%------------------------------------------------------------------------------
% 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    : FSM_Tests/0059_Simple_Convergence_Graph/prob_00121_005622__20602968_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2088 ( 594 unt; 812 typ;   0 def)
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%            Maximal formula atoms :   18 (   3 avg)
%            Number of connectives : 17842 (1076   ~;  74   |; 508   &;14149   @)
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%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :  172 ( 171 usr)
%            Number of type conns  : 1810 (1810   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  644 ( 641 usr;  27 con; 0-4 aty)
%            Number of variables   : 5238 ( 232   ^;4525   !; 481   ?;5238   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:29:40.591
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    set_list_list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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    set_Product_prod_o_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    product_prod_a_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    product_prod_nat_a: $tType ).

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

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
    list_o: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

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

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (641)
thf(sy_c_AList_Odelete__aux_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    delete3389553227512332702list_a: $o > list_P1915399164375025317list_a > list_P1915399164375025317list_a ).

thf(sy_c_AList_Odelete__aux_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Odelete__aux_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    delete8241016168524078360list_a: $o > list_P369347992569427103list_a > list_P369347992569427103list_a ).

thf(sy_c_AList_Odelete__aux__rel_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    delete1279353588773826763list_a: produc756509542915563956list_a > produc756509542915563956list_a > $o ).

thf(sy_c_AList_Odelete__aux__rel_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    delete4210044761207720821st_nat: produc5234193878020974990st_nat > produc5234193878020974990st_nat > $o ).

thf(sy_c_AList_Odelete__aux__rel_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    delete8886510592334333637list_a: produc2919922447449601198list_a > produc2919922447449601198list_a > $o ).

thf(sy_c_AList_Omap__default_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    map_de2428310187333209296list_a: $o > list_list_list_a > ( list_list_list_a > list_list_list_a ) > list_P1915399164375025317list_a > list_P1915399164375025317list_a ).

thf(sy_c_AList_Omap__default_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    map_de5183444947954434480st_nat: $o > list_list_nat > ( list_list_nat > list_list_nat ) > list_P3026376796534914631st_nat > list_P3026376796534914631st_nat ).

thf(sy_c_AList_Omap__default_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_AList_Omap__default__rel_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    map_de3558955406781211673list_a: produc4781535081606627195list_a > produc4781535081606627195list_a > $o ).

thf(sy_c_AList_Omap__default__rel_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Omap__default__rel_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    map_de8605239412455127315list_a: produc1131589114034611951list_a > produc1131589114034611951list_a > $o ).

thf(sy_c_AList_Omap__entry_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_AList_Omap__entry_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Omap__entry_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_AList_Omap__entry__rel_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_AList_Omap__entry__rel_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Omap__entry__rel_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    map_ra9120075116080084029list_a: ( $o > list_list_nat > list_list_list_a ) > list_P3026376796534914631st_nat > list_P1915399164375025317list_a ).

thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    map_ra5394953335731719811st_nat: ( $o > list_list_nat > list_list_nat ) > list_P3026376796534914631st_nat > list_P3026376796534914631st_nat ).

thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    map_ra8120288265412832055list_a: ( $o > list_list_nat > list_list_a ) > list_P3026376796534914631st_nat > list_P369347992569427103list_a ).

thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    map_ra8396789647597898279list_a: ( $o > list_list_a > list_list_list_a ) > list_P369347992569427103list_a > list_P1915399164375025317list_a ).

thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Omap__ran_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    map_ra3570251311891529249list_a: ( $o > list_list_a > list_list_a ) > list_P369347992569427103list_a > list_P369347992569427103list_a ).

thf(sy_c_AList_Oupdate_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_AList_Oupdate_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_AList_Oupdate_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    update_o_list_list_a: $o > list_list_a > list_P369347992569427103list_a > list_P369347992569427103list_a ).

thf(sy_c_AList_Oupdate__with__aux_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001_Eo,type,
    update1976731422980637706st_a_o: list_list_list_a > $o > ( list_list_list_a > list_list_list_a ) > list_P1915399164375025317list_a > list_P1915399164375025317list_a ).

