list_P5707943133018811711nt_int] :
            ( ( ( cons_P3334398858971670639nt_int @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append7030698103840186580nt_int @ Ys5 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_348_Cons__eq__append__conv,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ( ( cons_int @ X @ Xs )
        = ( append_int @ Ys @ Zs ) )
      = ( ( ( Ys = nil_int )
          & ( ( cons_int @ X @ Xs )
            = Zs ) )
        | ? [Ys5: list_int] :
            ( ( ( cons_int @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_int @ Ys5 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_349_rev__exhaust,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ~ ! [Ys3: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
            ( Xs
           != ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ Y3 @ nil_Pr2300489316682597567nt_int ) ) ) ) ).

% rev_exhaust
thf(fact_350_rev__exhaust,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ~ ! [Ys3: list_int,Y3: int] :
            ( Xs
           != ( append_int @ Ys3 @ ( cons_int @ Y3 @ nil_int ) ) ) ) ).

% rev_exhaust
thf(fact_351_rev__induct,axiom,
    ! [P2: list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int] :
      ( ( P2 @ nil_Pr2300489316682597567nt_int )
     => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
            ( ( P2 @ Xs3 )
           => ( P2 @ ( append7030698103840186580nt_int @ Xs3 @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_352_rev__induct,axiom,
    ! [P2: list_int > $o,Xs: list_int] :
      ( ( P2 @ nil_int )
     => ( ! [X3: int,Xs3: list_int] :
            ( ( P2 @ Xs3 )
           => ( P2 @ ( append_int @ Xs3 @ ( cons_int @ X3 @ nil_int ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_353_last__ConsR,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
        = ( last_P3305686521732843992nt_int @ Xs ) ) ) ).

% last_ConsR
thf(fact_354_last__ConsR,axiom,
    ! [Xs: list_int,X: int] :
      ( ( Xs != nil_int )
     => ( ( last_int @ ( cons_int @ X @ Xs ) )
        = ( last_int @ Xs ) ) ) ).

% last_ConsR
thf(fact_355_last__ConsL,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( Xs = nil_Pr2300489316682597567nt_int )
     => ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_356_last__ConsL,axiom,
    ! [Xs: list_int,X: int] :
      ( ( Xs = nil_int )
     => ( ( last_int @ ( cons_int @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_357_last_Osimps,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( Xs = nil_Pr2300489316682597567nt_int )
       => ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_Pr2300489316682597567nt_int )
       => ( ( last_P3305686521732843992nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
          = ( last_P3305686521732843992nt_int @ Xs ) ) ) ) ).

% last.simps
thf(fact_358_last_Osimps,axiom,
    ! [Xs: list_int,X: int] :
      ( ( ( Xs = nil_int )
       => ( ( last_int @ ( cons_int @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_int )
       => ( ( last_int @ ( cons_int @ X @ Xs ) )
          = ( last_int @ Xs ) ) ) ) ).

% last.simps
thf(fact_359_asymD,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ( asym_on_int @ top_top_set_int @ R )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
       => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_360_asym__iff,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ( asym_on_int @ top_top_set_int @ R )
      = ( ! [X4: int,Y5: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_361_maps__simps_I1_J,axiom,
    ! [F2: product_prod_int_int > list_int,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( maps_P493294916530037604nt_int @ F2 @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
      = ( append_int @ ( F2 @ X ) @ ( maps_P493294916530037604nt_int @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_362_maps__simps_I1_J,axiom,
    ! [F2: int > list_int,X: int,Xs: list_int] :
      ( ( maps_int_int @ F2 @ ( cons_int @ X @ Xs ) )
      = ( append_int @ ( F2 @ X ) @ ( maps_int_int @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_363_SuccD,axiom,
    ! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
      ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
     => ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_364_SuccD,axiom,
    ! [K: product_prod_int_int,Kl: set_li3855193629254082847nt_int,Kl2: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ K @ ( bNF_Gr2185362013987207252nt_int @ Kl @ Kl2 ) )
     => ( member2764346250752101224nt_int @ ( append7030698103840186580nt_int @ Kl2 @ ( cons_P3334398858971670639nt_int @ K @ nil_Pr2300489316682597567nt_int ) ) @ Kl ) ) ).

% SuccD
thf(fact_365_SuccD,axiom,
    ! [K: int,Kl: set_list_int,Kl2: list_int] :
      ( ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ Kl2 ) )
     => ( member_list_int @ ( append_int @ Kl2 @ ( cons_int @ K @ nil_int ) ) @ Kl ) ) ).

% SuccD
thf(fact_366_SuccI,axiom,
    ! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
     => ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_367_SuccI,axiom,
    ! [Kl2: list_P5707943133018811711nt_int,K: product_prod_int_int,Kl: set_li3855193629254082847nt_int] :
      ( ( member2764346250752101224nt_int @ ( append7030698103840186580nt_int @ Kl2 @ ( cons_P3334398858971670639nt_int @ K @ nil_Pr2300489316682597567nt_int ) ) @ Kl )
     => ( member5262025264175285858nt_int @ K @ ( bNF_Gr2185362013987207252nt_int @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_368_SuccI,axiom,
    ! [Kl2: list_int,K: int,Kl: set_list_int] :
      ( ( member_list_int @ ( append_int @ Kl2 @ ( cons_int @ K @ nil_int ) ) @ Kl )
     => ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_369_append__butlast__last__id,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ( ( append7030698103840186580nt_int @ ( butlas2390654596765537291nt_int @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( last_P3305686521732843992nt_int @ Xs ) @ nil_Pr2300489316682597567nt_int ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_370_append__butlast__last__id,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ( ( append_int @ ( butlast_int @ Xs ) @ ( cons_int @ ( last_int @ Xs ) @ nil_int ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_371_mirror2__aux_Oelims,axiom,
    ! [X: int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( mirror2_aux @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
         => ( Y != nil_Pr2300489316682597567nt_int ) )
       => ~ ! [S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
              ( ( Xa
                = ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) )
             => ( Y
               != ( cons_P3334398858971670639nt_int @ ( mirror2_square @ X @ S_i ) @ ( mirror2_aux @ X @ Ps2 ) ) ) ) ) ) ).

% mirror2_aux.elims
thf(fact_372_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) )
        = Ys )
      = ( ( Ys != nil_Pr2300489316682597567nt_int )
        & ( ( butlas2390654596765537291nt_int @ Ys )
          = Xs )
        & ( ( last_P3305686521732843992nt_int @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_373_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] :
      ( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
        = Ys )
      = ( ( Ys != nil_int )
        & ( ( butlast_int @ Ys )
          = Xs )
        & ( ( last_int @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_374_insert__Nil,axiom,
    ! [X: product_prod_int_int] :
      ( ( insert5765537519290168021nt_int @ X @ nil_Pr2300489316682597567nt_int )
      = ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ).

% insert_Nil
thf(fact_375_insert__Nil,axiom,
    ! [X: int] :
      ( ( insert_int @ X @ nil_int )
      = ( cons_int @ X @ nil_int ) ) ).

% insert_Nil
thf(fact_376_mirror1__aux_Oelims,axiom,
    ! [X: int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( mirror1_aux @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
         => ( Y != nil_Pr2300489316682597567nt_int ) )
       => ~ ! [S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
              ( ( Xa
                = ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) )
             => ( Y
               != ( cons_P3334398858971670639nt_int @ ( mirror1_square @ X @ S_i ) @ ( mirror1_aux @ X @ Ps2 ) ) ) ) ) ) ).

% mirror1_aux.elims
thf(fact_377_butlast__snoc,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( butlas2390654596765537291nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_378_butlast__snoc,axiom,
    ! [Xs: list_int,X: int] :
      ( ( butlast_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_379_butlast_Osimps_I1_J,axiom,
    ( ( butlas2390654596765537291nt_int @ nil_Pr2300489316682597567nt_int )
    = nil_Pr2300489316682597567nt_int ) ).

% butlast.simps(1)
thf(fact_380_butlast_Osimps_I1_J,axiom,
    ( ( butlast_int @ nil_int )
    = nil_int ) ).

% butlast.simps(1)
thf(fact_381_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( Xs = nil_Pr2300489316682597567nt_int )
       => ( ( butlas2390654596765537291nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
          = nil_Pr2300489316682597567nt_int ) )
      & ( ( Xs != nil_Pr2300489316682597567nt_int )
       => ( ( butlas2390654596765537291nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
          = ( cons_P3334398858971670639nt_int @ X @ ( butlas2390654596765537291nt_int @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_382_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_int,X: int] :
      ( ( ( Xs = nil_int )
       => ( ( butlast_int @ ( cons_int @ X @ Xs ) )
          = nil_int ) )
      & ( ( Xs != nil_int )
       => ( ( butlast_int @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( butlast_int @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_383_butlast__append,axiom,
    ! [Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( ( Ys = nil_Pr2300489316682597567nt_int )
       => ( ( butlas2390654596765537291nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
          = ( butlas2390654596765537291nt_int @ Xs ) ) )
      & ( ( Ys != nil_Pr2300489316682597567nt_int )
       => ( ( butlas2390654596765537291nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
          = ( append7030698103840186580nt_int @ Xs @ ( butlas2390654596765537291nt_int @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_384_butlast__append,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( ( Ys = nil_int )
       => ( ( butlast_int @ ( append_int @ Xs @ Ys ) )
          = ( butlast_int @ Xs ) ) )
      & ( ( Ys != nil_int )
       => ( ( butlast_int @ ( append_int @ Xs @ Ys ) )
          = ( append_int @ Xs @ ( butlast_int @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_385_mirror1__aux_Osimps_I2_J,axiom,
    ! [N: int,S_i2: product_prod_int_int,Ps: list_P5707943133018811711nt_int] :
      ( ( mirror1_aux @ N @ ( cons_P3334398858971670639nt_int @ S_i2 @ Ps ) )
      = ( cons_P3334398858971670639nt_int @ ( mirror1_square @ N @ S_i2 ) @ ( mirror1_aux @ N @ Ps ) ) ) ).

% mirror1_aux.simps(2)
thf(fact_386_mirror2__aux_Osimps_I2_J,axiom,
    ! [M: int,S_i2: product_prod_int_int,Ps: list_P5707943133018811711nt_int] :
      ( ( mirror2_aux @ M @ ( cons_P3334398858971670639nt_int @ S_i2 @ Ps ) )
      = ( cons_P3334398858971670639nt_int @ ( mirror2_square @ M @ S_i2 ) @ ( mirror2_aux @ M @ Ps ) ) ) ).

% mirror2_aux.simps(2)
thf(fact_387_empty__Shift,axiom,
    ! [Kl: set_list_nat,K: nat] :
      ( ( member_list_nat @ nil_nat @ Kl )
     => ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_388_empty__Shift,axiom,
    ! [Kl: set_li3855193629254082847nt_int,K: product_prod_int_int] :
      ( ( member2764346250752101224nt_int @ nil_Pr2300489316682597567nt_int @ Kl )
     => ( ( member5262025264175285858nt_int @ K @ ( bNF_Gr2185362013987207252nt_int @ Kl @ nil_Pr2300489316682597567nt_int ) )
       => ( member2764346250752101224nt_int @ nil_Pr2300489316682597567nt_int @ ( bNF_Gr9175161896313899224nt_int @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_389_empty__Shift,axiom,
    ! [Kl: set_list_int,K: int] :
      ( ( member_list_int @ nil_int @ Kl )
     => ( ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ nil_int ) )
       => ( member_list_int @ nil_int @ ( bNF_Gr1870224194279859149ft_int @ Kl @ K ) ) ) ) ).

% empty_Shift
thf(fact_390_product__lists_Osimps_I1_J,axiom,
    ( ( produc5568053154996169768nt_int @ nil_li8670148097206105925nt_int )
    = ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).

% product_lists.simps(1)
thf(fact_391_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_int @ nil_list_int )
    = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% product_lists.simps(1)
thf(fact_392_subseqs_Osimps_I1_J,axiom,
    ( ( subseq1357044202310323342nt_int @ nil_Pr2300489316682597567nt_int )
    = ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).

% subseqs.simps(1)
thf(fact_393_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_int @ nil_int )
    = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% subseqs.simps(1)
thf(fact_394_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) @ ( append7030698103840186580nt_int @ Ys @ ( cons_P3334398858971670639nt_int @ Y @ nil_Pr2300489316682597567nt_int ) ) ) @ ( listre1649617614542993432nt_int @ R ) )
      = ( ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_395_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_int,X: int,Ys: list_int,Y: int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) @ ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) ) @ ( listrel1_int @ R ) )
      = ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_396_mirror2__aux_Opelims,axiom,
    ! [X: int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( mirror2_aux @ X @ Xa )
        = Y )
     => ( ( accp_P2804173024491580863nt_int @ mirror2_aux_rel @ ( produc8814303788642274490nt_int @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
           => ( ( Y = nil_Pr2300489316682597567nt_int )
             => ~ ( accp_P2804173024491580863nt_int @ mirror2_aux_rel @ ( produc8814303788642274490nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) )
         => ~ ! [S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
                ( ( Xa
                  = ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) )
               => ( ( Y
                    = ( cons_P3334398858971670639nt_int @ ( mirror2_square @ X @ S_i ) @ ( mirror2_aux @ X @ Ps2 ) ) )
                 => ~ ( accp_P2804173024491580863nt_int @ mirror2_aux_rel @ ( produc8814303788642274490nt_int @ X @ ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) ) ) ) ) ) ) ) ).

% mirror2_aux.pelims
thf(fact_397_mirror1__aux_Opelims,axiom,
    ! [X: int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( mirror1_aux @ X @ Xa )
        = Y )
     => ( ( accp_P2804173024491580863nt_int @ mirror1_aux_rel @ ( produc8814303788642274490nt_int @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
           => ( ( Y = nil_Pr2300489316682597567nt_int )
             => ~ ( accp_P2804173024491580863nt_int @ mirror1_aux_rel @ ( produc8814303788642274490nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) )
         => ~ ! [S_i: product_prod_int_int,Ps2: list_P5707943133018811711nt_int] :
                ( ( Xa
                  = ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) )
               => ( ( Y
                    = ( cons_P3334398858971670639nt_int @ ( mirror1_square @ X @ S_i ) @ ( mirror1_aux @ X @ Ps2 ) ) )
                 => ~ ( accp_P2804173024491580863nt_int @ mirror1_aux_rel @ ( produc8814303788642274490nt_int @ X @ ( cons_P3334398858971670639nt_int @ S_i @ Ps2 ) ) ) ) ) ) ) ) ).

% mirror1_aux.pelims
thf(fact_398_concat__eq__append__conv,axiom,
    ! [Xss2: list_l1670014477004246597nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int] :
      ( ( ( concat4512918505337516154nt_int @ Xss2 )
        = ( append7030698103840186580nt_int @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_li8670148097206105925nt_int )
         => ( ( Ys = nil_Pr2300489316682597567nt_int )
            & ( Zs = nil_Pr2300489316682597567nt_int ) ) )
        & ( ( Xss2 != nil_li8670148097206105925nt_int )
         => ? [Xss1: list_l1670014477004246597nt_int,Xs4: list_P5707943133018811711nt_int,Xs5: list_P5707943133018811711nt_int,Xss22: list_l1670014477004246597nt_int] :
              ( ( Xss2
                = ( append1237393811882493146nt_int @ Xss1 @ ( cons_l7309679040211256053nt_int @ ( append7030698103840186580nt_int @ Xs4 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append7030698103840186580nt_int @ ( concat4512918505337516154nt_int @ Xss1 ) @ Xs4 ) )
              & ( Zs
                = ( append7030698103840186580nt_int @ Xs5 @ ( concat4512918505337516154nt_int @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_399_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_int,Ys: list_int,Zs: list_int] :
      ( ( ( concat_int @ Xss2 )
        = ( append_int @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_list_int )
         => ( ( Ys = nil_int )
            & ( Zs = nil_int ) ) )
        & ( ( Xss2 != nil_list_int )
         => ? [Xss1: list_list_int,Xs4: list_int,Xs5: list_int,Xss22: list_list_int] :
              ( ( Xss2
                = ( append_list_int @ Xss1 @ ( cons_list_int @ ( append_int @ Xs4 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_int @ ( concat_int @ Xss1 ) @ Xs4 ) )
              & ( Zs
                = ( append_int @ Xs5 @ ( concat_int @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_400_member__remove,axiom,
    ! [X: nat,Y: nat,A4: set_nat] :
      ( ( member_nat2 @ X @ ( remove_nat @ Y @ A4 ) )
      = ( ( member_nat2 @ X @ A4 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_401_concat__append,axiom,
    ! [Xs: list_list_int,Ys: list_list_int] :
      ( ( concat_int @ ( append_list_int @ Xs @ Ys ) )
      = ( append_int @ ( concat_int @ Xs ) @ ( concat_int @ Ys ) ) ) ).

% concat_append
thf(fact_402_Cons__listrel1__Cons,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre1649617614542993432nt_int @ R ) )
      = ( ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_403_Cons__listrel1__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_404_accp_Ocases,axiom,
    ! [R: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,A: produc661532565036771336nt_int] :
      ( ( accp_P2804173024491580863nt_int @ R @ A )
     => ! [Y6: produc661532565036771336nt_int] :
          ( ( R @ Y6 @ A )
         => ( accp_P2804173024491580863nt_int @ R @ Y6 ) ) ) ).

% accp.cases
thf(fact_405_accp_Ocases,axiom,
    ! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int] :
      ( ( accp_P1096762738010456898nt_int @ R @ A )
     => ! [Y6: product_prod_int_int] :
          ( ( R @ Y6 @ A )
         => ( accp_P1096762738010456898nt_int @ R @ Y6 ) ) ) ).

% accp.cases
thf(fact_406_accp_Osimps,axiom,
    ( accp_P2804173024491580863nt_int
    = ( ^ [R3: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,A5: produc661532565036771336nt_int] :
        ? [X4: produc661532565036771336nt_int] :
          ( ( A5 = X4 )
          & ! [Y5: produc661532565036771336nt_int] :
              ( ( R3 @ Y5 @ X4 )
             => ( accp_P2804173024491580863nt_int @ R3 @ Y5 ) ) ) ) ) ).

% accp.simps
thf(fact_407_accp_Osimps,axiom,
    ( accp_P1096762738010456898nt_int
    = ( ^ [R3: product_prod_int_int > product_prod_int_int > $o,A5: product_prod_int_int] :
        ? [X4: product_prod_int_int] :
          ( ( A5 = X4 )
          & ! [Y5: product_prod_int_int] :
              ( ( R3 @ Y5 @ X4 )
             => ( accp_P1096762738010456898nt_int @ R3 @ Y5 ) ) ) ) ) ).

% accp.simps
thf(fact_408_accpI,axiom,
    ! [R: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,X: produc661532565036771336nt_int] :
      ( ! [Y3: produc661532565036771336nt_int] :
          ( ( R @ Y3 @ X )
         => ( accp_P2804173024491580863nt_int @ R @ Y3 ) )
     => ( accp_P2804173024491580863nt_int @ R @ X ) ) ).

% accpI
thf(fact_409_accpI,axiom,
    ! [R: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
      ( ! [Y3: product_prod_int_int] :
          ( ( R @ Y3 @ X )
         => ( accp_P1096762738010456898nt_int @ R @ Y3 ) )
     => ( accp_P1096762738010456898nt_int @ R @ X ) ) ).

% accpI
thf(fact_410_accp__induct,axiom,
    ! [R: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,A: produc661532565036771336nt_int,P2: produc661532565036771336nt_int > $o] :
      ( ( accp_P2804173024491580863nt_int @ R @ A )
     => ( ! [X3: produc661532565036771336nt_int] :
            ( ( accp_P2804173024491580863nt_int @ R @ X3 )
           => ( ! [Y6: produc661532565036771336nt_int] :
                  ( ( R @ Y6 @ X3 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_411_accp__induct,axiom,
    ! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P2: product_prod_int_int > $o] :
      ( ( accp_P1096762738010456898nt_int @ R @ A )
     => ( ! [X3: product_prod_int_int] :
            ( ( accp_P1096762738010456898nt_int @ R @ X3 )
           => ( ! [Y6: product_prod_int_int] :
                  ( ( R @ Y6 @ X3 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_412_accp__downward,axiom,
    ! [R: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,B: produc661532565036771336nt_int,A: produc661532565036771336nt_int] :
      ( ( accp_P2804173024491580863nt_int @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_P2804173024491580863nt_int @ R @ A ) ) ) ).

% accp_downward
thf(fact_413_accp__downward,axiom,
    ! [R: product_prod_int_int > product_prod_int_int > $o,B: product_prod_int_int,A: product_prod_int_int] :
      ( ( accp_P1096762738010456898nt_int @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_P1096762738010456898nt_int @ R @ A ) ) ) ).

% accp_downward
thf(fact_414_not__accp__down,axiom,
    ! [R2: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,X: produc661532565036771336nt_int] :
      ( ~ ( accp_P2804173024491580863nt_int @ R2 @ X )
     => ~ ! [Z2: produc661532565036771336nt_int] :
            ( ( R2 @ Z2 @ X )
           => ( accp_P2804173024491580863nt_int @ R2 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_415_not__accp__down,axiom,
    ! [R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
      ( ~ ( accp_P1096762738010456898nt_int @ R2 @ X )
     => ~ ! [Z2: product_prod_int_int] :
            ( ( R2 @ Z2 @ X )
           => ( accp_P1096762738010456898nt_int @ R2 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_416_accp__induct__rule,axiom,
    ! [R: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,A: produc661532565036771336nt_int,P2: produc661532565036771336nt_int > $o] :
      ( ( accp_P2804173024491580863nt_int @ R @ A )
     => ( ! [X3: produc661532565036771336nt_int] :
            ( ( accp_P2804173024491580863nt_int @ R @ X3 )
           => ( ! [Y6: produc661532565036771336nt_int] :
                  ( ( R @ Y6 @ X3 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_417_accp__induct__rule,axiom,
    ! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P2: product_prod_int_int > $o] :
      ( ( accp_P1096762738010456898nt_int @ R @ A )
     => ( ! [X3: product_prod_int_int] :
            ( ( accp_P1096762738010456898nt_int @ R @ X3 )
           => ( ! [Y6: product_prod_int_int] :
                  ( ( R @ Y6 @ X3 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_418_listrel1I2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,X: product_prod_int_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) )
     => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ X @ Ys ) ) @ ( listre1649617614542993432nt_int @ R ) ) ) ).