thf(sy_c_AList_Oupdate__with__aux_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001_Eo,type,
    update2517193581708834468_nat_o: list_list_nat > $o > ( list_list_nat > list_list_nat ) > list_P3026376796534914631st_nat > list_P3026376796534914631st_nat ).

thf(sy_c_AList_Oupdate__with__aux_001t__List__Olist_It__List__Olist_Itf__a_J_J_001_Eo,type,
    update3143123273235729040st_a_o: list_list_a > $o > ( list_list_a > list_list_a ) > list_P369347992569427103list_a > list_P369347992569427103list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    bNF_Gr1120979937343206697list_a: set_list_list_list_a > list_list_a > set_list_list_list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__Nat__Onat_J,type,
    bNF_Gr9051742241863529473st_nat: set_list_list_nat > list_nat > set_list_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_Itf__a_J,type,
    bNF_Gr7042794125918077091list_a: set_list_list_a > list_a > set_list_list_a ).

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_001tf__a,type,
    bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    bNF_Gr6039074496463509157list_a: set_list_list_list_a > list_list_list_a > set_list_list_a ).

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

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_Itf__a_J,type,
    bNF_Gr4634511371912843295list_a: set_list_list_a > list_list_a > set_list_a ).

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_001tf__a,type,
    bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_If_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001tf__a,type,
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thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__ex1_001tf__a,type,
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thf(sy_c_List_Olistrel1_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olistrel1_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olistrel1_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olistrel1_001tf__a,type,
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thf(sy_c_List_Olistrel_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
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thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001tf__a,type,
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thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_List_Olistrel_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olistrel_001tf__a_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Olistrel_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
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thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Omaps_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Nat__Onat,type,
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thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
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thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_List_Omaps_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Oproduct__lists_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct__lists_001tf__a,type,
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thf(sy_c_List_Osubseqs_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Osubseqs_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
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thf(sy_c_List_Osubseqs_001tf__a,type,
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thf(sy_c_List_Otranspose_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Otranspose_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Otranspose_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
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thf(sy_c_List_Otranspose_001tf__a,type,
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thf(sy_c_List_Otranspose__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Otranspose__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Otranspose__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Otranspose__rel_001t__Nat__Onat,type,
    transpose_rel_nat: list_list_nat > list_list_nat > $o ).

thf(sy_c_List_Otranspose__rel_001tf__a,type,
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thf(sy_c_List_Ounion_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_List_Oupt,type,
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thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
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thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
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thf(sy_c_Set_Othe__elem_001tf__a,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper_001t__Nat__Onat,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper_001tf__a,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper__rel_001t__Nat__Onat,type,
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thf(sy_c_Simple__Convergence__Graph_Oprefixes__in__list__helper__rel_001tf__a,type,
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thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Sublist_Oprefixes_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
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thf(sy_c_Sublist_Oprefixes_001tf__a,type,
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thf(sy_c_Sublist_Osublists_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Sublist_Osublists_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Sublist_Osublists_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
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thf(sy_c_Sublist_Osublists_001tf__a,type,
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thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Sublist_Osuffixes_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
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thf(sy_c_Sublist_Osuffixes_001tf__a,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H_001t__Nat__Onat,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H_001tf__a,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H__rel_001t__Nat__Onat,type,
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thf(sy_c_Util_Olinear__order__from__list__position_H__rel_001tf__a,type,
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thf(sy_c_Util_Olist__ordered__pairs_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Olist__ordered__pairs_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Olist__ordered__pairs_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Olist__ordered__pairs_001t__Nat__Onat,type,
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thf(sy_c_Util_Olist__ordered__pairs_001tf__a,type,
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thf(sy_c_Util_Olist__ordered__pairs__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Olist__ordered__pairs__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Olist__ordered__pairs__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Olist__ordered__pairs__rel_001t__Nat__Onat,type,
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thf(sy_c_Util_Olist__ordered__pairs__rel_001tf__a,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H_001t__Nat__Onat,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H_001tf__a,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H__rel_001t__Nat__Onat,type,
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thf(sy_c_Util_Onon__sym__dist__pairs_H__rel_001tf__a,type,
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thf(sy_c_Util_Opow__list_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Opow__list_001t__List__Olist_It__Nat__Onat_J,type,
    pow_list_list_nat: list_list_nat > list_list_list_nat ).