% listrel1I2
thf(fact_419_listrel1I2,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int,X: int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ X @ Ys ) ) @ ( listrel1_int @ R ) ) ) ).

% listrel1I2
thf(fact_420_not__listrel1__Nil,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( listre1649617614542993432nt_int @ R ) ) ).

% not_listrel1_Nil
thf(fact_421_not__listrel1__Nil,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ nil_int ) @ ( listrel1_int @ R ) ) ).

% not_listrel1_Nil
thf(fact_422_not__Nil__listrel1,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Xs ) @ ( listre1649617614542993432nt_int @ R ) ) ).

% not_Nil_listrel1
thf(fact_423_not__Nil__listrel1,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Xs ) @ ( listrel1_int @ R ) ) ).

% not_Nil_listrel1
thf(fact_424_append__listrel1I,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int,Us2: list_int,Vs: list_int] :
      ( ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
          & ( Us2 = Vs ) )
        | ( ( Xs = Ys )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Us2 @ Vs ) @ ( listrel1_int @ R ) ) ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Us2 ) @ ( append_int @ Ys @ Vs ) ) @ ( listrel1_int @ R ) ) ) ).

% append_listrel1I
thf(fact_425_concat_Osimps_I1_J,axiom,
    ( ( concat4512918505337516154nt_int @ nil_li8670148097206105925nt_int )
    = nil_Pr2300489316682597567nt_int ) ).

% concat.simps(1)
thf(fact_426_concat_Osimps_I1_J,axiom,
    ( ( concat_int @ nil_list_int )
    = nil_int ) ).

% concat.simps(1)
thf(fact_427_concat_Osimps_I2_J,axiom,
    ! [X: list_int,Xs: list_list_int] :
      ( ( concat_int @ ( cons_list_int @ X @ Xs ) )
      = ( append_int @ X @ ( concat_int @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_428_Cons__listrel1E2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre1649617614542993432nt_int @ R ) )
     => ( ! [X3: product_prod_int_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ X3 @ Ys ) )
           => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_P5707943133018811711nt_int] :
              ( ( Xs
                = ( cons_P3334398858971670639nt_int @ Y @ Zs2 ) )
             => ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Zs2 @ Ys ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_429_Cons__listrel1E2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R ) )
     => ( ! [X3: int] :
            ( ( Xs
              = ( cons_int @ X3 @ Ys ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_int] :
              ( ( Xs
                = ( cons_int @ Y @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_430_Cons__listrel1E1,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys ) @ ( listre1649617614542993432nt_int @ R ) )
     => ( ! [Y3: product_prod_int_int] :
            ( ( Ys
              = ( cons_P3334398858971670639nt_int @ Y3 @ Xs ) )
           => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_P5707943133018811711nt_int] :
              ( ( Ys
                = ( cons_P3334398858971670639nt_int @ X @ Zs2 ) )
             => ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Zs2 ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_431_Cons__listrel1E1,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys ) @ ( listrel1_int @ R ) )
     => ( ! [Y3: int] :
            ( ( Ys
              = ( cons_int @ Y3 @ Xs ) )
           => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_int] :
              ( ( Ys
                = ( cons_int @ X @ Zs2 ) )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_432_listrel1I1,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
     => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) @ ( listre1649617614542993432nt_int @ R ) ) ) ).

% listrel1I1
thf(fact_433_listrel1I1,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Xs ) ) @ ( listrel1_int @ R ) ) ) ).

% listrel1I1
thf(fact_434_concat__eq__appendD,axiom,
    ! [Xss2: list_list_int,Ys: list_int,Zs: list_int] :
      ( ( ( concat_int @ Xss2 )
        = ( append_int @ Ys @ Zs ) )
     => ( ( Xss2 != nil_list_int )
       => ? [Xss12: list_list_int,Xs3: list_int,Xs2: list_int,Xss23: list_list_int] :
            ( ( Xss2
              = ( append_list_int @ Xss12 @ ( cons_list_int @ ( append_int @ Xs3 @ Xs2 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_int @ ( concat_int @ Xss12 ) @ Xs3 ) )
            & ( Zs
              = ( append_int @ Xs2 @ ( concat_int @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_435_listrel1E,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) )
     => ~ ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
            ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ R )
           => ! [Us3: list_P5707943133018811711nt_int,Vs2: list_P5707943133018811711nt_int] :
                ( ( Xs
                  = ( append7030698103840186580nt_int @ Us3 @ ( cons_P3334398858971670639nt_int @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append7030698103840186580nt_int @ Us3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_436_listrel1E,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
           => ! [Us3: list_int,Vs2: list_int] :
                ( ( Xs
                  = ( append_int @ Us3 @ ( cons_int @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_int @ Us3 @ ( cons_int @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_437_listrel1I,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int,Us2: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
     => ( ( Xs
          = ( append7030698103840186580nt_int @ Us2 @ ( cons_P3334398858971670639nt_int @ X @ Vs ) ) )
       => ( ( Ys
            = ( append7030698103840186580nt_int @ Us2 @ ( cons_P3334398858971670639nt_int @ Y @ Vs ) ) )
         => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) ) ) ) ) ).

% listrel1I
thf(fact_438_listrel1I,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Us2: list_int,Vs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
     => ( ( Xs
          = ( append_int @ Us2 @ ( cons_int @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_int @ Us2 @ ( cons_int @ Y @ Vs ) ) )
         => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).

% listrel1I
thf(fact_439_lexord__append__left__rightI,axiom,
    ! [A: product_prod_int_int,B: product_prod_int_int,R: set_Pr2560585780119916871nt_int,U: list_P5707943133018811711nt_int,X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A @ B ) @ R )
     => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ U @ ( cons_P3334398858971670639nt_int @ A @ X ) ) @ ( append7030698103840186580nt_int @ U @ ( cons_P3334398858971670639nt_int @ B @ Y ) ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_440_lexord__append__left__rightI,axiom,
    ! [A: int,B: int,R: set_Pr958786334691620121nt_int,U: list_int,X: list_int,Y: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ U @ ( cons_int @ A @ X ) ) @ ( append_int @ U @ ( cons_int @ B @ Y ) ) ) @ ( lexord_int @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_441_listrel_Osimps,axiom,
    ! [A1: list_P5707943133018811711nt_int,A22: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ A1 @ A22 ) @ ( listre7525836092462517041nt_int @ R ) )
      = ( ( ( A1 = nil_Pr2300489316682597567nt_int )
          & ( A22 = nil_Pr2300489316682597567nt_int ) )
        | ? [X4: product_prod_int_int,Y5: product_prod_int_int,Xs4: list_P5707943133018811711nt_int,Ys4: list_P5707943133018811711nt_int] :
            ( ( A1
              = ( cons_P3334398858971670639nt_int @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_P3334398858971670639nt_int @ Y5 @ Ys4 ) )
            & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X4 @ Y5 ) @ R )
            & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs4 @ Ys4 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_442_listrel_Osimps,axiom,
    ! [A1: list_P5707943133018811711nt_int,A22: list_int,R: set_Pr53225590989397322nt_int] :
      ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ A1 @ A22 ) @ ( listre2687791572143342772nt_int @ R ) )
      = ( ( ( A1 = nil_Pr2300489316682597567nt_int )
          & ( A22 = nil_int ) )
        | ? [X4: product_prod_int_int,Y5: int,Xs4: list_P5707943133018811711nt_int,Ys4: list_int] :
            ( ( A1
              = ( cons_P3334398858971670639nt_int @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_int @ Y5 @ Ys4 ) )
            & ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X4 @ Y5 ) @ R )
            & ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Xs4 @ Ys4 ) @ ( listre2687791572143342772nt_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_443_listrel_Osimps,axiom,
    ! [A1: list_int,A22: list_P5707943133018811711nt_int,R: set_Pr5025984505184017080nt_int] :
      ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ A1 @ A22 ) @ ( listre3351099077621233370nt_int @ R ) )
      = ( ( ( A1 = nil_int )
          & ( A22 = nil_Pr2300489316682597567nt_int ) )
        | ? [X4: int,Y5: product_prod_int_int,Xs4: list_int,Ys4: list_P5707943133018811711nt_int] :
            ( ( A1
              = ( cons_int @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_P3334398858971670639nt_int @ Y5 @ Ys4 ) )
            & ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X4 @ Y5 ) @ R )
            & ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Xs4 @ Ys4 ) @ ( listre3351099077621233370nt_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_444_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_set_int,R: set_Pr4810089274464741491et_int] :
      ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ A1 @ A22 ) @ ( listrel_nat_set_int @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_set_int ) )
        | ? [X4: nat,Y5: set_int,Xs4: list_nat,Ys4: list_set_int] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_set_int @ Y5 @ Ys4 ) )
            & ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X4 @ Y5 ) @ R )
            & ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs4 @ Ys4 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_445_listrel_Osimps,axiom,
    ! [A1: list_int,A22: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
      ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ A1 @ A22 ) @ ( listre1889173885445251040nt_int @ R ) )
      = ( ( ( A1 = nil_int )
          & ( A22 = nil_li8670148097206105925nt_int ) )
        | ? [X4: int,Y5: list_P5707943133018811711nt_int,Xs4: list_int,Ys4: list_l1670014477004246597nt_int] :
            ( ( A1
              = ( cons_int @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_l7309679040211256053nt_int @ Y5 @ Ys4 ) )
            & ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X4 @ Y5 ) @ R )
            & ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs4 @ Ys4 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_446_listrel_Osimps,axiom,
    ! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
      = ( ( ( A1 = nil_int )
          & ( A22 = nil_int ) )
        | ? [X4: int,Y5: int,Xs4: list_int,Ys4: list_int] :
            ( ( A1
              = ( cons_int @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_int @ Y5 @ Ys4 ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R )
            & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs4 @ Ys4 ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_447_listrel_Osimps,axiom,
    ! [A1: list_P1743416141875011707e_term,A22: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
      ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ A1 @ A22 ) @ ( listre7929554186369383643nt_int @ R ) )
      = ( ( ( A1 = nil_Pr2193041437141079013e_term )
          & ( A22 = nil_Pr2300489316682597567nt_int ) )
        | ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int,Xs4: list_P1743416141875011707e_term,Ys4: list_P5707943133018811711nt_int] :
            ( ( A1
              = ( cons_P2630085844062958645e_term @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_P3334398858971670639nt_int @ Y5 @ Ys4 ) )
            & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ R )
            & ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs4 @ Ys4 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_448_listrel_Osimps,axiom,
    ! [A1: list_i8448526496819171953e_term,A22: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
      ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ A1 @ A22 ) @ ( listre7622244038138809061nt_int @ R ) )
      = ( ( ( A1 = nil_in3447728264653889115e_term )
          & ( A22 = nil_Pr2300489316682597567nt_int ) )
        | ? [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int,Xs4: list_i8448526496819171953e_term,Ys4: list_P5707943133018811711nt_int] :
            ( ( A1
              = ( cons_i7166360444231718571e_term @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_P3334398858971670639nt_int @ Y5 @ Ys4 ) )
            & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ R )
            & ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs4 @ Ys4 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_449_listrel_Ocases,axiom,
    ! [A1: list_P5707943133018811711nt_int,A22: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ A1 @ A22 ) @ ( listre7525836092462517041nt_int @ R ) )
     => ( ( ( A1 = nil_Pr2300489316682597567nt_int )
         => ( A22 != nil_Pr2300489316682597567nt_int ) )
       => ~ ! [X3: product_prod_int_int,Y3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
              ( ( A1
                = ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
             => ! [Ys3: list_P5707943133018811711nt_int] :
                  ( ( A22
                    = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
                 => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y3 ) @ R )
                   => ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs3 @ Ys3 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_450_listrel_Ocases,axiom,
    ! [A1: list_P5707943133018811711nt_int,A22: list_int,R: set_Pr53225590989397322nt_int] :
      ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ A1 @ A22 ) @ ( listre2687791572143342772nt_int @ R ) )
     => ( ( ( A1 = nil_Pr2300489316682597567nt_int )
         => ( A22 != nil_int ) )
       => ~ ! [X3: product_prod_int_int,Y3: int,Xs3: list_P5707943133018811711nt_int] :
              ( ( A1
                = ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
             => ! [Ys3: list_int] :
                  ( ( A22
                    = ( cons_int @ Y3 @ Ys3 ) )
                 => ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X3 @ Y3 ) @ R )
                   => ~ ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Xs3 @ Ys3 ) @ ( listre2687791572143342772nt_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_451_listrel_Ocases,axiom,
    ! [A1: list_int,A22: list_P5707943133018811711nt_int,R: set_Pr5025984505184017080nt_int] :
      ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ A1 @ A22 ) @ ( listre3351099077621233370nt_int @ R ) )
     => ( ( ( A1 = nil_int )
         => ( A22 != nil_Pr2300489316682597567nt_int ) )
       => ~ ! [X3: int,Y3: product_prod_int_int,Xs3: list_int] :
              ( ( A1
                = ( cons_int @ X3 @ Xs3 ) )
             => ! [Ys3: list_P5707943133018811711nt_int] :
                  ( ( A22
                    = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
                 => ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X3 @ Y3 ) @ R )
                   => ~ ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Xs3 @ Ys3 ) @ ( listre3351099077621233370nt_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_452_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_set_int,R: set_Pr4810089274464741491et_int] :
      ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ A1 @ A22 ) @ ( listrel_nat_set_int @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_set_int ) )
       => ~ ! [X3: nat,Y3: set_int,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X3 @ Xs3 ) )
             => ! [Ys3: list_set_int] :
                  ( ( A22
                    = ( cons_set_int @ Y3 @ Ys3 ) )
                 => ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R )
                   => ~ ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs3 @ Ys3 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_453_listrel_Ocases,axiom,
    ! [A1: list_int,A22: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
      ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ A1 @ A22 ) @ ( listre1889173885445251040nt_int @ R ) )
     => ( ( ( A1 = nil_int )
         => ( A22 != nil_li8670148097206105925nt_int ) )
       => ~ ! [X3: int,Y3: list_P5707943133018811711nt_int,Xs3: list_int] :
              ( ( A1
                = ( cons_int @ X3 @ Xs3 ) )
             => ! [Ys3: list_l1670014477004246597nt_int] :
                  ( ( A22
                    = ( cons_l7309679040211256053nt_int @ Y3 @ Ys3 ) )
                 => ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R )
                   => ~ ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs3 @ Ys3 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_454_listrel_Ocases,axiom,
    ! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
     => ( ( ( A1 = nil_int )
         => ( A22 != nil_int ) )
       => ~ ! [X3: int,Y3: int,Xs3: list_int] :
              ( ( A1
                = ( cons_int @ X3 @ Xs3 ) )
             => ! [Ys3: list_int] :
                  ( ( A22
                    = ( cons_int @ Y3 @ Ys3 ) )
                 => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
                   => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys3 ) @ ( listrel_int_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_455_listrel_Ocases,axiom,
    ! [A1: list_P1743416141875011707e_term,A22: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
      ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ A1 @ A22 ) @ ( listre7929554186369383643nt_int @ R ) )
     => ( ( ( A1 = nil_Pr2193041437141079013e_term )
         => ( A22 != nil_Pr2300489316682597567nt_int ) )
       => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs3: list_P1743416141875011707e_term] :
              ( ( A1
                = ( cons_P2630085844062958645e_term @ X3 @ Xs3 ) )
             => ! [Ys3: list_P5707943133018811711nt_int] :
                  ( ( A22
                    = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
                 => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
                   => ~ ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs3 @ Ys3 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_456_listrel_Ocases,axiom,
    ! [A1: list_i8448526496819171953e_term,A22: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
      ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ A1 @ A22 ) @ ( listre7622244038138809061nt_int @ R ) )
     => ( ( ( A1 = nil_in3447728264653889115e_term )
         => ( A22 != nil_Pr2300489316682597567nt_int ) )
       => ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs3: list_i8448526496819171953e_term] :
              ( ( A1
                = ( cons_i7166360444231718571e_term @ X3 @ Xs3 ) )
             => ! [Ys3: list_P5707943133018811711nt_int] :
                  ( ( A22
                    = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
                 => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
                   => ~ ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs3 @ Ys3 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_457_lexord__Nil__left,axiom,
    ! [Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
      = ( ? [A5: product_prod_int_int,X4: list_P5707943133018811711nt_int] :
            ( Y
            = ( cons_P3334398858971670639nt_int @ A5 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_458_lexord__Nil__left,axiom,
    ! [Y: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Y ) @ ( lexord_int @ R ) )
      = ( ? [A5: int,X4: list_int] :
            ( Y
            = ( cons_int @ A5 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_459_lexord__cons__cons,axiom,
    ! [A: product_prod_int_int,X: list_P5707943133018811711nt_int,B: product_prod_int_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ A @ X ) @ ( cons_P3334398858971670639nt_int @ B @ Y ) ) @ ( lexord8886728381914977324nt_int @ R ) )
      = ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ A @ B ) @ R )
        | ( ( A = B )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_460_lexord__cons__cons,axiom,
    ! [A: int,X: list_int,B: int,Y: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ A @ X ) @ ( cons_int @ B @ Y ) ) @ ( lexord_int @ R ) )
      = ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
        | ( ( A = B )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_461_lex__append__leftD,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lex_int @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_462_lexord__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
     => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lexord_int @ R ) ) ) ).

% lexord_irreflexive
thf(fact_463_lexord__linear,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: list_int,Y: list_int] :
      ( ! [A3: int,B3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ B3 @ A3 ) @ R ) )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) )
        | ( X = Y )
        | ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Y @ X ) @ ( lexord_int @ R ) ) ) ) ).

% lexord_linear
thf(fact_464_listrel_ONil,axiom,
    ! [R: set_Pr2560585780119916871nt_int] : ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int ) @ ( listre7525836092462517041nt_int @ R ) ) ).

% listrel.Nil
thf(fact_465_listrel_ONil,axiom,
    ! [R: set_Pr53225590989397322nt_int] : ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ nil_Pr2300489316682597567nt_int @ nil_int ) @ ( listre2687791572143342772nt_int @ R ) ) ).

% listrel.Nil
thf(fact_466_listrel_ONil,axiom,
    ! [R: set_Pr5025984505184017080nt_int] : ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ nil_int @ nil_Pr2300489316682597567nt_int ) @ ( listre3351099077621233370nt_int @ R ) ) ).

% listrel.Nil
thf(fact_467_listrel_ONil,axiom,
    ! [R: set_Pr958786334691620121nt_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ nil_int ) @ ( listrel_int_int @ R ) ) ).

% listrel.Nil
thf(fact_468_listrel__Nil1,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Xs ) @ ( listre7525836092462517041nt_int @ R ) )
     => ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% listrel_Nil1
thf(fact_469_listrel__Nil1,axiom,
    ! [Xs: list_int,R: set_Pr53225590989397322nt_int] :
      ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ nil_Pr2300489316682597567nt_int @ Xs ) @ ( listre2687791572143342772nt_int @ R ) )
     => ( Xs = nil_int ) ) ).

% listrel_Nil1
thf(fact_470_listrel__Nil1,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr5025984505184017080nt_int] :
      ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ nil_int @ Xs ) @ ( listre3351099077621233370nt_int @ R ) )
     => ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% listrel_Nil1
thf(fact_471_listrel__Nil1,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Xs ) @ ( listrel_int_int @ R ) )
     => ( Xs = nil_int ) ) ).

% listrel_Nil1
thf(fact_472_listrel__Nil2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( listre7525836092462517041nt_int @ R ) )
     => ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% listrel_Nil2
thf(fact_473_listrel__Nil2,axiom,
    ! [Xs: list_int,R: set_Pr5025984505184017080nt_int] :
      ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( listre3351099077621233370nt_int @ R ) )
     => ( Xs = nil_int ) ) ).

% listrel_Nil2
thf(fact_474_listrel__Nil2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr53225590989397322nt_int] :
      ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Xs @ nil_int ) @ ( listre2687791572143342772nt_int @ R ) )
     => ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% listrel_Nil2
thf(fact_475_listrel__Nil2,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ nil_int ) @ ( listrel_int_int @ R ) )
     => ( Xs = nil_int ) ) ).

% listrel_Nil2
thf(fact_476_lexord__Nil__right,axiom,
    ! [X: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ nil_Pr2300489316682597567nt_int ) @ ( lexord8886728381914977324nt_int @ R ) ) ).

% lexord_Nil_right
thf(fact_477_lexord__Nil__right,axiom,
    ! [X: list_int,R: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ nil_int ) @ ( lexord_int @ R ) ) ).

% lexord_Nil_right
thf(fact_478_lexord__append__leftI,axiom,
    ! [U: list_int,V: list_int,R: set_Pr958786334691620121nt_int,X: list_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ U @ V ) @ ( lexord_int @ R ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ X @ U ) @ ( append_int @ X @ V ) ) @ ( lexord_int @ R ) ) ) ).

% lexord_append_leftI
thf(fact_479_Nil__notin__lex,axiom,
    ! [Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ).

% Nil_notin_lex
thf(fact_480_Nil__notin__lex,axiom,
    ! [Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ys ) @ ( lex_int @ R ) ) ).

% Nil_notin_lex
thf(fact_481_Nil2__notin__lex,axiom,
    ! [Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ nil_Pr2300489316682597567nt_int ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ).

% Nil2_notin_lex
thf(fact_482_Nil2__notin__lex,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ nil_int ) @ ( lex_int @ R ) ) ).

% Nil2_notin_lex
thf(fact_483_lex__append__leftI,axiom,
    ! [Ys: list_int,Zs: list_int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lex_int @ R ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R ) ) ) ).

% lex_append_leftI
thf(fact_484_listrel_OCons,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int,R: set_Pr2560585780119916871nt_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
     => ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre7525836092462517041nt_int @ R ) )
       => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_485_listrel_OCons,axiom,
    ! [X: product_prod_int_int,Y: int,R: set_Pr53225590989397322nt_int,Xs: list_P5707943133018811711nt_int,Ys: list_int] :
      ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X @ Y ) @ R )
     => ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Xs @ Ys ) @ ( listre2687791572143342772nt_int @ R ) )
       => ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listre2687791572143342772nt_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_486_listrel_OCons,axiom,
    ! [X: int,Y: product_prod_int_int,R: set_Pr5025984505184017080nt_int,Xs: list_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X @ Y ) @ R )
     => ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Xs @ Ys ) @ ( listre3351099077621233370nt_int @ R ) )
       => ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ ( cons_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre3351099077621233370nt_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_487_listrel_OCons,axiom,
    ! [X: nat,Y: set_int,R: set_Pr4810089274464741491et_int,Xs: list_nat,Ys: list_set_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y ) @ R )
     => ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs @ Ys ) @ ( listrel_nat_set_int @ R ) )
       => ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ ( cons_nat @ X @ Xs ) @ ( cons_set_int @ Y @ Ys ) ) @ ( listrel_nat_set_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_488_listrel_OCons,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int,R: set_Pr1843331182504696254nt_int,Xs: list_int,Ys: list_l1670014477004246597nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ R )
     => ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs @ Ys ) @ ( listre1889173885445251040nt_int @ R ) )
       => ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ ( cons_int @ X @ Xs ) @ ( cons_l7309679040211256053nt_int @ Y @ Ys ) ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_489_listrel_OCons,axiom,
    ! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_490_listrel_OCons,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,R: set_Pr9222295170931077689nt_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R )
     => ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs @ Ys ) @ ( listre7929554186369383643nt_int @ R ) )
       => ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ ( cons_P2630085844062958645e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_491_listrel_OCons,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,R: set_Pr1872883991513573699nt_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R )
     => ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs @ Ys ) @ ( listre7622244038138809061nt_int @ R ) )
       => ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ ( cons_i7166360444231718571e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ).

% listrel.Cons
thf(fact_492_listrel__Cons1,axiom,
    ! [Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) @ Xs ) @ ( listre7525836092462517041nt_int @ R ) )
     => ~ ! [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
           => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y @ Y3 ) @ R )
             => ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ys @ Ys3 ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_493_listrel__Cons1,axiom,
    ! [Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Xs: list_int,R: set_Pr53225590989397322nt_int] :
      ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) @ Xs ) @ ( listre2687791572143342772nt_int @ R ) )
     => ~ ! [Y3: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y3 @ Ys3 ) )
           => ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ Y @ Y3 ) @ R )
             => ~ ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Ys @ Ys3 ) @ ( listre2687791572143342772nt_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_494_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_P5707943133018811711nt_int,R: set_Pr5025984505184017080nt_int] :
      ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listre3351099077621233370nt_int @ R ) )
     => ~ ! [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
           => ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ Y @ Y3 ) @ R )
             => ~ ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Ys @ Ys3 ) @ ( listre3351099077621233370nt_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_495_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_set_int,R: set_Pr4810089274464741491et_int] :
      ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_set_int @ R ) )
     => ~ ! [Y3: set_int,Ys3: list_set_int] :
            ( ( Xs
              = ( cons_set_int @ Y3 @ Ys3 ) )
           => ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ Y @ Y3 ) @ R )
             => ~ ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Ys @ Ys3 ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_496_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
      ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listre1889173885445251040nt_int @ R ) )
     => ~ ! [Y3: list_P5707943133018811711nt_int,Ys3: list_l1670014477004246597nt_int] :
            ( ( Xs
              = ( cons_l7309679040211256053nt_int @ Y3 @ Ys3 ) )
           => ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ Y @ Y3 ) @ R )
             => ~ ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Ys @ Ys3 ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_497_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listrel_int_int @ R ) )
     => ~ ! [Y3: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y3 @ Ys3 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y3 ) @ R )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys3 ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_498_listrel__Cons1,axiom,
    ! [Y: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P1743416141875011707e_term,Xs: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
      ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ ( cons_P2630085844062958645e_term @ Y @ Ys ) @ Xs ) @ ( listre7929554186369383643nt_int @ R ) )
     => ~ ! [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
           => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y @ Y3 ) @ R )
             => ~ ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Ys @ Ys3 ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_499_listrel__Cons1,axiom,
    ! [Y: int > option6357759511663192854e_term,Ys: list_i8448526496819171953e_term,Xs: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
      ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ ( cons_i7166360444231718571e_term @ Y @ Ys ) @ Xs ) @ ( listre7622244038138809061nt_int @ R ) )
     => ~ ! [Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
           => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y @ Y3 ) @ R )
             => ~ ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Ys @ Ys3 ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_500_listrel__Cons2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre7525836092462517041nt_int @ R ) )
     => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
           => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y ) @ R )
             => ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs3 @ Ys ) @ ( listre7525836092462517041nt_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_501_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr5025984505184017080nt_int] :
      ( ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre3351099077621233370nt_int @ R ) )
     => ~ ! [X3: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs3 ) )
           => ( ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X3 @ Y ) @ R )
             => ~ ( member3512312383758291599nt_int @ ( produc8214783121078382506nt_int @ Xs3 @ Ys ) @ ( listre3351099077621233370nt_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_502_listrel__Cons2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Y: int,Ys: list_int,R: set_Pr53225590989397322nt_int] :
      ( ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listre2687791572143342772nt_int @ R ) )
     => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
           => ( ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X3 @ Y ) @ R )
             => ~ ( member1281411469525323969st_int @ ( produc4509345508061262116st_int @ Xs3 @ Ys ) @ ( listre2687791572143342772nt_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_503_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: set_int,Ys: list_set_int,R: set_Pr4810089274464741491et_int] :
      ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs @ ( cons_set_int @ Y @ Ys ) ) @ ( listrel_nat_set_int @ R ) )
     => ~ ! [X3: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs3 ) )
           => ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y ) @ R )
             => ~ ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs3 @ Ys ) @ ( listrel_nat_set_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_504_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: list_P5707943133018811711nt_int,Ys: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
      ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs @ ( cons_l7309679040211256053nt_int @ Y @ Ys ) ) @ ( listre1889173885445251040nt_int @ R ) )
     => ~ ! [X3: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs3 ) )
           => ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y ) @ R )
             => ~ ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs3 @ Ys ) @ ( listre1889173885445251040nt_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_505_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R ) )
     => ~ ! [X3: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs3 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys ) @ ( listrel_int_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_506_listrel__Cons2,axiom,
    ! [Xs: list_P1743416141875011707e_term,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
      ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre7929554186369383643nt_int @ R ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Xs3: list_P1743416141875011707e_term] :
            ( ( Xs
              = ( cons_P2630085844062958645e_term @ X3 @ Xs3 ) )
           => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ R )
             => ~ ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs3 @ Ys ) @ ( listre7929554186369383643nt_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_507_listrel__Cons2,axiom,
    ! [Xs: list_i8448526496819171953e_term,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
      ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( listre7622244038138809061nt_int @ R ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Xs3: list_i8448526496819171953e_term] :
            ( ( Xs
              = ( cons_i7166360444231718571e_term @ X3 @ Xs3 ) )
           => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ R )
             => ~ ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs3 @ Ys ) @ ( listre7622244038138809061nt_int @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_508_lexord__append__leftD,axiom,
    ! [X: list_int,U: list_int,V: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ X @ U ) @ ( append_int @ X @ V ) ) @ ( lexord_int @ R ) )
     => ( ! [A3: int] :
            ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ A3 ) @ R )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ U @ V ) @ ( lexord_int @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_509_lexord__append__rightI,axiom,
    ! [Y: list_P5707943133018811711nt_int,X: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ? [B6: product_prod_int_int,Z3: list_P5707943133018811711nt_int] :
          ( Y
          = ( cons_P3334398858971670639nt_int @ B6 @ Z3 ) )
     => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ ( append7030698103840186580nt_int @ X @ Y ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ).

% lexord_append_rightI
thf(fact_510_lexord__append__rightI,axiom,
    ! [Y: list_int,X: list_int,R: set_Pr958786334691620121nt_int] :
      ( ? [B6: int,Z3: list_int] :
          ( Y
          = ( cons_int @ B6 @ Z3 ) )
     => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ ( append_int @ X @ Y ) ) @ ( lexord_int @ R ) ) ) ).

% lexord_append_rightI
thf(fact_511_lex__append__left__iff,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lex_int @ R ) )
        = ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lex_int @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_512_lexord__same__pref__if__irrefl,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int,Zs: list_int] :
      ( ( irrefl_on_int @ top_top_set_int @ R )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lexord_int @ R ) )
        = ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lexord_int @ R ) ) ) ) ).

% lexord_same_pref_if_irrefl
thf(fact_513_splice_Opinduct,axiom,
    ! [A0: list_P5707943133018811711nt_int,A1: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
      ( ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ A0 @ A1 ) )
     => ( ! [Ys3: list_P5707943133018811711nt_int] :
            ( ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys3 ) )
           => ( P2 @ nil_Pr2300489316682597567nt_int @ Ys3 ) )
       => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Ys3: list_P5707943133018811711nt_int] :
              ( ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ Ys3 ) )
             => ( ( P2 @ Ys3 @ Xs3 )
               => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ Ys3 ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_514_splice_Opinduct,axiom,
    ! [A0: list_int,A1: list_int,P2: list_int > list_int > $o] :
      ( ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ A0 @ A1 ) )
     => ( ! [Ys3: list_int] :
            ( ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
           => ( P2 @ nil_int @ Ys3 ) )
       => ( ! [X3: int,Xs3: list_int,Ys3: list_int] :
              ( ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs3 ) @ Ys3 ) )
             => ( ( P2 @ Ys3 @ Xs3 )
               => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ Ys3 ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_515_shuffles_Opinduct,axiom,
    ! [A0: list_P5707943133018811711nt_int,A1: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
      ( ( accp_P1749757178599463408nt_int @ shuffl6524295016199058339nt_int @ ( produc1932183703851549015nt_int @ A0 @ A1 ) )
     => ( ! [Ys3: list_P5707943133018811711nt_int] :
            ( ( accp_P1749757178599463408nt_int @ shuffl6524295016199058339nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys3 ) )
           => ( P2 @ nil_Pr2300489316682597567nt_int @ Ys3 ) )
       => ( ! [Xs3: list_P5707943133018811711nt_int] :
              ( ( accp_P1749757178599463408nt_int @ shuffl6524295016199058339nt_int @ ( produc1932183703851549015nt_int @ Xs3 @ nil_Pr2300489316682597567nt_int ) )
             => ( P2 @ Xs3 @ nil_Pr2300489316682597567nt_int ) )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
                ( ( accp_P1749757178599463408nt_int @ shuffl6524295016199058339nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) )
               => ( ( P2 @ Xs3 @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) )
                 => ( ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ Ys3 )
                   => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_516_shuffles_Opinduct,axiom,
    ! [A0: list_int,A1: list_int,P2: list_int > list_int > $o] :
      ( ( accp_P7395280492732805858st_int @ shuffles_rel_int @ ( produc364263696895485585st_int @ A0 @ A1 ) )
     => ( ! [Ys3: list_int] :
            ( ( accp_P7395280492732805858st_int @ shuffles_rel_int @ ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
           => ( P2 @ nil_int @ Ys3 ) )
       => ( ! [Xs3: list_int] :
              ( ( accp_P7395280492732805858st_int @ shuffles_rel_int @ ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
             => ( P2 @ Xs3 @ nil_int ) )
         => ( ! [X3: int,Xs3: list_int,Y3: int,Ys3: list_int] :
                ( ( accp_P7395280492732805858st_int @ shuffles_rel_int @ ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
               => ( ( P2 @ Xs3 @ ( cons_int @ Y3 @ Ys3 ) )
                 => ( ( P2 @ ( cons_int @ X3 @ Xs3 ) @ Ys3 )
                   => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) ) )
           => ( P2 @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_517_Cons__in__lex,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
          & ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_518_Cons__in__lex,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
      = ( ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ R )
          & ( ( size_s5157815400016825771nt_int @ Xs )
            = ( size_s5157815400016825771nt_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_519_map__tailrec__rev_Opelims,axiom,
    ! [X: product_prod_int_int > product_prod_int_int,Xa: list_P5707943133018811711nt_int,Xb: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( map_ta5484600921387902261nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P2088104858098394414nt_int @ map_ta1851010244768962692nt_int @ ( produc2285083125503127081nt_int @ X @ ( produc1932183703851549015nt_int @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
           => ( ( Y = Xb )
             => ~ ( accp_P2088104858098394414nt_int @ map_ta1851010244768962692nt_int @ ( produc2285083125503127081nt_int @ X @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Xb ) ) ) ) )
         => ~ ! [A3: product_prod_int_int,As: list_P5707943133018811711nt_int] :
                ( ( Xa
                  = ( cons_P3334398858971670639nt_int @ A3 @ As ) )
               => ( ( Y
                    = ( map_ta5484600921387902261nt_int @ X @ As @ ( cons_P3334398858971670639nt_int @ ( X @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P2088104858098394414nt_int @ map_ta1851010244768962692nt_int @ ( produc2285083125503127081nt_int @ X @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_520_map__tailrec__rev_Opelims,axiom,
    ! [X: product_prod_int_int > int,Xa: list_P5707943133018811711nt_int,Xb: list_int,Y: list_int] :
      ( ( ( map_ta8640746543295758256nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1723189894724648166st_int @ map_ta7508373496359787297nt_int @ ( produc3771223888072323881st_int @ X @ ( produc4509345508061262116st_int @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
           => ( ( Y = Xb )
             => ~ ( accp_P1723189894724648166st_int @ map_ta7508373496359787297nt_int @ ( produc3771223888072323881st_int @ X @ ( produc4509345508061262116st_int @ nil_Pr2300489316682597567nt_int @ Xb ) ) ) ) )
         => ~ ! [A3: product_prod_int_int,As: list_P5707943133018811711nt_int] :
                ( ( Xa
                  = ( cons_P3334398858971670639nt_int @ A3 @ As ) )
               => ( ( Y
                    = ( map_ta8640746543295758256nt_int @ X @ As @ ( cons_int @ ( X @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P1723189894724648166st_int @ map_ta7508373496359787297nt_int @ ( produc3771223888072323881st_int @ X @ ( produc4509345508061262116st_int @ ( cons_P3334398858971670639nt_int @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_521_map__tailrec__rev_Opelims,axiom,
    ! [X: int > product_prod_int_int,Xa: list_int,Xb: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( map_ta80682011918873046nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7431833251743291682nt_int @ map_ta8171681001837677895nt_int @ ( produc1273925807858914917nt_int @ X @ ( produc8214783121078382506nt_int @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_int )
           => ( ( Y = Xb )
             => ~ ( accp_P7431833251743291682nt_int @ map_ta8171681001837677895nt_int @ ( produc1273925807858914917nt_int @ X @ ( produc8214783121078382506nt_int @ nil_int @ Xb ) ) ) ) )
         => ~ ! [A3: int,As: list_int] :
                ( ( Xa
                  = ( cons_int @ A3 @ As ) )
               => ( ( Y
                    = ( map_ta80682011918873046nt_int @ X @ As @ ( cons_P3334398858971670639nt_int @ ( X @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P7431833251743291682nt_int @ map_ta8171681001837677895nt_int @ ( produc1273925807858914917nt_int @ X @ ( produc8214783121078382506nt_int @ ( cons_int @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_522_map__tailrec__rev_Opelims,axiom,
    ! [X: int > int,Xa: list_int,Xb: list_int,Y: list_int] :
      ( ( ( map_ta8161051695879884879nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7217010622171553742st_int @ map_ta389364721648293406nt_int @ ( produc740395959522281929st_int @ X @ ( produc364263696895485585st_int @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_int )
           => ( ( Y = Xb )
             => ~ ( accp_P7217010622171553742st_int @ map_ta389364721648293406nt_int @ ( produc740395959522281929st_int @ X @ ( produc364263696895485585st_int @ nil_int @ Xb ) ) ) ) )
         => ~ ! [A3: int,As: list_int] :
                ( ( Xa
                  = ( cons_int @ A3 @ As ) )
               => ( ( Y
                    = ( map_ta8161051695879884879nt_int @ X @ As @ ( cons_int @ ( X @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P7217010622171553742st_int @ map_ta389364721648293406nt_int @ ( produc740395959522281929st_int @ X @ ( produc364263696895485585st_int @ ( cons_int @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_523_lexord__same__pref__iff,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Xs @ Zs ) ) @ ( lexord_int @ R ) )
      = ( ? [X4: int] :
            ( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R ) )
        | ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lexord_int @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_524_rotate1_Osimps_I2_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( rotate9137591727117064716nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
      = ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) ) ).

% rotate1.simps(2)
thf(fact_525_rotate1_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( rotate1_int @ ( cons_int @ X @ Xs ) )
      = ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) ) ).

% rotate1.simps(2)
thf(fact_526_append__eq__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int,Us2: list_int,Vs: list_int] :
      ( ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Ys ) )
        | ( ( size_size_list_int @ Us2 )
          = ( size_size_list_int @ Vs ) ) )
     => ( ( ( append_int @ Xs @ Us2 )
          = ( append_int @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_527_append__eq__append__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Us2: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int] :
      ( ( ( ( size_s5157815400016825771nt_int @ Xs )
          = ( size_s5157815400016825771nt_int @ Ys ) )
        | ( ( size_s5157815400016825771nt_int @ Us2 )
          = ( size_s5157815400016825771nt_int @ Vs ) ) )
     => ( ( ( append7030698103840186580nt_int @ Xs @ Us2 )
          = ( append7030698103840186580nt_int @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_528_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l1670014477004246597nt_int] :
      ( ( nil_Pr2300489316682597567nt_int
        = ( concat4512918505337516154nt_int @ Xss2 ) )
      = ( ! [X4: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X4 @ ( set_li2659200638379878868nt_int @ Xss2 ) )
           => ( X4 = nil_Pr2300489316682597567nt_int ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_529_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_list_int] :
      ( ( nil_int
        = ( concat_int @ Xss2 ) )
      = ( ! [X4: list_int] :
            ( ( member_list_int @ X4 @ ( set_list_int2 @ Xss2 ) )
           => ( X4 = nil_int ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_530_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l1670014477004246597nt_int] :
      ( ( ( concat4512918505337516154nt_int @ Xss2 )
        = nil_Pr2300489316682597567nt_int )
      = ( ! [X4: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X4 @ ( set_li2659200638379878868nt_int @ Xss2 ) )
           => ( X4 = nil_Pr2300489316682597567nt_int ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_531_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_list_int] :
      ( ( ( concat_int @ Xss2 )
        = nil_int )
      = ( ! [X4: list_int] :
            ( ( member_list_int @ X4 @ ( set_list_int2 @ Xss2 ) )
           => ( X4 = nil_int ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_532_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( ( rotate9137591727117064716nt_int @ Xs )
        = nil_Pr2300489316682597567nt_int )
      = ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% rotate1_is_Nil_conv
thf(fact_533_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( rotate1_int @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
thf(fact_534_set__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( set_int2 @ ( rotate1_int @ Xs ) )
      = ( set_int2 @ Xs ) ) ).

% set_rotate1
thf(fact_535_length__rotate1,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ ( rotate9137591727117064716nt_int @ Xs ) )
      = ( size_s5157815400016825771nt_int @ Xs ) ) ).

% length_rotate1
thf(fact_536_in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_537_in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( insert_int @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_538_not__in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = ( cons_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_539_not__in__set__insert,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ( ( insert5765537519290168021nt_int @ X @ Xs )
        = ( cons_P3334398858971670639nt_int @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_540_not__in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( insert_int @ X @ Xs )
        = ( cons_int @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_541_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_542_neq__if__length__neq,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
       != ( size_s5157815400016825771nt_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_543_in__set__product__lists__length,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Xss2: list_l1670014477004246597nt_int] :
      ( ( member2764346250752101224nt_int @ Xs @ ( set_li2659200638379878868nt_int @ ( produc5568053154996169768nt_int @ Xss2 ) ) )
     => ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s2969076144586574001nt_int @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_544_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: list_nat,X21: nat] :
      ( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
     => ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_545_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_int_int,X22: list_P5707943133018811711nt_int,X21: product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ X22 ) )
     => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_546_list_Oset__intros_I2_J,axiom,
    ! [Y: int,X22: list_int,X21: int] :
      ( ( member_int2 @ Y @ ( set_int2 @ X22 ) )
     => ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_547_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_548_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_int_int,X22: list_P5707943133018811711nt_int] : ( member5262025264175285858nt_int @ X21 @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_549_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X22: list_int] : ( member_int2 @ X21 @ ( set_int2 @ ( cons_int @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_550_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat2 @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_551_list_Oset__cases,axiom,
    ! [E: product_prod_int_int,A: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ E @ ( set_Pr2470121279949933262nt_int @ A ) )
     => ( ! [Z22: list_P5707943133018811711nt_int] :
            ( A
           != ( cons_P3334398858971670639nt_int @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_int_int,Z22: list_P5707943133018811711nt_int] :
              ( ( A
                = ( cons_P3334398858971670639nt_int @ Z1 @ Z22 ) )
             => ~ ( member5262025264175285858nt_int @ E @ ( set_Pr2470121279949933262nt_int @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_552_list_Oset__cases,axiom,
    ! [E: int,A: list_int] :
      ( ( member_int2 @ E @ ( set_int2 @ A ) )
     => ( ! [Z22: list_int] :
            ( A
           != ( cons_int @ E @ Z22 ) )
       => ~ ! [Z1: int,Z22: list_int] :
              ( ( A
                = ( cons_int @ Z1 @ Z22 ) )
             => ~ ( member_int2 @ E @ ( set_int2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_553_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_554_set__ConsD,axiom,
    ! [Y: product_prod_int_int,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_555_set__ConsD,axiom,
    ! [Y: int,X: int,Xs: list_int] :
      ( ( member_int2 @ Y @ ( set_int2 @ ( cons_int @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_int2 @ Y @ ( set_int2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_556_in__set__butlastD,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
     => ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_557_in__set__butlastD,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ ( butlast_int @ Xs ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_558_irrefl__onD,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( irrefl_on_nat @ A4 @ R )
     => ( ( member_nat2 @ A @ A4 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_559_irrefl__onD,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int,A: int] :
      ( ( irrefl_on_int @ A4 @ R )
     => ( ( member_int2 @ A @ A4 )
       => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_560_irrefl__onI,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat] :
          ( ( member_nat2 @ A3 @ A4 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R ) )
     => ( irrefl_on_nat @ A4 @ R ) ) ).