thf(sy_c_Util_Opow__list_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Opow__list_001t__Nat__Onat,type,
    pow_list_nat: list_nat > list_list_nat ).

thf(sy_c_Util_Opow__list_001tf__a,type,
    pow_list_a: list_a > list_list_a ).

thf(sy_c_Util_Opow__list__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Opow__list__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Opow__list__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Opow__list__rel_001t__Nat__Onat,type,
    pow_list_rel_nat: list_nat > list_nat > $o ).

thf(sy_c_Util_Opow__list__rel_001tf__a,type,
    pow_list_rel_a: list_a > list_a > $o ).

thf(sy_c_Util_Oprefix__pairs_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Oprefix__pairs_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Oprefix__pairs_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Oprefix__pairs_001t__Nat__Onat,type,
    prefix_pairs_nat: list_nat > list_P7940050157051400743st_nat ).

thf(sy_c_Util_Oprefix__pairs_001tf__a,type,
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thf(sy_c_Util_Oprefix__pairs__rel_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Oprefix__pairs__rel_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Oprefix__pairs__rel_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Util_Oprefix__pairs__rel_001t__Nat__Onat,type,
    prefix_pairs_rel_nat: list_nat > list_nat > $o ).

thf(sy_c_Util_Oprefix__pairs__rel_001tf__a,type,
    prefix_pairs_rel_a: list_a > list_a > $o ).

thf(sy_c_Util_Osuffixes_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Util_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Util_Osuffixes_001t__List__Olist_Itf__a_J,type,
    suffixes_list_a2: list_list_a > list_list_list_a ).

thf(sy_c_Util_Osuffixes_001t__Nat__Onat,type,
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thf(sy_c_Util_Osuffixes_001tf__a,type,
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    member4889668945541975382list_a: produc8685980395799941037list_a > set_Pr2070066670564046349list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_resa____,type,
    resa: produc2922778490715282969list_a ).

thf(sy_v_x,type,
    x: a ).

thf(sy_v_y____,type,
    y: a ).

thf(sy_v_ys_H____,type,
    ys: list_a ).

thf(sy_v_ys____,type,
    ys2: list_a ).

thf(sy_v_yssa____,type,
    yssa: list_list_a ).

% Relevant facts (1256)
thf(fact_0_True,axiom,
    x = y ).

% True
thf(fact_1_local_OCons,axiom,
    ( ys2
    = ( cons_a @ y @ ys ) ) ).