% irrefl_onI
thf(fact_561_irrefl__onI,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int] :
      ( ! [A3: int] :
          ( ( member_int2 @ A3 @ A4 )
         => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ A3 ) @ R ) )
     => ( irrefl_on_int @ A4 @ R ) ) ).

% irrefl_onI
thf(fact_562_irrefl__on__def,axiom,
    ( irrefl_on_int
    = ( ^ [A6: set_int,R3: set_Pr958786334691620121nt_int] :
        ! [X4: int] :
          ( ( member_int2 @ X4 @ A6 )
         => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ X4 ) @ R3 ) ) ) ) ).

% irrefl_on_def
thf(fact_563_length__mirror1__aux,axiom,
    ! [N: int] :
      ( size_s5157815400016825771nt_int
      = ( ^ [Ps3: list_P5707943133018811711nt_int] : ( size_s5157815400016825771nt_int @ ( mirror1_aux @ N @ Ps3 ) ) ) ) ).

% length_mirror1_aux
thf(fact_564_length__mirror2__aux,axiom,
    ! [N: int] :
      ( size_s5157815400016825771nt_int
      = ( ^ [Ps3: list_P5707943133018811711nt_int] : ( size_s5157815400016825771nt_int @ ( mirror2_aux @ N @ Ps3 ) ) ) ) ).

% length_mirror2_aux
thf(fact_565_rotate1_Osimps_I1_J,axiom,
    ( ( rotate9137591727117064716nt_int @ nil_Pr2300489316682597567nt_int )
    = nil_Pr2300489316682597567nt_int ) ).

% rotate1.simps(1)
thf(fact_566_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_int @ nil_int )
    = nil_int ) ).

% rotate1.simps(1)
thf(fact_567_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P5: nat > $o,Xs4: list_nat] :
        ? [X4: nat] :
          ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs4 ) )
          & ( P5 @ X4 )
          & ! [Y5: nat] :
              ( ( ( member_nat2 @ Y5 @ ( set_nat2 @ Xs4 ) )
                & ( P5 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_568_list__ex1__iff,axiom,
    ( list_ex1_int
    = ( ^ [P5: int > $o,Xs4: list_int] :
        ? [X4: int] :
          ( ( member_int2 @ X4 @ ( set_int2 @ Xs4 ) )
          & ( P5 @ X4 )
          & ! [Y5: int] :
              ( ( ( member_int2 @ Y5 @ ( set_int2 @ Xs4 ) )
                & ( P5 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_569_length__mirror1,axiom,
    ( size_s5157815400016825771nt_int
    = ( ^ [Ps3: list_P5707943133018811711nt_int] : ( size_s5157815400016825771nt_int @ ( mirror1 @ Ps3 ) ) ) ) ).

% length_mirror1
thf(fact_570_in__set__member,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( member_nat @ Xs @ X ) ) ).

% in_set_member
thf(fact_571_in__set__member,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( member_int @ Xs @ X ) ) ).

% in_set_member
thf(fact_572_length__mirror2,axiom,
    ( size_s5157815400016825771nt_int
    = ( ^ [Ps3: list_P5707943133018811711nt_int] : ( size_s5157815400016825771nt_int @ ( mirror2 @ Ps3 ) ) ) ) ).

% length_mirror2
thf(fact_573_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,P2: list_int > list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P2 @ nil_int @ nil_int @ nil_int )
         => ( ! [X3: int,Xs3: list_int,Y3: int,Ys3: list_int,Z2: int,Zs2: list_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_int @ Ys3 ) )
               => ( ( ( size_size_list_int @ Ys3 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_574_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_P5707943133018811711nt_int,P2: list_int > list_int > list_P5707943133018811711nt_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_s5157815400016825771nt_int @ Zs ) )
       => ( ( P2 @ nil_int @ nil_int @ nil_Pr2300489316682597567nt_int )
         => ( ! [X3: int,Xs3: list_int,Y3: int,Ys3: list_int,Z2: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_size_list_int @ Ys3 ) )
               => ( ( ( size_size_list_int @ Ys3 )
                    = ( size_s5157815400016825771nt_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_P3334398858971670639nt_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_575_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int,Zs: list_int,P2: list_int > list_P5707943133018811711nt_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( size_s5157815400016825771nt_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P2 @ nil_int @ nil_Pr2300489316682597567nt_int @ nil_int )
         => ( ! [X3: int,Xs3: list_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int,Z2: int,Zs2: list_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_s5157815400016825771nt_int @ Ys3 ) )
               => ( ( ( size_s5157815400016825771nt_int @ Ys3 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_576_list__induct3,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,P2: list_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( size_s5157815400016825771nt_int @ Ys )
          = ( size_s5157815400016825771nt_int @ Zs ) )
       => ( ( P2 @ nil_int @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
         => ( ! [X3: int,Xs3: list_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int,Z2: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
                ( ( ( size_size_list_int @ Xs3 )
                  = ( size_s5157815400016825771nt_int @ Ys3 ) )
               => ( ( ( size_s5157815400016825771nt_int @ Ys3 )
                    = ( size_s5157815400016825771nt_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) @ ( cons_P3334398858971670639nt_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_577_list__induct3,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int,Zs: list_int,P2: list_P5707943133018811711nt_int > list_int > list_int > $o] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_int @ nil_int )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: int,Ys3: list_int,Z2: int,Zs2: list_int] :
                ( ( ( size_s5157815400016825771nt_int @ Xs3 )
                  = ( size_size_list_int @ Ys3 ) )
               => ( ( ( size_size_list_int @ Ys3 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_578_list__induct3,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int,Zs: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_int > list_P5707943133018811711nt_int > $o] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_s5157815400016825771nt_int @ Zs ) )
       => ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_int @ nil_Pr2300489316682597567nt_int )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: int,Ys3: list_int,Z2: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
                ( ( ( size_s5157815400016825771nt_int @ Xs3 )
                  = ( size_size_list_int @ Ys3 ) )
               => ( ( ( size_size_list_int @ Ys3 )
                    = ( size_s5157815400016825771nt_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_P3334398858971670639nt_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_579_list__induct3,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > list_int > $o] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( size_s5157815400016825771nt_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int @ nil_int )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int,Z2: int,Zs2: list_int] :
                ( ( ( size_s5157815400016825771nt_int @ Xs3 )
                  = ( size_s5157815400016825771nt_int @ Ys3 ) )
               => ( ( ( size_s5157815400016825771nt_int @ Ys3 )
                    = ( size_size_list_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_580_list__induct3,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( ( size_s5157815400016825771nt_int @ Ys )
          = ( size_s5157815400016825771nt_int @ Zs ) )
       => ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
         => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int,Z2: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
                ( ( ( size_s5157815400016825771nt_int @ Xs3 )
                  = ( size_s5157815400016825771nt_int @ Ys3 ) )
               => ( ( ( size_s5157815400016825771nt_int @ Ys3 )
                    = ( size_s5157815400016825771nt_int @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) @ ( cons_P3334398858971670639nt_int @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_581_list__induct2,axiom,
    ! [Xs: list_int,Ys: list_int,P2: list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P2 @ nil_int @ nil_int )
       => ( ! [X3: int,Xs3: list_int,Y3: int,Ys3: list_int] :
              ( ( ( size_size_list_int @ Xs3 )
                = ( size_size_list_int @ Ys3 ) )
             => ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_582_list__induct2,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int,P2: list_int > list_P5707943133018811711nt_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( P2 @ nil_int @ nil_Pr2300489316682597567nt_int )
       => ( ! [X3: int,Xs3: list_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
              ( ( ( size_size_list_int @ Xs3 )
                = ( size_s5157815400016825771nt_int @ Ys3 ) )
             => ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_583_list__induct2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int,P2: list_P5707943133018811711nt_int > list_int > $o] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_int )
       => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: int,Ys3: list_int] :
              ( ( ( size_s5157815400016825771nt_int @ Xs3 )
                = ( size_size_list_int @ Ys3 ) )
             => ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_584_list__induct2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,P2: list_P5707943133018811711nt_int > list_P5707943133018811711nt_int > $o] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( P2 @ nil_Pr2300489316682597567nt_int @ nil_Pr2300489316682597567nt_int )
       => ( ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys3: list_P5707943133018811711nt_int] :
              ( ( ( size_s5157815400016825771nt_int @ Xs3 )
                = ( size_s5157815400016825771nt_int @ Ys3 ) )
             => ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ ( cons_P3334398858971670639nt_int @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_585_split__list__first__prop__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ( ? [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys4: list_P5707943133018811711nt_int,X4: product_prod_int_int] :
            ( ? [Zs3: list_P5707943133018811711nt_int] :
                ( Xs
                = ( append7030698103840186580nt_int @ Ys4 @ ( cons_P3334398858971670639nt_int @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y5 @ ( set_Pr2470121279949933262nt_int @ Ys4 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_586_split__list__first__prop__iff,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ( ? [X4: int] :
            ( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys4: list_int,X4: int] :
            ( ? [Zs3: list_int] :
                ( Xs
                = ( append_int @ Ys4 @ ( cons_int @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: int] :
                ( ( member_int2 @ Y5 @ ( set_int2 @ Ys4 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_587_split__list__last__prop__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ( ? [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys4: list_P5707943133018811711nt_int,X4: product_prod_int_int,Zs3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( append7030698103840186580nt_int @ Ys4 @ ( cons_P3334398858971670639nt_int @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: product_prod_int_int] :
                ( ( member5262025264175285858nt_int @ Y5 @ ( set_Pr2470121279949933262nt_int @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_588_split__list__last__prop__iff,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ( ? [X4: int] :
            ( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys4: list_int,X4: int,Zs3: list_int] :
            ( ( Xs
              = ( append_int @ Ys4 @ ( cons_int @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: int] :
                ( ( member_int2 @ Y5 @ ( set_int2 @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_589_in__set__conv__decomp__first,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys4: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat2 @ X @ ( set_nat2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_590_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
      = ( ? [Ys4: list_P5707943133018811711nt_int,Zs3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( append7030698103840186580nt_int @ Ys4 @ ( cons_P3334398858971670639nt_int @ X @ Zs3 ) ) )
            & ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_591_in__set__conv__decomp__first,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( ? [Ys4: list_int,Zs3: list_int] :
            ( ( Xs
              = ( append_int @ Ys4 @ ( cons_int @ X @ Zs3 ) ) )
            & ~ ( member_int2 @ X @ ( set_int2 @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_592_in__set__conv__decomp__last,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys4: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat2 @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_593_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
      = ( ? [Ys4: list_P5707943133018811711nt_int,Zs3: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( append7030698103840186580nt_int @ Ys4 @ ( cons_P3334398858971670639nt_int @ X @ Zs3 ) ) )
            & ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_594_in__set__conv__decomp__last,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( ? [Ys4: list_int,Zs3: list_int] :
            ( ( Xs
              = ( append_int @ Ys4 @ ( cons_int @ X @ Zs3 ) ) )
            & ~ ( member_int2 @ X @ ( set_int2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_595_split__list__first__propE,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ? [X5: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X5 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P5707943133018811711nt_int,X3: product_prod_int_int] :
            ( ? [Zs2: list_P5707943133018811711nt_int] :
                ( Xs
                = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa2: product_prod_int_int] :
                    ( ( member5262025264175285858nt_int @ Xa2 @ ( set_Pr2470121279949933262nt_int @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_596_split__list__first__propE,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_int,X3: int] :
            ( ? [Zs2: list_int] :
                ( Xs
                = ( append_int @ Ys3 @ ( cons_int @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa2: int] :
                    ( ( member_int2 @ Xa2 @ ( set_int2 @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_597_split__list__last__propE,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ? [X5: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X5 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P5707943133018811711nt_int,X3: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa2: product_prod_int_int] :
                    ( ( member5262025264175285858nt_int @ Xa2 @ ( set_Pr2470121279949933262nt_int @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_598_split__list__last__propE,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_int,X3: int,Zs2: list_int] :
            ( ( Xs
              = ( append_int @ Ys3 @ ( cons_int @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa2: int] :
                    ( ( member_int2 @ Xa2 @ ( set_int2 @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_599_split__list__first__prop,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ? [X5: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X5 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P5707943133018811711nt_int,X3: product_prod_int_int] :
          ( ? [Zs2: list_P5707943133018811711nt_int] :
              ( Xs
              = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa2: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ Xa2 @ ( set_Pr2470121279949933262nt_int @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_600_split__list__first__prop,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_int,X3: int] :
          ( ? [Zs2: list_int] :
              ( Xs
              = ( append_int @ Ys3 @ ( cons_int @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa2: int] :
              ( ( member_int2 @ Xa2 @ ( set_int2 @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_601_split__list__last__prop,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ? [X5: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X5 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P5707943133018811711nt_int,X3: product_prod_int_int,Zs2: list_P5707943133018811711nt_int] :
          ( ( Xs
            = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa2: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ Xa2 @ ( set_Pr2470121279949933262nt_int @ Zs2 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_602_split__list__last__prop,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_int,X3: int,Zs2: list_int] :
          ( ( Xs
            = ( append_int @ Ys3 @ ( cons_int @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa2: int] :
              ( ( member_int2 @ Xa2 @ ( set_int2 @ Zs2 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_603_in__set__conv__decomp,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys4: list_nat,Zs3: list_nat] :
            ( Xs
            = ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_604_in__set__conv__decomp,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
      = ( ? [Ys4: list_P5707943133018811711nt_int,Zs3: list_P5707943133018811711nt_int] :
            ( Xs
            = ( append7030698103840186580nt_int @ Ys4 @ ( cons_P3334398858971670639nt_int @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_605_in__set__conv__decomp,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( ? [Ys4: list_int,Zs3: list_int] :
            ( Xs
            = ( append_int @ Ys4 @ ( cons_int @ X @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_606_append__Cons__eq__iff,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Xs6: list_nat,Ys6: list_nat] :
      ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
            = ( append_nat @ Xs6 @ ( cons_nat @ X @ Ys6 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_607_append__Cons__eq__iff,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Xs6: list_P5707943133018811711nt_int,Ys6: list_P5707943133018811711nt_int] :
      ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Ys ) )
       => ( ( ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ Ys ) )
            = ( append7030698103840186580nt_int @ Xs6 @ ( cons_P3334398858971670639nt_int @ X @ Ys6 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_608_append__Cons__eq__iff,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,Xs6: list_int,Ys6: list_int] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ~ ( member_int2 @ X @ ( set_int2 @ Ys ) )
       => ( ( ( append_int @ Xs @ ( cons_int @ X @ Ys ) )
            = ( append_int @ Xs6 @ ( cons_int @ X @ Ys6 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_609_split__list__propE,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ? [X5: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X5 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P5707943133018811711nt_int,X3: product_prod_int_int] :
            ( ? [Zs2: list_P5707943133018811711nt_int] :
                ( Xs
                = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X3 @ Zs2 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_610_split__list__propE,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_int,X3: int] :
            ( ? [Zs2: list_int] :
                ( Xs
                = ( append_int @ Ys3 @ ( cons_int @ X3 @ Zs2 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_611_split__list__first,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
          & ~ ( member_nat2 @ X @ ( set_nat2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_612_split__list__first,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ? [Ys3: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
          ( ( Xs
            = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X @ Zs2 ) ) )
          & ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_613_split__list__first,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ? [Ys3: list_int,Zs2: list_int] :
          ( ( Xs
            = ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
          & ~ ( member_int2 @ X @ ( set_int2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_614_split__list__prop,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ? [X5: product_prod_int_int] :
          ( ( member5262025264175285858nt_int @ X5 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P5707943133018811711nt_int,X3: product_prod_int_int] :
          ( ? [Zs2: list_P5707943133018811711nt_int] :
              ( Xs
              = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_615_split__list__prop,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ? [X5: int] :
          ( ( member_int2 @ X5 @ ( set_int2 @ Xs ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_int,X3: int] :
          ( ? [Zs2: list_int] :
              ( Xs
              = ( append_int @ Ys3 @ ( cons_int @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_616_split__list__last,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
          & ~ ( member_nat2 @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_617_split__list__last,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ? [Ys3: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
          ( ( Xs
            = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X @ Zs2 ) ) )
          & ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_618_split__list__last,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ? [Ys3: list_int,Zs2: list_int] :
          ( ( Xs
            = ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) )
          & ~ ( member_int2 @ X @ ( set_int2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_619_split__list,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs2: list_nat] :
          ( Xs
          = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_620_split__list,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ? [Ys3: list_P5707943133018811711nt_int,Zs2: list_P5707943133018811711nt_int] :
          ( Xs
          = ( append7030698103840186580nt_int @ Ys3 @ ( cons_P3334398858971670639nt_int @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_621_split__list,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ? [Ys3: list_int,Zs2: list_int] :
          ( Xs
          = ( append_int @ Ys3 @ ( cons_int @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_622_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: product_prod_int_int > product_prod_int_int,A: product_prod_int_int,As2: list_P5707943133018811711nt_int,Bs: list_P5707943133018811711nt_int] :
      ( ( map_ta5484600921387902261nt_int @ F2 @ ( cons_P3334398858971670639nt_int @ A @ As2 ) @ Bs )
      = ( map_ta5484600921387902261nt_int @ F2 @ As2 @ ( cons_P3334398858971670639nt_int @ ( F2 @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_623_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: product_prod_int_int > int,A: product_prod_int_int,As2: list_P5707943133018811711nt_int,Bs: list_int] :
      ( ( map_ta8640746543295758256nt_int @ F2 @ ( cons_P3334398858971670639nt_int @ A @ As2 ) @ Bs )
      = ( map_ta8640746543295758256nt_int @ F2 @ As2 @ ( cons_int @ ( F2 @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_624_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: int > product_prod_int_int,A: int,As2: list_int,Bs: list_P5707943133018811711nt_int] :
      ( ( map_ta80682011918873046nt_int @ F2 @ ( cons_int @ A @ As2 ) @ Bs )
      = ( map_ta80682011918873046nt_int @ F2 @ As2 @ ( cons_P3334398858971670639nt_int @ ( F2 @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_625_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: int > int,A: int,As2: list_int,Bs: list_int] :
      ( ( map_ta8161051695879884879nt_int @ F2 @ ( cons_int @ A @ As2 ) @ Bs )
      = ( map_ta8161051695879884879nt_int @ F2 @ As2 @ ( cons_int @ ( F2 @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_626_last__in__set,axiom,
    ! [As2: list_nat] :
      ( ( As2 != nil_nat )
     => ( member_nat2 @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).

% last_in_set
thf(fact_627_last__in__set,axiom,
    ! [As2: list_P5707943133018811711nt_int] :
      ( ( As2 != nil_Pr2300489316682597567nt_int )
     => ( member5262025264175285858nt_int @ ( last_P3305686521732843992nt_int @ As2 ) @ ( set_Pr2470121279949933262nt_int @ As2 ) ) ) ).

% last_in_set
thf(fact_628_last__in__set,axiom,
    ! [As2: list_int] :
      ( ( As2 != nil_int )
     => ( member_int2 @ ( last_int @ As2 ) @ ( set_int2 @ As2 ) ) ) ).

% last_in_set
thf(fact_629_in__set__butlast__appendI,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
        | ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
     => ( member_nat2 @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_630_in__set__butlast__appendI,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] :
      ( ( ( member_int2 @ X @ ( set_int2 @ ( butlast_int @ Xs ) ) )
        | ( member_int2 @ X @ ( set_int2 @ ( butlast_int @ Ys ) ) ) )
     => ( member_int2 @ X @ ( set_int2 @ ( butlast_int @ ( append_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_631_listrel1__eq__len,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) )
     => ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_632_listrel__eq__len,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre7525836092462517041nt_int @ R ) )
     => ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_633_Cons__in__subseqsD,axiom,
    ! [Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member2764346250752101224nt_int @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) @ ( set_li2659200638379878868nt_int @ ( subseq1357044202310323342nt_int @ Xs ) ) )
     => ( member2764346250752101224nt_int @ Ys @ ( set_li2659200638379878868nt_int @ ( subseq1357044202310323342nt_int @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_634_Cons__in__subseqsD,axiom,
    ! [Y: int,Ys: list_int,Xs: list_int] :
      ( ( member_list_int @ ( cons_int @ Y @ Ys ) @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) )
     => ( member_list_int @ Ys @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_635_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X4: nat,Xs4: list_nat] : ( if_list_nat @ ( member_nat2 @ X4 @ ( set_nat2 @ Xs4 ) ) @ Xs4 @ ( cons_nat @ X4 @ Xs4 ) ) ) ) ).

% List.insert_def
thf(fact_636_List_Oinsert__def,axiom,
    ( insert5765537519290168021nt_int
    = ( ^ [X4: product_prod_int_int,Xs4: list_P5707943133018811711nt_int] : ( if_lis8883190402267401221nt_int @ ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs4 ) ) @ Xs4 @ ( cons_P3334398858971670639nt_int @ X4 @ Xs4 ) ) ) ) ).

% List.insert_def
thf(fact_637_List_Oinsert__def,axiom,
    ( insert_int
    = ( ^ [X4: int,Xs4: list_int] : ( if_list_int @ ( member_int2 @ X4 @ ( set_int2 @ Xs4 ) ) @ Xs4 @ ( cons_int @ X4 @ Xs4 ) ) ) ) ).

% List.insert_def
thf(fact_638_irreflD,axiom,
    ! [R: set_Pr958786334691620121nt_int,X: int] :
      ( ( irrefl_on_int @ top_top_set_int @ R )
     => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_639_irreflI,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ! [A3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ A3 ) @ R )
     => ( irrefl_on_int @ top_top_set_int @ R ) ) ).

% irreflI
thf(fact_640_same__length__different,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Ys ) )
       => ? [Pre: list_int,X3: int,Xs2: list_int,Y3: int,Ys2: list_int] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_int @ Pre @ ( append_int @ ( cons_int @ X3 @ nil_int ) @ Xs2 ) ) )
            & ( Ys
              = ( append_int @ Pre @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ Ys2 ) ) ) ) ) ) ).

% same_length_different
thf(fact_641_same__length__different,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( Xs != Ys )
     => ( ( ( size_s5157815400016825771nt_int @ Xs )
          = ( size_s5157815400016825771nt_int @ Ys ) )
       => ? [Pre: list_P5707943133018811711nt_int,X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int,Y3: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append7030698103840186580nt_int @ Pre @ ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ nil_Pr2300489316682597567nt_int ) @ Xs2 ) ) )
            & ( Ys
              = ( append7030698103840186580nt_int @ Pre @ ( append7030698103840186580nt_int @ ( cons_P3334398858971670639nt_int @ Y3 @ nil_Pr2300489316682597567nt_int ) @ Ys2 ) ) ) ) ) ) ).

% same_length_different
thf(fact_642_lexord__partial__trans,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_643_lexord__partial__trans,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int,Ys: list_int,Zs: list_int] :
      ( ! [X3: int,Y3: int,Z2: int] :
          ( ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
         => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y3 @ Z2 ) @ R )
             => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Z2 ) @ R ) ) ) )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lexord_int @ R ) )
       => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Zs ) @ ( lexord_int @ R ) )
         => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs ) @ ( lexord_int @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_644_lexord__sufE,axiom,
    ! [Xs: list_int,Zs: list_int,Ys: list_int,Qs: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Zs ) @ ( append_int @ Ys @ Qs ) ) @ ( lexord_int @ R ) )
     => ( ( Xs != Ys )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) )
         => ( ( ( size_size_list_int @ Zs )
              = ( size_size_list_int @ Qs ) )
           => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lexord_int @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_645_lexord__sufE,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Zs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Qs: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Zs ) @ ( append7030698103840186580nt_int @ Ys @ Qs ) ) @ ( lexord8886728381914977324nt_int @ R ) )
     => ( ( Xs != Ys )
       => ( ( ( size_s5157815400016825771nt_int @ Xs )
            = ( size_s5157815400016825771nt_int @ Ys ) )
         => ( ( ( size_s5157815400016825771nt_int @ Zs )
              = ( size_s5157815400016825771nt_int @ Qs ) )
           => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_646_lex__append__rightI,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int,Vs: list_int,Us2: list_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) )
     => ( ( ( size_size_list_int @ Vs )
          = ( size_size_list_int @ Us2 ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ Us2 ) @ ( append_int @ Ys @ Vs ) ) @ ( lex_int @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_647_lex__append__rightI,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,Vs: list_P5707943133018811711nt_int,Us2: list_P5707943133018811711nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
     => ( ( ( size_s5157815400016825771nt_int @ Vs )
          = ( size_s5157815400016825771nt_int @ Us2 ) )
       => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Xs @ Us2 ) @ ( append7030698103840186580nt_int @ Ys @ Vs ) ) @ ( lex_Pr5393148144989827363nt_int @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_648_lexord__lex,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
      = ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
        & ( ( size_s5157815400016825771nt_int @ X )
          = ( size_s5157815400016825771nt_int @ Y ) ) ) ) ).

% lexord_lex
thf(fact_649_map__tailrec__rev_Oelims,axiom,
    ! [X: product_prod_int_int > product_prod_int_int,Xa: list_P5707943133018811711nt_int,Xb: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( map_ta5484600921387902261nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
         => ( Y != Xb ) )
       => ~ ! [A3: product_prod_int_int,As: list_P5707943133018811711nt_int] :
              ( ( Xa
                = ( cons_P3334398858971670639nt_int @ A3 @ As ) )
             => ( Y
               != ( map_ta5484600921387902261nt_int @ X @ As @ ( cons_P3334398858971670639nt_int @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_650_map__tailrec__rev_Oelims,axiom,
    ! [X: product_prod_int_int > int,Xa: list_P5707943133018811711nt_int,Xb: list_int,Y: list_int] :
      ( ( ( map_ta8640746543295758256nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_Pr2300489316682597567nt_int )
         => ( Y != Xb ) )
       => ~ ! [A3: product_prod_int_int,As: list_P5707943133018811711nt_int] :
              ( ( Xa
                = ( cons_P3334398858971670639nt_int @ A3 @ As ) )
             => ( Y
               != ( map_ta8640746543295758256nt_int @ X @ As @ ( cons_int @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_651_map__tailrec__rev_Oelims,axiom,
    ! [X: int > product_prod_int_int,Xa: list_int,Xb: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( map_ta80682011918873046nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( Y != Xb ) )
       => ~ ! [A3: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A3 @ As ) )
             => ( Y
               != ( map_ta80682011918873046nt_int @ X @ As @ ( cons_P3334398858971670639nt_int @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_652_map__tailrec__rev_Oelims,axiom,
    ! [X: int > int,Xa: list_int,Xb: list_int,Y: list_int] :
      ( ( ( map_ta8161051695879884879nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( Y != Xb ) )
       => ~ ! [A3: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A3 @ As ) )
             => ( Y
               != ( map_ta8161051695879884879nt_int @ X @ As @ ( cons_int @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_653_is__empty__set,axiom,
    ! [Xs: list_int] :
      ( ( is_empty_int @ ( set_int2 @ Xs ) )
      = ( null_int @ Xs ) ) ).

% is_empty_set
thf(fact_654_can__select__set__list__ex1,axiom,
    ! [P2: int > $o,A4: list_int] :
      ( ( can_select_int @ P2 @ ( set_int2 @ A4 ) )
      = ( list_ex1_int @ P2 @ A4 ) ) ).

% can_select_set_list_ex1
thf(fact_655_splice_Opelims,axiom,
    ! [X: list_P5707943133018811711nt_int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( splice6983101402924261266nt_int @ X @ Xa )
        = Y )
     => ( ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ X @ Xa ) )
       => ( ( ( X = nil_Pr2300489316682597567nt_int )
           => ( ( Y = Xa )
             => ~ ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Xa ) ) ) )
         => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
                ( ( X
                  = ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( cons_P3334398858971670639nt_int @ X3 @ ( splice6983101402924261266nt_int @ Xa @ Xs3 ) ) )
                 => ~ ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_656_splice_Opelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( ( splice_int @ X @ Xa )
        = Y )
     => ( ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ X @ Xa ) )
       => ( ( ( X = nil_int )
           => ( ( Y = Xa )
             => ~ ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ nil_int @ Xa ) ) ) )
         => ~ ! [X3: int,Xs3: list_int] :
                ( ( X
                  = ( cons_int @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( cons_int @ X3 @ ( splice_int @ Xa @ Xs3 ) ) )
                 => ~ ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_657_lenlex__append2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,Us2: list_int,Xs: list_int,Ys: list_int] :
      ( ( irrefl_on_int @ top_top_set_int @ R2 )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Us2 @ Xs ) @ ( append_int @ Us2 @ Ys ) ) @ ( lenlex_int @ R2 ) )
        = ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lenlex_int @ R2 ) ) ) ) ).

% lenlex_append2
thf(fact_658_the__elem__set,axiom,
    ! [X: product_prod_int_int] :
      ( ( the_el8326832613380209454nt_int @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ nil_Pr2300489316682597567nt_int ) ) )
      = X ) ).

% the_elem_set
thf(fact_659_the__elem__set,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( set_int2 @ ( cons_int @ X @ nil_int ) ) )
      = X ) ).

% the_elem_set
thf(fact_660_splice_Opsimps_I1_J,axiom,
    ! [Ys: list_P5707943133018811711nt_int] :
      ( ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ys ) )
     => ( ( splice6983101402924261266nt_int @ nil_Pr2300489316682597567nt_int @ Ys )
        = Ys ) ) ).

% splice.psimps(1)
thf(fact_661_splice_Opsimps_I1_J,axiom,
    ! [Ys: list_int] :
      ( ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ nil_int @ Ys ) )
     => ( ( splice_int @ nil_int @ Ys )
        = Ys ) ) ).

% splice.psimps(1)
thf(fact_662_splice__Nil2,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( splice6983101402924261266nt_int @ Xs @ nil_Pr2300489316682597567nt_int )
      = Xs ) ).

% splice_Nil2
thf(fact_663_splice__Nil2,axiom,
    ! [Xs: list_int] :
      ( ( splice_int @ Xs @ nil_int )
      = Xs ) ).

% splice_Nil2
thf(fact_664_split__Nil__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( splice6983101402924261266nt_int @ Xs @ Ys )
        = nil_Pr2300489316682597567nt_int )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        & ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% split_Nil_iff
thf(fact_665_split__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( splice_int @ Xs @ Ys )
        = nil_int )
      = ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% split_Nil_iff
thf(fact_666_Nil__lenlex__iff1,axiom,
    ! [Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ nil_Pr2300489316682597567nt_int @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) )
      = ( Ns != nil_Pr2300489316682597567nt_int ) ) ).

% Nil_lenlex_iff1
thf(fact_667_Nil__lenlex__iff1,axiom,
    ! [Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ns ) @ ( lenlex_int @ R ) )
      = ( Ns != nil_int ) ) ).

% Nil_lenlex_iff1
thf(fact_668_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P5: nat > $o,A6: set_nat] :
        ? [X4: nat] :
          ( ( member_nat2 @ X4 @ A6 )
          & ( P5 @ X4 )
          & ! [Y5: nat] :
              ( ( ( member_nat2 @ Y5 @ A6 )
                & ( P5 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_669_splice_Osimps_I2_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( splice6983101402924261266nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys )
      = ( cons_P3334398858971670639nt_int @ X @ ( splice6983101402924261266nt_int @ Ys @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_670_splice_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] :
      ( ( splice_int @ ( cons_int @ X @ Xs ) @ Ys )
      = ( cons_int @ X @ ( splice_int @ Ys @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_671_splice_Osimps_I1_J,axiom,
    ! [Ys: list_P5707943133018811711nt_int] :
      ( ( splice6983101402924261266nt_int @ nil_Pr2300489316682597567nt_int @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_672_splice_Osimps_I1_J,axiom,
    ! [Ys: list_int] :
      ( ( splice_int @ nil_int @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_673_lenlex__irreflexive,axiom,
    ! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
     => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_674_Nil__lenlex__iff2,axiom,
    ! [Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ~ ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ns @ nil_Pr2300489316682597567nt_int ) @ ( lenlex6370358691973319492nt_int @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_675_Nil__lenlex__iff2,axiom,
    ! [Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ns @ nil_int ) @ ( lenlex_int @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_676_splice_Oelims,axiom,
    ! [X: list_P5707943133018811711nt_int,Xa: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( splice6983101402924261266nt_int @ X @ Xa )
        = Y )
     => ( ( ( X = nil_Pr2300489316682597567nt_int )
         => ( Y != Xa ) )
       => ~ ! [X3: product_prod_int_int,Xs3: list_P5707943133018811711nt_int] :
              ( ( X
                = ( cons_P3334398858971670639nt_int @ X3 @ Xs3 ) )
             => ( Y
               != ( cons_P3334398858971670639nt_int @ X3 @ ( splice6983101402924261266nt_int @ Xa @ Xs3 ) ) ) ) ) ) ).

% splice.elims
thf(fact_677_splice_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( ( splice_int @ X @ Xa )
        = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ~ ! [X3: int,Xs3: list_int] :
              ( ( X
                = ( cons_int @ X3 @ Xs3 ) )
             => ( Y
               != ( cons_int @ X3 @ ( splice_int @ Xa @ Xs3 ) ) ) ) ) ) ).

% splice.elims
thf(fact_678_lenlex__append1,axiom,
    ! [Us2: list_int,Xs: list_int,R2: set_Pr958786334691620121nt_int,Vs: list_int,Ys: list_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Us2 @ Xs ) @ ( lenlex_int @ R2 ) )
     => ( ( ( size_size_list_int @ Vs )
          = ( size_size_list_int @ Ys ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Us2 @ Vs ) @ ( append_int @ Xs @ Ys ) ) @ ( lenlex_int @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_679_lenlex__append1,axiom,
    ! [Us2: list_P5707943133018811711nt_int,Xs: list_P5707943133018811711nt_int,R2: set_Pr2560585780119916871nt_int,Vs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Us2 @ Xs ) @ ( lenlex6370358691973319492nt_int @ R2 ) )
     => ( ( ( size_s5157815400016825771nt_int @ Vs )
          = ( size_s5157815400016825771nt_int @ Ys ) )
       => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ Us2 @ Vs ) @ ( append7030698103840186580nt_int @ Xs @ Ys ) ) @ ( lenlex6370358691973319492nt_int @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_680_splice_Opsimps_I2_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( accp_P1749757178599463408nt_int @ splice6470080718302970981nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys ) )
     => ( ( splice6983101402924261266nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ Ys )
        = ( cons_P3334398858971670639nt_int @ X @ ( splice6983101402924261266nt_int @ Ys @ Xs ) ) ) ) ).

% splice.psimps(2)
thf(fact_681_splice_Opsimps_I2_J,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] :
      ( ( accp_P7395280492732805858st_int @ splice_rel_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys ) )
     => ( ( splice_int @ ( cons_int @ X @ Xs ) @ Ys )
        = ( cons_int @ X @ ( splice_int @ Ys @ Xs ) ) ) ) ).

% splice.psimps(2)
thf(fact_682_Cons__lenlex__iff,axiom,
    ! [M: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
      = ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
        | ( ( ( size_size_list_int @ Ms )
            = ( size_size_list_int @ Ns ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_683_Cons__lenlex__iff,axiom,
    ! [M: product_prod_int_int,Ms: list_P5707943133018811711nt_int,N: product_prod_int_int,Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( cons_P3334398858971670639nt_int @ M @ Ms ) @ ( cons_P3334398858971670639nt_int @ N @ Ns ) ) @ ( lenlex6370358691973319492nt_int @ R ) )
      = ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) )
        | ( ( ( size_s5157815400016825771nt_int @ Ms )
            = ( size_s5157815400016825771nt_int @ Ns ) )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_684_last__zip,axiom,
    ! [Xs: list_nat,Ys: list_set_int] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_set_int )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_s763970886550748294et_int @ Ys ) )
         => ( ( last_P2058727373477175858et_int @ ( zip_nat_set_int @ Xs @ Ys ) )
            = ( produc29655638201817675et_int @ ( last_nat @ Xs ) @ ( last_set_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_685_last__zip,axiom,
    ! [Xs: list_int,Ys: list_l1670014477004246597nt_int] :
      ( ( Xs != nil_int )
     => ( ( Ys != nil_li8670148097206105925nt_int )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_s2969076144586574001nt_int @ Ys ) )
         => ( ( last_P2520469823115007657nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) )
            = ( produc8814303788642274490nt_int @ ( last_int @ Xs ) @ ( last_l5818330359162608606nt_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_686_last__zip,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs != nil_int )
     => ( ( Ys != nil_int )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) )
         => ( ( last_P3305686521732843992nt_int @ ( zip_int_int @ Xs @ Ys ) )
            = ( product_Pair_int_int @ ( last_int @ Xs ) @ ( last_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_687_last__zip,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_int )
     => ( ( Ys != nil_Pr2300489316682597567nt_int )
       => ( ( ( size_size_list_int @ Xs )
            = ( size_s5157815400016825771nt_int @ Ys ) )
         => ( ( last_P1404851042722871203nt_int @ ( zip_in865470750896630164nt_int @ Xs @ Ys ) )
            = ( produc346731203614136500nt_int @ ( last_int @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_688_last__zip,axiom,
    ! [Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_Pr2193041437141079013e_term )
     => ( ( Ys != nil_Pr2300489316682597567nt_int )
       => ( ( ( size_s3817825649112772879e_term @ Xs )
            = ( size_s5157815400016825771nt_int @ Ys ) )
         => ( ( last_P7958387789370520228nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) )
            = ( produc5700946648718959541nt_int @ ( last_P8974135382477003148e_term @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_689_last__zip,axiom,
    ! [Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_in3447728264653889115e_term )
     => ( ( Ys != nil_Pr2300489316682597567nt_int )
       => ( ( ( size_s8881356780637927685e_term @ Xs )
            = ( size_s5157815400016825771nt_int @ Ys ) )
         => ( ( last_P858631638320919470nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) )
            = ( produc4305682042979456191nt_int @ ( last_i813233280922151042e_term @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_690_last__zip,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ( ( Ys != nil_int )
       => ( ( ( size_s5157815400016825771nt_int @ Xs )
            = ( size_size_list_int @ Ys ) )
         => ( ( last_P7399753664740188981nt_int @ ( zip_Pr202163245418739566nt_int @ Xs @ Ys ) )
            = ( produc8906795734991021710nt_int @ ( last_P3305686521732843992nt_int @ Xs ) @ ( last_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_691_last__zip,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( Xs != nil_Pr2300489316682597567nt_int )
     => ( ( Ys != nil_Pr2300489316682597567nt_int )
       => ( ( ( size_s5157815400016825771nt_int @ Xs )
            = ( size_s5157815400016825771nt_int @ Ys ) )
         => ( ( last_P6195391842956826118nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) )
            = ( produc3646306378393792727nt_int @ ( last_P3305686521732843992nt_int @ Xs ) @ ( last_P3305686521732843992nt_int @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_692_length__n__lists__elem,axiom,
    ! [Ys: list_P5707943133018811711nt_int,N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( member2764346250752101224nt_int @ Ys @ ( set_li2659200638379878868nt_int @ ( n_list2671089462871817415nt_int @ N @ Xs ) ) )
     => ( ( size_s5157815400016825771nt_int @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_693_length__product,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( size_s6770063216428074713nt_int @ ( produc1028813369992947845nt_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% length_product
thf(fact_694_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_695_lexord__sufI,axiom,
    ! [U: list_int,W: list_int,R: set_Pr958786334691620121nt_int,V: list_int,Z: list_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ U @ W ) @ ( lexord_int @ R ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_int @ W ) @ ( size_size_list_int @ U ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ U @ V ) @ ( append_int @ W @ Z ) ) @ ( lexord_int @ R ) ) ) ) ).

% lexord_sufI
thf(fact_696_lexord__sufI,axiom,
    ! [U: list_P5707943133018811711nt_int,W: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int,V: list_P5707943133018811711nt_int,Z: list_P5707943133018811711nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ U @ W ) @ ( lexord8886728381914977324nt_int @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ W ) @ ( size_s5157815400016825771nt_int @ U ) )
       => ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ ( append7030698103840186580nt_int @ U @ V ) @ ( append7030698103840186580nt_int @ W @ Z ) ) @ ( lexord8886728381914977324nt_int @ R ) ) ) ) ).

% lexord_sufI
thf(fact_697_transpose__empty,axiom,
    ! [Xs: list_l1670014477004246597nt_int] :
      ( ( ( transp5579559677893564233nt_int @ Xs )
        = nil_li8670148097206105925nt_int )
      = ( ! [X4: list_P5707943133018811711nt_int] :
            ( ( member2764346250752101224nt_int @ X4 @ ( set_li2659200638379878868nt_int @ Xs ) )
           => ( X4 = nil_Pr2300489316682597567nt_int ) ) ) ) ).

% transpose_empty
thf(fact_698_transpose__empty,axiom,
    ! [Xs: list_list_int] :
      ( ( ( transpose_int @ Xs )
        = nil_list_int )
      = ( ! [X4: list_int] :
            ( ( member_list_int @ X4 @ ( set_list_int2 @ Xs ) )
           => ( X4 = nil_int ) ) ) ) ).

% transpose_empty
thf(fact_699_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_700_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_701_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_702_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_703_zip__append,axiom,
    ! [Xs: list_int,Us2: list_int,Ys: list_int,Vs: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Us2 ) )
     => ( ( zip_int_int @ ( append_int @ Xs @ Ys ) @ ( append_int @ Us2 @ Vs ) )
        = ( append7030698103840186580nt_int @ ( zip_int_int @ Xs @ Us2 ) @ ( zip_int_int @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_704_zip__append,axiom,
    ! [Xs: list_int,Us2: list_P5707943133018811711nt_int,Ys: list_int,Vs: list_P5707943133018811711nt_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Us2 ) )
     => ( ( zip_in865470750896630164nt_int @ ( append_int @ Xs @ Ys ) @ ( append7030698103840186580nt_int @ Us2 @ Vs ) )
        = ( append6084095472372872615nt_int @ ( zip_in865470750896630164nt_int @ Xs @ Us2 ) @ ( zip_in865470750896630164nt_int @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_705_zip__append,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Us2: list_int,Ys: list_P5707943133018811711nt_int,Vs: list_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_size_list_int @ Us2 ) )
     => ( ( zip_Pr202163245418739566nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ ( append_int @ Us2 @ Vs ) )
        = ( append2855626057535414585nt_int @ ( zip_Pr202163245418739566nt_int @ Xs @ Us2 ) @ ( zip_Pr202163245418739566nt_int @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_706_zip__append,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Us2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Vs: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Us2 ) )
     => ( ( zip_Pr2148635498163357687nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ ( append7030698103840186580nt_int @ Us2 @ Vs ) )
        = ( append9111546825139864322nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Us2 ) @ ( zip_Pr2148635498163357687nt_int @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_707_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( zip_Pr2148635498163357687nt_int @ Xs @ Ys )
        = nil_Pr2941907410405800557nt_int )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        | ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_708_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int] :
      ( ( ( zip_Pr202163245418739566nt_int @ Xs @ Ys )
        = nil_Pr4975111619762539406nt_int )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_709_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( zip_in865470750896630164nt_int @ Xs @ Ys )
        = nil_Pr8203581034599997436nt_int )
      = ( ( Xs = nil_int )
        | ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_710_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = nil_Pr2300489316682597567nt_int )
      = ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_711_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( nil_Pr2941907410405800557nt_int
        = ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        | ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_712_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int] :
      ( ( nil_Pr4975111619762539406nt_int
        = ( zip_Pr202163245418739566nt_int @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr2300489316682597567nt_int )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_713_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int] :
      ( ( nil_Pr8203581034599997436nt_int
        = ( zip_in865470750896630164nt_int @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_Pr2300489316682597567nt_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_714_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_Pr2300489316682597567nt_int
        = ( zip_int_int @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_715_zip__Nil,axiom,
    ! [Ys: list_int] :
      ( ( zip_int_int @ nil_int @ Ys )
      = nil_Pr2300489316682597567nt_int ) ).