% local.Cons
thf(fact_2_prefixes__in__list__helper_Osimps_I3_J,axiom,
    ! [X: list_nat,Y: list_nat,Ys: list_list_nat,Yss: list_list_list_nat,Res: produc3585458782675957319st_nat] :
      ( ( ( X = Y )
       => ( ( simple6281051842103134061st_nat @ X @ ( cons_list_list_nat @ ( cons_list_nat @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple6281051842103134061st_nat @ X @ Yss @ ( produc3734841530647658561st_nat @ ( produc5989645662155510613st_nat @ Res ) @ ( cons_list_list_nat @ Ys @ ( produc5745704060670869907st_nat @ Res ) ) ) ) ) )
      & ( ( X != Y )
       => ( ( simple6281051842103134061st_nat @ X @ ( cons_list_list_nat @ ( cons_list_nat @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple6281051842103134061st_nat @ X @ Yss @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(3)
thf(fact_3_prefixes__in__list__helper_Osimps_I3_J,axiom,
    ! [X: list_list_a,Y: list_list_a,Ys: list_list_list_a,Yss: list_l7815035709764258326list_a,Res: produc2144844930413989541list_a] :
      ( ( ( X = Y )
       => ( ( simple7980934897880681149list_a @ X @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple7980934897880681149list_a @ X @ Yss @ ( produc3597484984460465685list_a @ ( produc7804899884738361857list_a @ Res ) @ ( cons_l8577867181958743494list_a @ Ys @ ( produc8615863343057196611list_a @ Res ) ) ) ) ) )
      & ( ( X != Y )
       => ( ( simple7980934897880681149list_a @ X @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple7980934897880681149list_a @ X @ Yss @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(3)
thf(fact_4_prefixes__in__list__helper_Osimps_I3_J,axiom,
    ! [X: list_a,Y: list_a,Ys: list_list_a,Yss: list_list_list_a,Res: produc2086622085225315487list_a] :
      ( ( ( X = Y )
       => ( ( simple5839407689009508407list_a @ X @ ( cons_list_list_a @ ( cons_list_a @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple5839407689009508407list_a @ X @ Yss @ ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ Res ) @ ( cons_list_list_a @ Ys @ ( produc1153002459082627517list_a @ Res ) ) ) ) ) )
      & ( ( X != Y )
       => ( ( simple5839407689009508407list_a @ X @ ( cons_list_list_a @ ( cons_list_a @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple5839407689009508407list_a @ X @ Yss @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(3)
thf(fact_5_prefixes__in__list__helper_Osimps_I3_J,axiom,
    ! [X: nat,Y: nat,Ys: list_nat,Yss: list_list_nat,Res: produc7974263126369132087st_nat] :
      ( ( ( X = Y )
       => ( ( simple1541803723984754397er_nat @ X @ ( cons_list_nat @ ( cons_nat @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple1541803723984754397er_nat @ X @ Yss @ ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ Res ) @ ( cons_list_nat @ Ys @ ( produc4024048262274189571st_nat @ Res ) ) ) ) ) )
      & ( ( X != Y )
       => ( ( simple1541803723984754397er_nat @ X @ ( cons_list_nat @ ( cons_nat @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple1541803723984754397er_nat @ X @ Yss @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(3)
thf(fact_6_prefixes__in__list__helper_Osimps_I3_J,axiom,
    ! [X: a,Y: a,Ys: list_a,Yss: list_list_a,Res: produc2922778490715282969list_a] :
      ( ( ( X = Y )
       => ( ( simple2673724192819300657lper_a @ X @ ( cons_list_a @ ( cons_a @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple2673724192819300657lper_a @ X @ Yss @ ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ Res ) @ ( cons_list_a @ Ys @ ( produc4969414473911996599list_a @ Res ) ) ) ) ) )
      & ( ( X != Y )
       => ( ( simple2673724192819300657lper_a @ X @ ( cons_list_a @ ( cons_a @ Y @ Ys ) @ Yss ) @ Res )
          = ( simple2673724192819300657lper_a @ X @ Yss @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(3)
thf(fact_7_prod_Ocollapse,axiom,
    ! [Prod: produc7974263126369132087st_nat] :
      ( ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ Prod ) @ ( produc4024048262274189571st_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8_prod_Ocollapse,axiom,
    ! [Prod: produc2086622085225315487list_a] :
      ( ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ Prod ) @ ( produc1153002459082627517list_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_9_prod_Ocollapse,axiom,
    ! [Prod: produc2922778490715282969list_a] :
      ( ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ Prod ) @ ( produc4969414473911996599list_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_10_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7974263126369132087st_nat] :
      ( Prod
      = ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ Prod ) @ ( produc4024048262274189571st_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_11_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2086622085225315487list_a] :
      ( Prod
      = ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ Prod ) @ ( produc1153002459082627517list_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_12_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2922778490715282969list_a] :
      ( Prod
      = ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ Prod ) @ ( produc4969414473911996599list_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_13_exI__realizer,axiom,
    ! [P: list_list_nat > $o > $o,Y: list_list_nat,X: $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc4024048262274189571st_nat @ ( produc8892817347036706481st_nat @ X @ Y ) ) @ ( produc3357746125391881157st_nat @ ( produc8892817347036706481st_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_14_exI__realizer,axiom,
    ! [P: list_list_list_a > $o > $o,Y: list_list_list_a,X: $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc1153002459082627517list_a @ ( produc8548274197880810127list_a @ X @ Y ) ) @ ( produc2785537127425143931list_a @ ( produc8548274197880810127list_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_15_exI__realizer,axiom,
    ! [P: list_list_a > $o > $o,Y: list_list_a,X: $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc4969414473911996599list_a @ ( produc7347373609238049417list_a @ X @ Y ) ) @ ( produc3726311466776224373list_a @ ( produc7347373609238049417list_a @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_16_conjI__realizer,axiom,
    ! [P: $o > $o,P2: $o,Q: list_list_nat > $o,Q2: list_list_nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3357746125391881157st_nat @ ( produc8892817347036706481st_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4024048262274189571st_nat @ ( produc8892817347036706481st_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_17_conjI__realizer,axiom,
    ! [P: $o > $o,P2: $o,Q: list_list_list_a > $o,Q2: list_list_list_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc2785537127425143931list_a @ ( produc8548274197880810127list_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc1153002459082627517list_a @ ( produc8548274197880810127list_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_18_conjI__realizer,axiom,
    ! [P: $o > $o,P2: $o,Q: list_list_a > $o,Q2: list_list_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc3726311466776224373list_a @ ( produc7347373609238049417list_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc4969414473911996599list_a @ ( produc7347373609238049417list_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_19_surjective__pairing,axiom,
    ! [T: produc7974263126369132087st_nat] :
      ( T
      = ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ T ) @ ( produc4024048262274189571st_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_20_surjective__pairing,axiom,
    ! [T: produc2086622085225315487list_a] :
      ( T
      = ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ T ) @ ( produc1153002459082627517list_a @ T ) ) ) ).