% zip_Nil
thf(fact_716_in__measures_I2_J,axiom,
    ! [X: int,Y: int,F2: int > nat,Fs: list_int_nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_717_zip__Cons__Cons,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
      ( ( zip_Pr2148635498163357687nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) )
      = ( cons_P9091998269994194461nt_int @ ( produc3646306378393792727nt_int @ X @ Y ) @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_718_zip__Cons__Cons,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: int,Ys: list_int] :
      ( ( zip_Pr202163245418739566nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P4449357501642116062nt_int @ ( produc8906795734991021710nt_int @ X @ Y ) @ ( zip_Pr202163245418739566nt_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_719_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
      ( ( zip_in865470750896630164nt_int @ ( cons_int @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) )
      = ( cons_P7677826916479574092nt_int @ ( produc346731203614136500nt_int @ X @ Y ) @ ( zip_in865470750896630164nt_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_720_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: set_int,Ys: list_set_int] :
      ( ( zip_nat_set_int @ ( cons_nat @ X @ Xs ) @ ( cons_set_int @ Y @ Ys ) )
      = ( cons_P5529800768674571465et_int @ ( produc29655638201817675et_int @ X @ Y ) @ ( zip_nat_set_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_721_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: list_P5707943133018811711nt_int,Ys: list_l1670014477004246597nt_int] :
      ( ( zip_in8018307065301790106nt_int @ ( cons_int @ X @ Xs ) @ ( cons_l7309679040211256053nt_int @ Y @ Ys ) )
      = ( cons_P9095479431110121170nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_722_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_723_zip__Cons__Cons,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Xs: list_P1743416141875011707e_term,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
      ( ( zip_Pr4168994715204986005nt_int @ ( cons_P2630085844062958645e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) )
      = ( cons_P6018425551955479501nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_724_zip__Cons__Cons,axiom,
    ! [X: int > option6357759511663192854e_term,Xs: list_i8448526496819171953e_term,Y: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
      ( ( zip_in8766932505889695135nt_int @ ( cons_i7166360444231718571e_term @ X @ Xs ) @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) )
      = ( cons_P2743708091642732631nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_725_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_726_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_727_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_728_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_729_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_730_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_731_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_732_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_733_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_734_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_735_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_736_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_737_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_738_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_739_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_740_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y7: int,Z4: int] : ( Y7 = Z4 ) )
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_eq_int @ X4 @ Y5 )
          & ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_741_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_742_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_743_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_744_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_745_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_746_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_747_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_748_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_749_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_750_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_751_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_752_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_753_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_754_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_755_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_756_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_757_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_758_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_759_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_760_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_761_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_762_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X3 )
             => ( P2 @ Y6 ) )
         => ( P2 @ X3 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_763_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_764_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_765_linorder__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_766_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_767_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_768_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_769_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y7: int,Z4: int] : ( Y7 = Z4 ) )
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ B5 @ A5 )
          & ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_770_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 ) )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_771_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_eq_int @ X4 @ Y5 )
          & ~ ( ord_less_eq_int @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_772_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_773_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_774_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_775_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_776_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_777_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_778_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_779_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_780_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_781_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_782_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_783_exists__least__iff,axiom,
    ( ( ^ [P6: nat > $o] :
        ? [X6: nat] : ( P6 @ X6 ) )
    = ( ^ [P5: nat > $o] :
        ? [N3: nat] :
          ( ( P5 @ N3 )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
             => ~ ( P5 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_784_linorder__less__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P2 @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_785_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_786_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_787_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_788_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_789_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_790_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_791_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_792_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_793_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_794_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_795_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_796_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
          & ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_797_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_798_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_799_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_800_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_801_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_802_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_less_int @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_803_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_nat @ B5 @ A5 )
          | ( A5 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_804_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_less_eq_int @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_805_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ( A5 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_806_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_807_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_808_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_809_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_810_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_811_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_812_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_less_eq_int @ B5 @ A5 )
          & ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_813_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_814_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_815_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_816_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_817_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_818_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_819_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_820_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_821_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_822_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y7: int,Z4: int] : ( Y7 = Z4 ) )
    = ( ^ [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
          & ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_823_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y7: nat,Z4: nat] : ( Y7 = Z4 ) )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_824_order__subst1,axiom,
    ! [A: int,F2: int > int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_825_order__subst1,axiom,
    ! [A: int,F2: nat > int,B: nat,C: nat] :
      ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_826_order__subst1,axiom,
    ! [A: nat,F2: int > nat,B: int,C: int] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_827_order__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_828_order__subst2,axiom,
    ! [A: int,B: int,F2: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_829_order__subst2,axiom,
    ! [A: int,B: int,F2: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_830_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_831_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_832_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_833_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_834_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_835_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_836_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_int @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_837_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_838_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y5: int] :
          ( ( ord_less_eq_int @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_839_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_840_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_841_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_842_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_843_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_844_ord__eq__le__subst,axiom,
    ! [A: int,F2: int > int,B: int,C: int] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_845_ord__eq__le__subst,axiom,
    ! [A: nat,F2: int > nat,B: int,C: int] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_846_ord__eq__le__subst,axiom,
    ! [A: int,F2: nat > int,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_847_ord__eq__le__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_848_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F2: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_849_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F2: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_850_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_851_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_852_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_853_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_854_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_855_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_856_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_857_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_858_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_859_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_860_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_861_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_862_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_863_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_864_ord__eq__less__subst,axiom,
    ! [A: int,F2: int > int,B: int,C: int] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_865_ord__eq__less__subst,axiom,
    ! [A: nat,F2: int > nat,B: int,C: int] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_866_ord__eq__less__subst,axiom,
    ! [A: int,F2: nat > int,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_867_ord__eq__less__subst,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_868_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F2: int > int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_869_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F2: int > nat,C: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_870_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_871_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_872_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_873_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_874_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_875_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_876_order__less__subst1,axiom,
    ! [A: int,F2: int > int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_877_order__less__subst1,axiom,
    ! [A: int,F2: nat > int,B: nat,C: nat] :
      ( ( ord_less_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_878_order__less__subst1,axiom,
    ! [A: nat,F2: int > nat,B: int,C: int] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_879_order__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_880_order__less__subst2,axiom,
    ! [A: int,B: int,F2: int > int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_881_order__less__subst2,axiom,
    ! [A: int,B: int,F2: int > nat,C: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_882_order__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_883_order__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_884_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_885_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_886_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_887_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_888_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_889_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_890_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_891_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_892_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_893_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_894_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_895_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_896_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_897_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_898_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_899_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_900_order__le__less__subst1,axiom,
    ! [A: int,F2: int > int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_901_order__le__less__subst1,axiom,
    ! [A: int,F2: nat > int,B: nat,C: nat] :
      ( ( ord_less_eq_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_902_order__le__less__subst1,axiom,
    ! [A: nat,F2: int > nat,B: int,C: int] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_903_order__le__less__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_904_order__le__less__subst2,axiom,
    ! [A: int,B: int,F2: int > int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_905_order__le__less__subst2,axiom,
    ! [A: int,B: int,F2: int > nat,C: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_906_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > int,C: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_907_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_908_order__less__le__subst1,axiom,
    ! [A: int,F2: int > int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_909_order__less__le__subst1,axiom,
    ! [A: nat,F2: int > nat,B: int,C: int] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_910_order__less__le__subst1,axiom,
    ! [A: int,F2: nat > int,B: nat,C: nat] :
      ( ( ord_less_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_911_order__less__le__subst1,axiom,
    ! [A: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_912_order__less__le__subst2,axiom,
    ! [A: int,B: int,F2: int > int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_913_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > int,C: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_914_order__less__le__subst2,axiom,
    ! [A: int,B: int,F2: int > nat,C: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_915_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_916_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_917_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_918_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_919_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_920_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_921_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_922_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_923_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_924_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_925_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_926_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_927_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_928_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B5: int] : ( times_times_int @ B5 @ A5 ) ) ) ).

% mult.commute
thf(fact_929_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_930_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D2 @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_931_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_932_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_933_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_934_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D2 @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_935_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_936_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_937_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_938_length__induct,axiom,
    ! [P2: list_P5707943133018811711nt_int > $o,Xs: list_P5707943133018811711nt_int] :
      ( ! [Xs3: list_P5707943133018811711nt_int] :
          ( ! [Ys7: list_P5707943133018811711nt_int] :
              ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ Ys7 ) @ ( size_s5157815400016825771nt_int @ Xs3 ) )
             => ( P2 @ Ys7 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_939_set__zip__rightD,axiom,
    ! [X: nat,Y: set_int,Xs: list_nat,Ys: list_set_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y ) @ ( set_Pr3697907889984883496et_int @ ( zip_nat_set_int @ Xs @ Ys ) ) )
     => ( member_set_int @ Y @ ( set_set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_940_set__zip__rightD,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int,Xs: list_int,Ys: list_l1670014477004246597nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ ( set_Pr867874496877641011nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) ) )
     => ( member2764346250752101224nt_int @ Y @ ( set_li2659200638379878868nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_941_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_942_set__zip__rightD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_943_set__zip__rightD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_944_set__zip__leftD,axiom,
    ! [X: nat,Y: set_int,Xs: list_nat,Ys: list_set_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y ) @ ( set_Pr3697907889984883496et_int @ ( zip_nat_set_int @ Xs @ Ys ) ) )
     => ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_945_set__zip__leftD,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int,Xs: list_int,Ys: list_l1670014477004246597nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ ( set_Pr867874496877641011nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_946_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_947_set__zip__leftD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ( member3222579708246209666e_term @ X @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_948_set__zip__leftD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ( member8845023287901829240e_term @ X @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_949_in__set__zipE,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_950_in__set__zipE,axiom,
    ! [X: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( set_Pr1470767568048878706at_int @ ( zip_nat_int @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_951_in__set__zipE,axiom,
    ! [X: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( set_Pr6647972299459129970nt_nat @ ( zip_int_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_952_in__set__zipE,axiom,
    ! [X: nat,Y: set_int,Xs: list_nat,Ys: list_set_int] :
      ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y ) @ ( set_Pr3697907889984883496et_int @ ( zip_nat_set_int @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_set_int @ Y @ ( set_set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_953_in__set__zipE,axiom,
    ! [X: int,Y: list_P5707943133018811711nt_int,Xs: list_int,Ys: list_l1670014477004246597nt_int] :
      ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y ) @ ( set_Pr867874496877641011nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) ) )
     => ~ ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
         => ~ ( member2764346250752101224nt_int @ Y @ ( set_li2659200638379878868nt_int @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_954_in__set__zipE,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ~ ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
         => ~ ( member_int2 @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_955_in__set__zipE,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ~ ( ( member3222579708246209666e_term @ X @ ( set_Pr16608062948090134e_term @ Xs ) )
         => ~ ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_956_in__set__zipE,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ~ ( ( member8845023287901829240e_term @ X @ ( set_in5217446777445088012e_term @ Xs ) )
         => ~ ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_957_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat2 @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_958_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
      = ( ( member_int2 @ A @ ( set_int2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_959_zip_Osimps_I1_J,axiom,
    ! [Xs: list_int] :
      ( ( zip_int_int @ Xs @ nil_int )
      = nil_Pr2300489316682597567nt_int ) ).

% zip.simps(1)
thf(fact_960_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
     => ( Xs
       != ( cons_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_961_impossible__Cons,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys ) )
     => ( Xs
       != ( cons_P3334398858971670639nt_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_962_measures__less,axiom,
    ! [F2: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_963_measures__lesseq,axiom,
    ! [F2: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_964_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_set_int,Y: set_int] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s763970886550748294et_int @ Ys ) )
     => ( ( member_set_int @ Y @ ( set_set_int2 @ Ys ) )
       => ~ ! [X3: nat] :
              ~ ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y ) @ ( set_Pr3697907889984883496et_int @ ( zip_nat_set_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_965_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_l1670014477004246597nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s2969076144586574001nt_int @ Ys ) )
     => ( ( member2764346250752101224nt_int @ Y @ ( set_li2659200638379878868nt_int @ Ys ) )
       => ~ ! [X3: int] :
              ~ ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y ) @ ( set_Pr867874496877641011nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_966_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_int,Ys: list_int,Y: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_967_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( ( size_s3817825649112772879e_term @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) )
       => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term] :
              ~ ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_968_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( ( size_s8881356780637927685e_term @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) )
       => ~ ! [X3: int > option6357759511663192854e_term] :
              ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_969_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_nat,Y: nat] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat2 @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: product_prod_int_int] :
              ~ ( member5907334269321869775nt_nat @ ( produc8909286205500071986nt_nat @ X3 @ Y ) @ ( set_Pr1171648944650862179nt_nat @ ( zip_Pr204653715927789842nt_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_970_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int,Y: int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: product_prod_int_int] :
              ~ ( member1729483249812673067nt_int @ ( produc8906795734991021710nt_int @ X3 @ Y ) @ ( set_Pr6217169961996441279nt_int @ ( zip_Pr202163245418739566nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_971_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ Y @ ( set_Pr2470121279949933262nt_int @ Ys ) )
       => ~ ! [X3: product_prod_int_int] :
              ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X3 @ Y ) @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_972_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_set_int,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s763970886550748294et_int @ Ys ) )
     => ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: set_int] :
              ~ ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X @ Y3 ) @ ( set_Pr3697907889984883496et_int @ ( zip_nat_set_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_973_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_l1670014477004246597nt_int,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s2969076144586574001nt_int @ Ys ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: list_P5707943133018811711nt_int] :
              ~ ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X @ Y3 ) @ ( set_Pr867874496877641011nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_974_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: int] :
              ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_975_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_P5707943133018811711nt_int,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member1920110580185997301nt_int @ ( produc6532261156004778512nt_int @ X @ Y3 ) @ ( set_Pr6407797292369765513nt_int @ ( zip_na7051000703287272176nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_976_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member4957952664650131097nt_int @ ( produc346731203614136500nt_int @ X @ Y3 ) @ ( set_Pr222267339979123501nt_int @ ( zip_in865470750896630164nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_977_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int,X: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( ( size_s3817825649112772879e_term @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member3222579708246209666e_term @ X @ ( set_Pr16608062948090134e_term @ Xs ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_978_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int,X: int > option6357759511663192854e_term] :
      ( ( ( size_s8881356780637927685e_term @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member8845023287901829240e_term @ X @ ( set_in5217446777445088012e_term @ Xs ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_979_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
       => ~ ! [Y3: product_prod_int_int] :
              ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ X @ Y3 ) @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_980_zip__eq__ConsE,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,Xy: produc1219242969750017639nt_int,Xys: list_P2336717926344734829nt_int] :
      ( ( ( zip_Pr2148635498163357687nt_int @ Xs @ Ys )
        = ( cons_P9091998269994194461nt_int @ Xy @ Xys ) )
     => ~ ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) )
           => ! [Y3: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
                ( ( Ys
                  = ( cons_P3334398858971670639nt_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc3646306378393792727nt_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr2148635498163357687nt_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_981_zip__eq__ConsE,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_int,Xy: produc6754623152930779156nt_int,Xys: list_P3467955165070007716nt_int] :
      ( ( ( zip_Pr202163245418739566nt_int @ Xs @ Ys )
        = ( cons_P4449357501642116062nt_int @ Xy @ Xys ) )
     => ~ ! [X3: product_prod_int_int,Xs2: list_P5707943133018811711nt_int] :
            ( ( Xs
              = ( cons_P3334398858971670639nt_int @ X3 @ Xs2 ) )
           => ! [Y3: int,Ys2: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc8906795734991021710nt_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr202163245418739566nt_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_982_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_P5707943133018811711nt_int,Xy: produc759720530913461378nt_int,Xys: list_P8440714079264627474nt_int] :
      ( ( ( zip_in865470750896630164nt_int @ Xs @ Ys )
        = ( cons_P7677826916479574092nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs2 ) )
           => ! [Y3: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
                ( ( Ys
                  = ( cons_P3334398858971670639nt_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc346731203614136500nt_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_in865470750896630164nt_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_983_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_set_int,Xy: produc9133624956312949779et_int,Xys: list_P7388728034488748185et_int] :
      ( ( ( zip_nat_set_int @ Xs @ Ys )
        = ( cons_P5529800768674571465et_int @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ! [Y3: set_int,Ys2: list_set_int] :
                ( ( Ys
                  = ( cons_set_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc29655638201817675et_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_set_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_984_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_l1670014477004246597nt_int,Xy: produc661532565036771336nt_int,Xys: list_P6957225279449687320nt_int] :
      ( ( ( zip_in8018307065301790106nt_int @ Xs @ Ys )
        = ( cons_P9095479431110121170nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs2 ) )
           => ! [Y3: list_P5707943133018811711nt_int,Ys2: list_l1670014477004246597nt_int] :
                ( ( Ys
                  = ( cons_l7309679040211256053nt_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc8814303788642274490nt_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_in8018307065301790106nt_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_985_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs2: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs2 ) )
           => ! [Y3: int,Ys2: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_986_zip__eq__ConsE,axiom,
    ! [Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int,Xy: produc2285326912895808259nt_int,Xys: list_P651320350408439699nt_int] :
      ( ( ( zip_Pr4168994715204986005nt_int @ Xs @ Ys )
        = ( cons_P6018425551955479501nt_int @ Xy @ Xys ) )
     => ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Xs2: list_P1743416141875011707e_term] :
            ( ( Xs
              = ( cons_P2630085844062958645e_term @ X3 @ Xs2 ) )
           => ! [Y3: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
                ( ( Ys
                  = ( cons_P3334398858971670639nt_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr4168994715204986005nt_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_987_zip__eq__ConsE,axiom,
    ! [Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int,Xy: produc7773217078559923341nt_int,Xys: list_P8915022641806594461nt_int] :
      ( ( ( zip_in8766932505889695135nt_int @ Xs @ Ys )
        = ( cons_P2743708091642732631nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int > option6357759511663192854e_term,Xs2: list_i8448526496819171953e_term] :
            ( ( Xs
              = ( cons_i7166360444231718571e_term @ X3 @ Xs2 ) )
           => ! [Y3: product_prod_int_int,Ys2: list_P5707943133018811711nt_int] :
                ( ( Ys
                  = ( cons_P3334398858971670639nt_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_in8766932505889695135nt_int @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_988_List_Otranspose_Osimps_I2_J,axiom,
    ! [Xss2: list_l1670014477004246597nt_int] :
      ( ( transp5579559677893564233nt_int @ ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss2 ) )
      = ( transp5579559677893564233nt_int @ Xss2 ) ) ).

% List.transpose.simps(2)
thf(fact_989_List_Otranspose_Osimps_I2_J,axiom,
    ! [Xss2: list_list_int] :
      ( ( transpose_int @ ( cons_list_int @ nil_int @ Xss2 ) )
      = ( transpose_int @ Xss2 ) ) ).

% List.transpose.simps(2)
thf(fact_990_lenlex__length,axiom,
    ! [Ms: list_P5707943133018811711nt_int,Ns: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Ms @ Ns ) @ ( lenlex6370358691973319492nt_int @ R ) )
     => ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Ms ) @ ( size_s5157815400016825771nt_int @ Ns ) ) ) ).

% lenlex_length
thf(fact_991_in__measure,axiom,
    ! [X: int,Y: int,F2: int > nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_992_List_Otranspose_Opsimps_I2_J,axiom,
    ! [Xss2: list_l1670014477004246597nt_int] :
      ( ( accp_l7878653158910992590nt_int @ transp3175463746559578478nt_int @ ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss2 ) )
     => ( ( transp5579559677893564233nt_int @ ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ Xss2 ) )
        = ( transp5579559677893564233nt_int @ Xss2 ) ) ) ).

% List.transpose.psimps(2)
thf(fact_993_List_Otranspose_Opsimps_I2_J,axiom,
    ! [Xss2: list_list_int] :
      ( ( accp_list_list_int @ transpose_rel_int @ ( cons_list_int @ nil_int @ Xss2 ) )
     => ( ( transpose_int @ ( cons_list_int @ nil_int @ Xss2 ) )
        = ( transpose_int @ Xss2 ) ) ) ).

% List.transpose.psimps(2)
thf(fact_994_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_995_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_996_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_997_accp__subset,axiom,
    ! [R1: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,R22: produc661532565036771336nt_int > produc661532565036771336nt_int > $o] :
      ( ( ord_le6253525219522735894_int_o @ R1 @ R22 )
     => ( ord_le4985975449105673191_int_o @ ( accp_P2804173024491580863nt_int @ R22 ) @ ( accp_P2804173024491580863nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_998_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_999_subrelI,axiom,
    ! [R: set_Pr4810089274464741491et_int,S: set_Pr4810089274464741491et_int] :
      ( ! [X3: nat,Y3: set_int] :
          ( ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ R )
         => ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le8255767777184198675et_int @ R @ S ) ) ).

% subrelI
thf(fact_1000_subrelI,axiom,
    ! [R: set_Pr1843331182504696254nt_int,S: set_Pr1843331182504696254nt_int] :
      ( ! [X3: int,Y3: list_P5707943133018811711nt_int] :
          ( ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ R )
         => ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le5985196928369334814nt_int @ R @ S ) ) ).

% subrelI
thf(fact_1001_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R @ S ) ) ).

% subrelI
thf(fact_1002_subrelI,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le8725513860283290265nt_int @ R @ S ) ) ).

% subrelI
thf(fact_1003_subrelI,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le135402666524580259nt_int @ R @ S ) ) ).

% subrelI
thf(fact_1004_image__mono,axiom,
    ! [A4: set_nat,B4: set_nat,F2: nat > nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ ( image_nat_nat @ F2 @ B4 ) ) ) ).

% image_mono
thf(fact_1005_image__subsetI,axiom,
    ! [A4: set_nat,F2: nat > nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat2 @ X3 @ A4 )
         => ( member_nat2 @ ( F2 @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_1006_subset__imageE,axiom,
    ! [B4: set_nat,F2: nat > nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A4 ) )
     => ~ ! [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A4 )
           => ( B4
             != ( image_nat_nat @ F2 @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_1007_image__subset__iff,axiom,
    ! [F2: nat > nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ A4 )
           => ( member_nat2 @ ( F2 @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_1008_subset__image__iff,axiom,
    ! [B4: set_nat,F2: nat > nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A4 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A4 )
            & ( B4
              = ( image_nat_nat @ F2 @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1009_image__diff__subset,axiom,
    ! [F2: nat > nat,A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ ( image_nat_nat @ F2 @ B4 ) ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% image_diff_subset
thf(fact_1010_accp__subset__induct,axiom,
    ! [D3: produc661532565036771336nt_int > $o,R2: produc661532565036771336nt_int > produc661532565036771336nt_int > $o,X: produc661532565036771336nt_int,P2: produc661532565036771336nt_int > $o] :
      ( ( ord_le4985975449105673191_int_o @ D3 @ ( accp_P2804173024491580863nt_int @ R2 ) )
     => ( ! [X3: produc661532565036771336nt_int,Z2: produc661532565036771336nt_int] :
            ( ( D3 @ X3 )
           => ( ( R2 @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: produc661532565036771336nt_int] :
                ( ( D3 @ X3 )
               => ( ! [Z3: produc661532565036771336nt_int] :
                      ( ( R2 @ Z3 @ X3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1011_accp__subset__induct,axiom,
    ! [D3: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D3 @ ( accp_P1096762738010456898nt_int @ R2 ) )
     => ( ! [X3: product_prod_int_int,Z2: product_prod_int_int] :
            ( ( D3 @ X3 )
           => ( ( R2 @ Z2 @ X3 )
             => ( D3 @ Z2 ) ) )
       => ( ( D3 @ X )
         => ( ! [X3: product_prod_int_int] :
                ( ( D3 @ X3 )
               => ( ! [Z3: product_prod_int_int] :
                      ( ( R2 @ Z3 @ X3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1012_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
           => ( member_nat2 @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1013_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B4 )
      = ( ! [X4: int] :
            ( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
           => ( member_int2 @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1014_range__subsetD,axiom,
    ! [F2: nat > nat,B4: set_nat,I3: nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ B4 )
     => ( member_nat2 @ ( F2 @ I3 ) @ B4 ) ) ).

% range_subsetD
thf(fact_1015_set__subset__Cons,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( set_Pr2470121279949933262nt_int @ Xs ) @ ( set_Pr2470121279949933262nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1016_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1017_subset__code_I2_J,axiom,
    ! [A4: set_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( coset_nat @ Ys ) )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
           => ~ ( member_nat2 @ X4 @ A4 ) ) ) ) ).

% subset_code(2)
thf(fact_1018_subset__code_I2_J,axiom,
    ! [A4: set_int,Ys: list_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( coset_int @ Ys ) )
      = ( ! [X4: int] :
            ( ( member_int2 @ X4 @ ( set_int2 @ Ys ) )
           => ~ ( member_int2 @ X4 @ A4 ) ) ) ) ).

% subset_code(2)
thf(fact_1019_subset__code_I3_J,axiom,
    ~ ( ord_le2843351958646193337nt_int @ ( coset_3631019413092825772nt_int @ nil_Pr2300489316682597567nt_int ) @ ( set_Pr2470121279949933262nt_int @ nil_Pr2300489316682597567nt_int ) ) ).

% subset_code(3)
thf(fact_1020_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_int @ ( coset_int @ nil_int ) @ ( set_int2 @ nil_int ) ) ).

% subset_code(3)
thf(fact_1021_subset__subseqs,axiom,
    ! [X7: set_int,Xs: list_int] :
      ( ( ord_less_eq_set_int @ X7 @ ( set_int2 @ Xs ) )
     => ( member_set_int @ X7 @ ( image_3606813740839090725et_int @ set_int2 @ ( set_list_int2 @ ( subseqs_int @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_1022_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1023_inf__period_I2_J,axiom,
    ! [P2: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X5: int,K3: int] :
            ( ( ( P2 @ X5 )
              | ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) )
              | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_1024_inf__period_I1_J,axiom,
    ! [P2: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
       => ! [X5: int,K3: int] :
            ( ( ( P2 @ X5 )
              & ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) )
              & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_1025_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list2671089462871817415nt_int @ N @ nil_Pr2300489316682597567nt_int )
          = ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list2671089462871817415nt_int @ N @ nil_Pr2300489316682597567nt_int )
          = nil_li8670148097206105925nt_int ) ) ) ).

% n_lists_Nil
thf(fact_1026_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = ( cons_list_int @ nil_int @ nil_list_int ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = nil_list_int ) ) ) ).

% n_lists_Nil
thf(fact_1027_in__set__zip,axiom,
    ! [P: produc1219242969750017639nt_int,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( member8566619992076573584nt_int @ P @ ( set_Pr5878228222108503548nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) ) )
      = ( ? [N3: nat] :
            ( ( ( nth_Pr4439495888332055232nt_int @ Xs @ N3 )
              = ( produc698254169746827971nt_int @ P ) )
            & ( ( nth_Pr4439495888332055232nt_int @ Ys @ N3 )
              = ( produc3892743399831173125nt_int @ P ) )
            & ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Xs ) )
            & ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1028_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat2 @ X3 @ A4 )
         => ( member_nat2 @ X3 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_1029_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1030_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1031_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1032_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1033_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1034_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1035_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1036_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_1037_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1038_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1039_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1040_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_int ) ) ).

% length_0_conv
thf(fact_1041_length__0__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% length_0_conv
thf(fact_1042_nth__Cons__0,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( nth_Pr4439495888332055232nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_1043_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_1044_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
      = ( Xs != nil_int ) ) ).

% length_greater_0_conv
thf(fact_1045_length__greater__0__conv,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s5157815400016825771nt_int @ Xs ) )
      = ( Xs != nil_Pr2300489316682597567nt_int ) ) ).

% length_greater_0_conv
thf(fact_1046_nth__append__length,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] :
      ( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_1047_nth__append__length,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys: list_P5707943133018811711nt_int] :
      ( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ Ys ) ) @ ( size_s5157815400016825771nt_int @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_1048_nth__zip,axiom,
    ! [I3: nat,Xs: list_nat,Ys: list_set_int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s763970886550748294et_int @ Ys ) )
       => ( ( nth_Pr4328007639516273434et_int @ ( zip_nat_set_int @ Xs @ Ys ) @ I3 )
          = ( produc29655638201817675et_int @ ( nth_nat @ Xs @ I3 ) @ ( nth_set_int @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_1049_nth__zip,axiom,
    ! [I3: nat,Xs: list_int,Ys: list_l1670014477004246597nt_int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s2969076144586574001nt_int @ Ys ) )
       => ( ( nth_Pr4686562112285265857nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) @ I3 )
          = ( produc8814303788642274490nt_int @ ( nth_int @ Xs @ I3 ) @ ( nth_li549691060785553606nt_int @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_1050_nth__zip,axiom,
    ! [I3: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys ) @ I3 )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_1051_nth__zip,axiom,
    ! [I3: nat,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ I3 @ ( size_s3817825649112772879e_term @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Ys ) )
       => ( ( nth_Pr5768845505533887420nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I3 )
          = ( produc5700946648718959541nt_int @ ( nth_Pr6123904574653836964e_term @ Xs @ I3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_1052_nth__zip,axiom,
    ! [I3: nat,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ I3 @ ( size_s8881356780637927685e_term @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Ys ) )
       => ( ( nth_Pr7284623736940738758nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I3 )
          = ( produc4305682042979456191nt_int @ ( nth_in8840386337573990810e_term @ Xs @ I3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_1053_nth__zip,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Ys ) )
       => ( ( nth_Pr7104135640663552750nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) @ I3 )
          = ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_1054_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat2 @ X @ A4 )
       => ( member_nat2 @ X @ B4 ) ) ) ).

% in_mono
thf(fact_1055_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat2 @ C @ A4 )
       => ( member_nat2 @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1056_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
        ! [X4: nat] :
          ( ( member_nat2 @ X4 @ A6 )
         => ( member_nat2 @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1057_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
        ! [T2: nat] :
          ( ( member_nat2 @ T2 @ A6 )
         => ( member_nat2 @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1058_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ? [B3: nat] : ( member_nat2 @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1059_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1060_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_1061_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1062_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1063_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_1064_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y7: int,Z4: int] : ( Y7 = Z4 ) )
    = ( ^ [A5: int,B5: int] :
          ( ( minus_minus_int @ A5 @ B5 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1065_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1066_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1067_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1068_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1069_nth__equal__first__eq,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,N: nat] :
      ( ~ ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
       => ( ( ( nth_Pr4439495888332055232nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1070_nth__equalityI,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( size_s5157815400016825771nt_int @ Xs )
        = ( size_s5157815400016825771nt_int @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs ) )
           => ( ( nth_Pr4439495888332055232nt_int @ Xs @ I )
              = ( nth_Pr4439495888332055232nt_int @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_1071_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > product_prod_int_int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X8: product_prod_int_int] : ( P2 @ I4 @ X8 ) ) )
      = ( ? [Xs4: list_P5707943133018811711nt_int] :
            ( ( ( size_s5157815400016825771nt_int @ Xs4 )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P2 @ I4 @ ( nth_Pr4439495888332055232nt_int @ Xs4 @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1072_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y7: list_P5707943133018811711nt_int,Z4: list_P5707943133018811711nt_int] : ( Y7 = Z4 ) )
    = ( ^ [Xs4: list_P5707943133018811711nt_int,Ys4: list_P5707943133018811711nt_int] :
          ( ( ( size_s5157815400016825771nt_int @ Xs4 )
            = ( size_s5157815400016825771nt_int @ Ys4 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs4 ) )
             => ( ( nth_Pr4439495888332055232nt_int @ Xs4 @ I4 )
                = ( nth_Pr4439495888332055232nt_int @ Ys4 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1073_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1074_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1075_list_Osize_I3_J,axiom,
    ( ( size_size_list_int @ nil_int )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1076_list_Osize_I3_J,axiom,
    ( ( size_s5157815400016825771nt_int @ nil_Pr2300489316682597567nt_int )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1077_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P2: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int2 @ X4 @ ( set_int2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( P2 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1078_all__set__conv__all__nth,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ( ! [X4: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X4 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
           => ( P2 @ ( nth_Pr4439495888332055232nt_int @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1079_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: nat > $o,X: nat] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I ) ) )
     => ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1080_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P2: int > $o,X: int] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
         => ( P2 @ ( nth_int @ Xs @ I ) ) )
     => ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1081_all__nth__imp__all__set,axiom,
    ! [Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o,X: product_prod_int_int] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs ) )
         => ( P2 @ ( nth_Pr4439495888332055232nt_int @ Xs @ I ) ) )
     => ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1082_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1083_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1084_in__set__conv__nth,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5157815400016825771nt_int @ Xs ) )
            & ( ( nth_Pr4439495888332055232nt_int @ Xs @ I4 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1085_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P2: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X3: int] :
            ( ( member_int2 @ X3 @ ( set_int2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1086_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int,P2: product_prod_int_int > $o] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ ( set_Pr2470121279949933262nt_int @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1087_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1088_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int2 @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_1089_nth__mem,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( member5262025264175285858nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ N ) @ ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ).

% nth_mem
thf(fact_1090_nth__append,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
          = ( nth_int @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
          = ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1091_nth__append,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ N )
          = ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ N )
          = ( nth_Pr4439495888332055232nt_int @ Ys @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1092_length__code,axiom,
    ( size_s5157815400016825771nt_int
    = ( gen_le8428774395332151372nt_int @ zero_zero_nat ) ) ).

% length_code
thf(fact_1093_nth__butlast,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ ( butlas2390654596765537291nt_int @ Xs ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( butlas2390654596765537291nt_int @ Xs ) @ N )
        = ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ) ).

% nth_butlast
thf(fact_1094_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1095_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1096_length__pos__if__in__set,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5157815400016825771nt_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1097_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_set_int,R: set_Pr4810089274464741491et_int] :
      ( ( member4311088582351766364et_int @ ( produc6467668960084403947et_int @ Xs @ Ys ) @ ( listrel_nat_set_int @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s763970886550748294et_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member1292241183792264892et_int @ ( produc29655638201817675et_int @ ( nth_nat @ Xs @ N3 ) @ ( nth_set_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_1098_listrel__iff__nth,axiom,
    ! [Xs: list_int,Ys: list_l1670014477004246597nt_int,R: set_Pr1843331182504696254nt_int] :
      ( ( member4298863958545693205nt_int @ ( produc1403559775789145904nt_int @ Xs @ Ys ) @ ( listre1889173885445251040nt_int @ R ) )
      = ( ( ( size_size_list_int @ Xs )
          = ( size_s2969076144586574001nt_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
           => ( member3263328415308337823nt_int @ ( produc8814303788642274490nt_int @ ( nth_int @ Xs @ N3 ) @ ( nth_li549691060785553606nt_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_1099_listrel__iff__nth,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) )
      = ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N3 ) @ ( nth_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_1100_listrel__iff__nth,axiom,
    ! [Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int,R: set_Pr9222295170931077689nt_int] :
      ( ( member1869596255364795664nt_int @ ( produc8723725229082728235nt_int @ Xs @ Ys ) @ ( listre7929554186369383643nt_int @ R ) )
      = ( ( ( size_s3817825649112772879e_term @ Xs )
          = ( size_s5157815400016825771nt_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3817825649112772879e_term @ Xs ) )
           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ ( nth_Pr6123904574653836964e_term @ Xs @ N3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_1101_listrel__iff__nth,axiom,
    ! [Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int,R: set_Pr1872883991513573699nt_int] :
      ( ( member6768764179952475034nt_int @ ( produc2259991554960072885nt_int @ Xs @ Ys ) @ ( listre7622244038138809061nt_int @ R ) )
      = ( ( ( size_s8881356780637927685e_term @ Xs )
          = ( size_s5157815400016825771nt_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s8881356780637927685e_term @ Xs ) )
           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ ( nth_in8840386337573990810e_term @ Xs @ N3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_1102_listrel__iff__nth,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre7525836092462517041nt_int @ R ) )
      = ( ( ( size_s5157815400016825771nt_int @ Xs )
          = ( size_s5157815400016825771nt_int @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Xs ) )
           => ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ N3 ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_1103_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( n_list2671089462871817415nt_int @ zero_zero_nat @ Xs )
      = ( cons_l7309679040211256053nt_int @ nil_Pr2300489316682597567nt_int @ nil_li8670148097206105925nt_int ) ) ).

% n_lists.simps(1)
thf(fact_1104_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_int] :
      ( ( n_lists_int @ zero_zero_nat @ Xs )
      = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% n_lists.simps(1)
thf(fact_1105_all__subset__image,axiom,
    ! [F2: nat > nat,A4: set_nat,P2: set_nat > $o] :
      ( ( ! [B7: set_nat] :
            ( ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F2 @ A4 ) )
           => ( P2 @ B7 ) ) )
      = ( ! [B7: set_nat] :
            ( ( ord_less_eq_set_nat @ B7 @ A4 )
           => ( P2 @ ( image_nat_nat @ F2 @ B7 ) ) ) ) ) ).

% all_subset_image
thf(fact_1106_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
      = ( ? [Y5: int,N3: nat] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N3 ) @ Y5 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs ) )
            & ( Ys
              = ( list_update_int @ Xs @ N3 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1107_listrel1__iff__update,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( listre1649617614542993432nt_int @ R ) )
      = ( ? [Y5: product_prod_int_int,N3: nat] :
            ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ N3 ) @ Y5 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s5157815400016825771nt_int @ Xs ) )
            & ( Ys
              = ( list_u3002344382305578791nt_int @ Xs @ N3 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1108_lex__take__index,axiom,
    ! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
             => ( ( ( take_int @ I @ Xs )
                  = ( take_int @ I @ Ys ) )
               => ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_1109_lex__take__index,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ Xs @ Ys ) @ ( lex_Pr5393148144989827363nt_int @ R ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s5157815400016825771nt_int @ Ys ) )
             => ( ( ( take_P8218740963776755879nt_int @ I @ Xs )
                  = ( take_P8218740963776755879nt_int @ I @ Ys ) )
               => ~ ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I ) @ ( nth_Pr4439495888332055232nt_int @ Ys @ I ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_1110_list__update__nonempty,axiom,
    ! [Xs: list_P5707943133018811711nt_int,K: nat,X: product_prod_int_int] :
      ( ( ( list_u3002344382305578791nt_int @ Xs @ K @ X )
        = nil_Pr2300489316682597567nt_int )
      = ( Xs = nil_Pr2300489316682597567nt_int ) ) ).

% list_update_nonempty
thf(fact_1111_list__update__nonempty,axiom,
    ! [Xs: list_int,K: nat,X: int] :
      ( ( ( list_update_int @ Xs @ K @ X )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% list_update_nonempty
thf(fact_1112_length__list__update,axiom,
    ! [Xs: list_P5707943133018811711nt_int,I3: nat,X: product_prod_int_int] :
      ( ( size_s5157815400016825771nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I3 @ X ) )
      = ( size_s5157815400016825771nt_int @ Xs ) ) ).

% length_list_update
thf(fact_1113_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( nil_Pr2300489316682597567nt_int
        = ( take_P8218740963776755879nt_int @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_Pr2300489316682597567nt_int ) ) ) ).

% take_eq_Nil2
thf(fact_1114_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( nil_int
        = ( take_int @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_int ) ) ) ).

% take_eq_Nil2
thf(fact_1115_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( ( take_P8218740963776755879nt_int @ N @ Xs )
        = nil_Pr2300489316682597567nt_int )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_Pr2300489316682597567nt_int ) ) ) ).

% take_eq_Nil
thf(fact_1116_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ( take_int @ N @ Xs )
        = nil_int )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_int ) ) ) ).

% take_eq_Nil
thf(fact_1117_take0,axiom,
    ( ( take_P8218740963776755879nt_int @ zero_zero_nat )
    = ( ^ [Xs4: list_P5707943133018811711nt_int] : nil_Pr2300489316682597567nt_int ) ) ).

% take0
thf(fact_1118_take0,axiom,
    ( ( take_int @ zero_zero_nat )
    = ( ^ [Xs4: list_int] : nil_int ) ) ).

% take0
thf(fact_1119_take__all__iff,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( ( take_P8218740963776755879nt_int @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_1120_take__all,axiom,
    ! [Xs: list_P5707943133018811711nt_int,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N )
     => ( ( take_P8218740963776755879nt_int @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_1121_list__update__beyond,axiom,
    ! [Xs: list_P5707943133018811711nt_int,I3: nat,X: product_prod_int_int] :
      ( ( ord_less_eq_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ I3 )
     => ( ( list_u3002344382305578791nt_int @ Xs @ I3 @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_1122_take__append,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] :
      ( ( take_int @ N @ ( append_int @ Xs @ Ys ) )
      = ( append_int @ ( take_int @ N @ Xs ) @ ( take_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1123_take__append,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( take_P8218740963776755879nt_int @ N @ ( append7030698103840186580nt_int @ Xs @ Ys ) )
      = ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) @ ( take_P8218740963776755879nt_int @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1124_list__update__length,axiom,
    ! [Xs: list_int,X: int,Ys: list_int,Y: int] :
      ( ( list_update_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) @ Y )
      = ( append_int @ Xs @ ( cons_int @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1125_list__update__length,axiom,
    ! [Xs: list_P5707943133018811711nt_int,X: product_prod_int_int,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ X @ Ys ) ) @ ( size_s5157815400016825771nt_int @ Xs ) @ Y )
      = ( append7030698103840186580nt_int @ Xs @ ( cons_P3334398858971670639nt_int @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1126_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I3 @ X ) @ I3 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1127_set__swap,axiom,
    ! [I3: nat,Xs: list_int,J3: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I3 @ ( nth_int @ Xs @ J3 ) ) @ J3 @ ( nth_int @ Xs @ I3 ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_1128_set__swap,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int,J3: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_s5157815400016825771nt_int @ Xs ) )
       => ( ( set_Pr2470121279949933262nt_int @ ( list_u3002344382305578791nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I3 @ ( nth_Pr4439495888332055232nt_int @ Xs @ J3 ) ) @ J3 @ ( nth_Pr4439495888332055232nt_int @ Xs @ I3 ) ) )
          = ( set_Pr2470121279949933262nt_int @ Xs ) ) ) ) ).

% set_swap
thf(fact_1129_plusinfinity,axiom,
    ! [D2: int,P7: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P7 @ X3 )
            = ( P7 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z3 @ X3 )
             => ( ( P2 @ X3 )
                = ( P7 @ X3 ) ) )
         => ( ? [X_12: int] : ( P7 @ X_12 )
           => ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_1130_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z3 )
             => ( ( P2 @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P2 @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_1131_decr__mult__lemma,axiom,
    ! [D2: int,P2: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P2 @ X5 )
             => ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_1132_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( member_nat2 @ C @ A4 )
       => ( member_nat2 @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_1133_zip__update,axiom,
    ! [Xs: list_nat,I3: nat,X: nat,Ys: list_set_int,Y: set_int] :
      ( ( zip_nat_set_int @ ( list_update_nat @ Xs @ I3 @ X ) @ ( list_update_set_int @ Ys @ I3 @ Y ) )
      = ( list_u7092962573381667713et_int @ ( zip_nat_set_int @ Xs @ Ys ) @ I3 @ ( produc29655638201817675et_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1134_zip__update,axiom,
    ! [Xs: list_int,I3: nat,X: int,Ys: list_l1670014477004246597nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( zip_in8018307065301790106nt_int @ ( list_update_int @ Xs @ I3 @ X ) @ ( list_u1995434435150319021nt_int @ Ys @ I3 @ Y ) )
      = ( list_u6063430246987531802nt_int @ ( zip_in8018307065301790106nt_int @ Xs @ Ys ) @ I3 @ ( produc8814303788642274490nt_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1135_zip__update,axiom,
    ! [Xs: list_int,I3: nat,X: int,Ys: list_int,Y: int] :
      ( ( zip_int_int @ ( list_update_int @ Xs @ I3 @ X ) @ ( list_update_int @ Ys @ I3 @ Y ) )
      = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I3 @ ( product_Pair_int_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1136_zip__update,axiom,
    ! [Xs: list_P1743416141875011707e_term,I3: nat,X: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I3 @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I3 @ Y ) )
      = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I3 @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1137_zip__update,axiom,
    ! [Xs: list_i8448526496819171953e_term,I3: nat,X: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I3 @ X ) @ ( list_u3002344382305578791nt_int @ Ys @ I3 @ Y ) )
      = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I3 @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_1138_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_P8218740963776755879nt_int @ N @ nil_Pr2300489316682597567nt_int )
      = nil_Pr2300489316682597567nt_int ) ).

% take_Nil
thf(fact_1139_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_int @ N @ nil_int )
      = nil_int ) ).

% take_Nil
thf(fact_1140_refl__onD,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member_nat2 @ A @ A4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_1141_refl__onD,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int,A: int] :
      ( ( refl_on_int @ A4 @ R )
     => ( ( member_int2 @ A @ A4 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_1142_refl__onD1,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member_nat2 @ X @ A4 ) ) ) ).

% refl_onD1
thf(fact_1143_refl__onD1,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ( refl_on_int @ A4 @ R )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
       => ( member_int2 @ X @ A4 ) ) ) ).

% refl_onD1
thf(fact_1144_refl__onD2,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member_nat2 @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_1145_refl__onD2,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int,X: int,Y: int] :
      ( ( refl_on_int @ A4 @ R )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
       => ( member_int2 @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_1146_list__update_Osimps_I1_J,axiom,
    ! [I3: nat,V: product_prod_int_int] :
      ( ( list_u3002344382305578791nt_int @ nil_Pr2300489316682597567nt_int @ I3 @ V )
      = nil_Pr2300489316682597567nt_int ) ).

% list_update.simps(1)
thf(fact_1147_list__update_Osimps_I1_J,axiom,
    ! [I3: nat,V: int] :
      ( ( list_update_int @ nil_int @ I3 @ V )
      = nil_int ) ).

% list_update.simps(1)
thf(fact_1148_list__update__code_I1_J,axiom,
    ! [I3: nat,Y: product_prod_int_int] :
      ( ( list_u3002344382305578791nt_int @ nil_Pr2300489316682597567nt_int @ I3 @ Y )
      = nil_Pr2300489316682597567nt_int ) ).

% list_update_code(1)
thf(fact_1149_list__update__code_I1_J,axiom,
    ! [I3: nat,Y: int] :
      ( ( list_update_int @ nil_int @ I3 @ Y )
      = nil_int ) ).

% list_update_code(1)
thf(fact_1150_in__set__takeD,axiom,
    ! [X: nat,N: nat,Xs: list_nat] :
      ( ( member_nat2 @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
     => ( member_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1151_in__set__takeD,axiom,
    ! [X: int,N: nat,Xs: list_int] :
      ( ( member_int2 @ X @ ( set_int2 @ ( take_int @ N @ Xs ) ) )
     => ( member_int2 @ X @ ( set_int2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1152_take__0,axiom,
    ! [Xs: list_P5707943133018811711nt_int] :
      ( ( take_P8218740963776755879nt_int @ zero_zero_nat @ Xs )
      = nil_Pr2300489316682597567nt_int ) ).

% take_0
thf(fact_1153_take__0,axiom,
    ! [Xs: list_int] :
      ( ( take_int @ zero_zero_nat @ Xs )
      = nil_int ) ).

% take_0
thf(fact_1154_set__take__subset,axiom,
    ! [N: nat,Xs: list_int] : ( ord_less_eq_set_int @ ( set_int2 @ ( take_int @ N @ Xs ) ) @ ( set_int2 @ Xs ) ) ).

% set_take_subset
thf(fact_1155_list__update__code_I2_J,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,Y: product_prod_int_int] :
      ( ( list_u3002344382305578791nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_P3334398858971670639nt_int @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_1156_list__update__code_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
      = ( cons_int @ Y @ Xs ) ) ).

% list_update_code(2)
thf(fact_1157_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,X: nat,I3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
     => ( ( member_nat2 @ X @ A4 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I3 @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_1158_set__update__subsetI,axiom,
    ! [Xs: list_int,A4: set_int,X: int,I3: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
     => ( ( member_int2 @ X @ A4 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I3 @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_1159_reflI,axiom,
    ! [R: set_Pr958786334691620121nt_int] :
      ( ! [X3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
     => ( refl_on_int @ top_top_set_int @ R ) ) ).

% reflI
thf(fact_1160_reflD,axiom,
    ! [R: set_Pr958786334691620121nt_int,A: int] :
      ( ( refl_on_int @ top_top_set_int @ R )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ R ) ) ).

% reflD
thf(fact_1161_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_set_int @ ( set_int2 @ ( take_int @ M @ Xs ) ) @ ( set_int2 @ ( take_int @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1162_take__butlast,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( take_P8218740963776755879nt_int @ N @ ( butlas2390654596765537291nt_int @ Xs ) )
        = ( take_P8218740963776755879nt_int @ N @ Xs ) ) ) ).

% take_butlast
thf(fact_1163_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1164_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int2 @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1165_set__update__memI,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( member5262025264175285858nt_int @ X @ ( set_Pr2470121279949933262nt_int @ ( list_u3002344382305578791nt_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1166_list__update__append1,axiom,
    ! [I3: nat,Xs: list_int,Ys: list_int,X: int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
     => ( ( list_update_int @ ( append_int @ Xs @ Ys ) @ I3 @ X )
        = ( append_int @ ( list_update_int @ Xs @ I3 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1167_list__update__append1,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ I3 @ X )
        = ( append7030698103840186580nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I3 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1168_list__update__append,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int,X: int] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( list_update_int @ ( append_int @ Xs @ Ys ) @ N @ X )
          = ( append_int @ ( list_update_int @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( ( list_update_int @ ( append_int @ Xs @ Ys ) @ N @ X )
          = ( append_int @ Xs @ ( list_update_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1169_list__update__append,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
       => ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ N @ X )
          = ( append7030698103840186580nt_int @ ( list_u3002344382305578791nt_int @ Xs @ N @ X ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) )
       => ( ( list_u3002344382305578791nt_int @ ( append7030698103840186580nt_int @ Xs @ Ys ) @ N @ X )
          = ( append7030698103840186580nt_int @ Xs @ ( list_u3002344382305578791nt_int @ Ys @ ( minus_minus_nat @ N @ ( size_s5157815400016825771nt_int @ Xs ) ) @ X ) ) ) ) ) ).

% list_update_append
thf(fact_1170_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int,X: product_prod_int_int] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( ( list_u3002344382305578791nt_int @ Xs @ I3 @ X )
          = Xs )
        = ( ( nth_Pr4439495888332055232nt_int @ Xs @ I3 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1171_nth__list__update,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int,J3: nat,X: product_prod_int_int] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( ( I3 = J3 )
         => ( ( nth_Pr4439495888332055232nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I3 @ X ) @ J3 )
            = X ) )
        & ( ( I3 != J3 )
         => ( ( nth_Pr4439495888332055232nt_int @ ( list_u3002344382305578791nt_int @ Xs @ I3 @ X ) @ J3 )
            = ( nth_Pr4439495888332055232nt_int @ Xs @ J3 ) ) ) ) ) ).

% nth_list_update
thf(fact_1172_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ord_less_eq_nat @ K @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s5157815400016825771nt_int @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ K )
             => ( ( nth_Pr4439495888332055232nt_int @ Xs @ I )
                = ( nth_Pr4439495888332055232nt_int @ Ys @ I ) ) )
         => ( ( take_P8218740963776755879nt_int @ K @ Xs )
            = ( take_P8218740963776755879nt_int @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1173_refl__on__domain,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( ( member_nat2 @ A @ A4 )
          & ( member_nat2 @ B @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_1174_refl__on__domain,axiom,
    ! [A4: set_int,R: set_Pr958786334691620121nt_int,A: int,B: int] :
      ( ( refl_on_int @ A4 @ R )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ R )
       => ( ( member_int2 @ A @ A4 )
          & ( member_int2 @ B @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_1175_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
     => ( ( take_int @ ( suc @ I3 ) @ Xs )
        = ( append_int @ ( take_int @ I3 @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I3 ) @ nil_int ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1176_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( ord_less_nat @ I3 @ ( size_s5157815400016825771nt_int @ Xs ) )
     => ( ( take_P8218740963776755879nt_int @ ( suc @ I3 ) @ Xs )
        = ( append7030698103840186580nt_int @ ( take_P8218740963776755879nt_int @ I3 @ Xs ) @ ( cons_P3334398858971670639nt_int @ ( nth_Pr4439495888332055232nt_int @ Xs @ I3 ) @ nil_Pr2300489316682597567nt_int ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1177_lexord__take__index__conv,axiom,
    ! [X: list_int,Y: list_int,R: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ X @ Y ) @ ( lexord_int @ R ) )
      = ( ( ( ord_less_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) )
          & ( ( take_int @ ( size_size_list_int @ X ) @ Y )
            = X ) )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_size_list_int @ X ) @ ( size_size_list_int @ Y ) ) )
            & ( ( take_int @ I4 @ X )
              = ( take_int @ I4 @ Y ) )
            & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ X @ I4 ) @ ( nth_int @ Y @ I4 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1178_lexord__take__index__conv,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int,R: set_Pr2560585780119916871nt_int] :
      ( ( member1390679175989562640nt_int @ ( produc1932183703851549015nt_int @ X @ Y ) @ ( lexord8886728381914977324nt_int @ R ) )
      = ( ( ( ord_less_nat @ ( size_s5157815400016825771nt_int @ X ) @ ( size_s5157815400016825771nt_int @ Y ) )
          & ( ( take_P8218740963776755879nt_int @ ( size_s5157815400016825771nt_int @ X ) @ Y )
            = X ) )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_s5157815400016825771nt_int @ X ) @ ( size_s5157815400016825771nt_int @ Y ) ) )
            & ( ( take_P8218740963776755879nt_int @ I4 @ X )
              = ( take_P8218740963776755879nt_int @ I4 @ Y ) )
            & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ ( nth_Pr4439495888332055232nt_int @ X @ I4 ) @ ( nth_Pr4439495888332055232nt_int @ Y @ I4 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1179_min_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_1180_min_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_1181_min_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_1182_min_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_1183_min_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_1184_min_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% min.bounded_iff
thf(fact_1185_min__less__iff__conj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        & ( ord_less_int @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_1186_min__less__iff__conj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        & ( ord_less_nat @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_1187_min_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_1188_min_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_1189_min_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_1190_min_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_1191_nth__Cons__Suc,axiom,
    ! [X: product_prod_int_int,Xs: list_P5707943133018811711nt_int,N: nat] :
      ( ( nth_Pr4439495888332055232nt_int @ ( cons_P3334398858971670639nt_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr4439495888332055232nt_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1192_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1193_take__Suc__Cons,axiom,
    ! [N: nat,X: product_prod_int_int,Xs: list_P5707943133018811711nt_int] :
      ( ( take_P8218740963776755879nt_int @ ( suc @ N ) @ ( cons_P3334398858971670639nt_int @ X @ Xs ) )
      = ( cons_P3334398858971670639nt_int @ X @ ( take_P8218740963776755879nt_int @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_1194_take__Suc__Cons,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( take_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
      = ( cons_int @ X @ ( take_int @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_1195_length__take,axiom,
    ! [N: nat,Xs: list_P5707943133018811711nt_int] :
      ( ( size_s5157815400016825771nt_int @ ( take_P8218740963776755879nt_int @ N @ Xs ) )
      = ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ N ) ) ).

% length_take
thf(fact_1196_length__zip,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( size_s6770063216428074713nt_int @ ( zip_Pr2148635498163357687nt_int @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_s5157815400016825771nt_int @ Xs ) @ ( size_s5157815400016825771nt_int @ Ys ) ) ) ).

% length_zip
thf(fact_1197_length__zip,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( zip_int_int @ Xs @ Ys ) )
      = ( ord_min_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_zip
thf(fact_1198_min__less__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        | ( ord_less_int @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_1199_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        | ( ord_less_nat @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_1200_min_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( ord_min_int @ B @ C ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_1201_min_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C ) ) ) ).

% min.strict_boundedE
thf(fact_1202_min_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] :
          ( ( A5
            = ( ord_min_int @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% min.strict_order_iff
thf(fact_1203_min_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( A5
            = ( ord_min_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% min.strict_order_iff
thf(fact_1204_min_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ C )
     => ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_1205_min_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ A @ C )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.strict_coboundedI1
thf(fact_1206_min_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ B @ C )
     => ( ord_less_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_1207_min_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ B @ C )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.strict_coboundedI2
thf(fact_1208_board__exec__aux_Ocases,axiom,
    ! [X: produc9133624956312949779et_int] :
      ( ! [M5: set_int] :
          ( X
         != ( produc29655638201817675et_int @ zero_zero_nat @ M5 ) )
     => ~ ! [V2: nat,M5: set_int] :
            ( X
           != ( produc29655638201817675et_int @ ( suc @ V2 ) @ M5 ) ) ) ).

% board_exec_aux.cases
thf(fact_1209_row__exec_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [V2: nat] :
            ( X
           != ( suc @ V2 ) ) ) ).

% row_exec.cases
thf(fact_1210_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_1211_min__le__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        | ( ord_less_eq_int @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_1212_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        | ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_1213_min_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ B @ C )
     => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_1214_min_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.coboundedI2
thf(fact_1215_min_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_1216_min_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C ) ) ).

% min.coboundedI1
thf(fact_1217_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_min_int @ A5 @ B5 )
          = B5 ) ) ) ).

% min.absorb_iff2
thf(fact_1218_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_min_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% min.absorb_iff2
thf(fact_1219_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_min_int @ A5 @ B5 )
          = A5 ) ) ) ).

% min.absorb_iff1
thf(fact_1220_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_min_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% min.absorb_iff1
thf(fact_1221_min_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_1222_min_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_1223_min_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_1224_min_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_1225_min_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( A5
          = ( ord_min_int @ A5 @ B5 ) ) ) ) ).

% min.order_iff
thf(fact_1226_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( A5
          = ( ord_min_nat @ A5 @ B5 ) ) ) ) ).

% min.order_iff
thf(fact_1227_min_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C )
       => ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_1228_min_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) ) ) ) ).

% min.boundedI
thf(fact_1229_min_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C ) ) ) ).

% min.boundedE
thf(fact_1230_min_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% min.boundedE
thf(fact_1231_min_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_min_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% min.orderI
thf(fact_1232_min_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_min_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% min.orderI
thf(fact_1233_min_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( ord_min_int @ A @ B ) ) ) ).

% min.orderE
thf(fact_1234_min_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( ord_min_nat @ A @ B ) ) ) ).

% min.orderE
thf(fact_1235_int__distrib_I4_J,axiom,
    ! [W: int,Z12: int,Z23: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z12 @ Z23 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z12 ) @ ( times_times_int @ W @ Z23 ) ) ) ).

% int_distrib(4)
thf(fact_1236_int__distrib_I3_J,axiom,
    ! [Z12: int,Z23: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z12 @ Z23 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z12 @ W ) @ ( times_times_int @ Z23 @ W ) ) ) ).

% int_distrib(3)
thf(fact_1237_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_1238_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_1239_image__Suc__atLeastLessThan,axiom,
    ! [I3: nat,J3: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I3 @ J3 ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I3 ) @ ( suc @ J3 ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_1240_ex__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
            & ( P2 @ M4 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1241_all__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
           => ( P2 @ M4 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat2 @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1242_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_1243_int__le__induct,axiom,
    ! [I3: int,K: int,P2: int > $o] :
      ( ( ord_less_eq_int @ I3 @ K )
     => ( ( P2 @ K )
       => ( ! [I: int] :
              ( ( ord_less_eq_int @ I @ K )
             => ( ( P2 @ I )
               => ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
         => ( P2 @ I3 ) ) ) ) ).

% int_le_induct
thf(fact_1244_int__less__induct,axiom,
    ! [I3: int,K: int,P2: int > $o] :
      ( ( ord_less_int @ I3 @ K )
     => ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I: int] :
              ( ( ord_less_int @ I @ K )
             => ( ( P2 @ I )
               => ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
         => ( P2 @ I3 ) ) ) ) ).

% int_less_induct
thf(fact_1245_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J4: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J4 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J4 @ one_one_int ) @ ( cons_int @ J4 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_1246_int__induct,axiom,
    ! [P2: int > $o,K: int,I3: int] :
      ( ( P2 @ K )
     => ( ! [I: int] :
            ( ( ord_less_eq_int @ K @ I )
           => ( ( P2 @ I )
             => ( P2 @ ( plus_plus_int @ I @ one_one_int ) ) ) )
       => ( ! [I: int] :
              ( ( ord_less_eq_int @ I @ K )
             => ( ( P2 @ I )
               => ( P2 @ ( minus_minus_int @ I @ one_one_int ) ) ) )
         => ( P2 @ I3 ) ) ) ) ).

% int_induct
thf(fact_1247_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I: int,J: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
           => ( ( ( ord_less_eq_int @ I @ J )
               => ( P2 @ ( plus_plus_int @ I @ one_one_int ) @ J ) )
             => ( P2 @ I @ J ) ) )
       => ( P2 @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_1248_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: nat > int,K: int] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F2 @ I ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
         => ? [I: nat] :
              ( ( ord_less_eq_nat @ I @ N )
              & ( ( F2 @ I )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_1249_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_1250_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_1251_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F2: nat > int,K: int] :
      ( ! [I: nat] :
          ( ( ( ord_less_eq_nat @ M @ I )
            & ( ord_less_nat @ I @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I ) ) @ ( F2 @ I ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F2 @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
           => ? [I: nat] :
                ( ( ord_less_eq_nat @ M @ I )
                & ( ord_less_eq_nat @ I @ N )
                & ( ( F2 @ I )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_1252_nat__ivt__aux,axiom,
    ! [N: nat,F2: nat > int,K: int] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I ) ) @ ( F2 @ I ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
         => ? [I: nat] :
              ( ( ord_less_eq_nat @ I @ N )
              & ( ( F2 @ I )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_1253_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_1254_upto_Opsimps,axiom,
    ! [I3: int,J3: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J3 ) )
     => ( ( ( ord_less_eq_int @ I3 @ J3 )
         => ( ( upto @ I3 @ J3 )
            = ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) )
        & ( ~ ( ord_less_eq_int @ I3 @ J3 )
         => ( ( upto @ I3 @ J3 )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_1255_upto__empty,axiom,
    ! [J3: int,I3: int] :
      ( ( ord_less_int @ J3 @ I3 )
     => ( ( upto @ I3 @ J3 )
        = nil_int ) ) ).

% upto_empty
thf(fact_1256_upto__Nil2,axiom,
    ! [I3: int,J3: int] :
      ( ( nil_int
        = ( upto @ I3 @ J3 ) )
      = ( ord_less_int @ J3 @ I3 ) ) ).

% upto_Nil2
thf(fact_1257_upto__Nil,axiom,
    ! [I3: int,J3: int] :
      ( ( ( upto @ I3 @ J3 )
        = nil_int )
      = ( ord_less_int @ J3 @ I3 ) ) ).

% upto_Nil
thf(fact_1258_upto__single,axiom,
    ! [I3: int] :
      ( ( upto @ I3 @ I3 )
      = ( cons_int @ I3 @ nil_int ) ) ).

% upto_single
thf(fact_1259_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I4: int,J4: int] : ( append_int @ ( upto @ I4 @ J4 ) ) ) ) ).

% upto_aux_def
thf(fact_1260_upto__code,axiom,
    ( upto
    = ( ^ [I4: int,J4: int] : ( upto_aux @ I4 @ J4 @ nil_int ) ) ) ).

% upto_code
thf(fact_1261_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I4: int,J4: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J4 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_1262_upto__split2,axiom,
    ! [I3: int,J3: int,K: int] :
      ( ( ord_less_eq_int @ I3 @ J3 )
     => ( ( ord_less_eq_int @ J3 @ K )
       => ( ( upto @ I3 @ K )
          = ( append_int @ ( upto @ I3 @ J3 ) @ ( upto @ ( plus_plus_int @ J3 @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_1263_upto__split1,axiom,
    ! [I3: int,J3: int,K: int] :
      ( ( ord_less_eq_int @ I3 @ J3 )
     => ( ( ord_less_eq_int @ J3 @ K )
       => ( ( upto @ I3 @ K )
          = ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) @ ( upto @ J3 @ K ) ) ) ) ) ).

% upto_split1
thf(fact_1264_upto__rec1,axiom,
    ! [I3: int,J3: int] :
      ( ( ord_less_eq_int @ I3 @ J3 )
     => ( ( upto @ I3 @ J3 )
        = ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).

% upto_rec1
thf(fact_1265_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J4: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J4 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J4 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_1266_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_1267_upto__rec2,axiom,
    ! [I3: int,J3: int] :
      ( ( ord_less_eq_int @ I3 @ J3 )
     => ( ( upto @ I3 @ J3 )
        = ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) @ ( cons_int @ J3 @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_1268_upto__split3,axiom,
    ! [I3: int,J3: int,K: int] :
      ( ( ord_less_eq_int @ I3 @ J3 )
     => ( ( ord_less_eq_int @ J3 @ K )
       => ( ( upto @ I3 @ K )
          = ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) @ ( cons_int @ J3 @ ( upto @ ( plus_plus_int @ J3 @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_1269_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_1270_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_1271_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_1272_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)

% Helper facts (7)
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( if_lis8883190402267401221nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X: list_P5707943133018811711nt_int,Y: list_P5707943133018811711nt_int] :
      ( ( if_lis8883190402267401221nt_int @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( last_P3305686521732843992nt_int @ ( mirror2_aux @ m @ ps ) )
    = ( product_Pair_int_int @ i @ ( minus_minus_int @ m @ j ) ) ) ).

%------------------------------------------------------------------------------