% surjective_pairing
thf(fact_21_surjective__pairing,axiom,
    ! [T: produc2922778490715282969list_a] :
      ( T
      = ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ T ) @ ( produc4969414473911996599list_a @ T ) ) ) ).

% surjective_pairing
thf(fact_22_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: $o > list_list_nat > $o,X: $o,Y: list_list_nat,A: produc7974263126369132087st_nat] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc8892817347036706481st_nat @ X @ Y ) )
       => ( P @ ( produc3357746125391881157st_nat @ A ) @ ( produc4024048262274189571st_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_23_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: $o > list_list_list_a > $o,X: $o,Y: list_list_list_a,A: produc2086622085225315487list_a] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc8548274197880810127list_a @ X @ Y ) )
       => ( P @ ( produc2785537127425143931list_a @ A ) @ ( produc1153002459082627517list_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_24_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: $o > list_list_a > $o,X: $o,Y: list_list_a,A: produc2922778490715282969list_a] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc7347373609238049417list_a @ X @ Y ) )
       => ( P @ ( produc3726311466776224373list_a @ A ) @ ( produc4969414473911996599list_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_25_list_Oinject,axiom,
    ! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
      ( ( ( cons_list_nat @ X21 @ X22 )
        = ( cons_list_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_26_list_Oinject,axiom,
    ! [X21: list_list_a,X22: list_list_list_a,Y21: list_list_a,Y22: list_list_list_a] :
      ( ( ( cons_list_list_a @ X21 @ X22 )
        = ( cons_list_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_27_list_Oinject,axiom,
    ! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
      ( ( ( cons_list_a @ X21 @ X22 )
        = ( cons_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_28_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_29_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_30_prod_Oinject,axiom,
    ! [X1: $o,X2: list_list_a,Y1: $o,Y2: list_list_a] :
      ( ( ( produc7347373609238049417list_a @ X1 @ X2 )
        = ( produc7347373609238049417list_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_31_prod_Oinject,axiom,
    ! [X1: $o,X2: list_list_nat,Y1: $o,Y2: list_list_nat] :
      ( ( ( produc8892817347036706481st_nat @ X1 @ X2 )
        = ( produc8892817347036706481st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_32_prod_Oinject,axiom,
    ! [X1: $o,X2: list_list_list_a,Y1: $o,Y2: list_list_list_a] :
      ( ( ( produc8548274197880810127list_a @ X1 @ X2 )
        = ( produc8548274197880810127list_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_33_old_Oprod_Oinject,axiom,
    ! [A: $o,B: list_list_a,A2: $o,B2: list_list_a] :
      ( ( ( produc7347373609238049417list_a @ A @ B )
        = ( produc7347373609238049417list_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
    ! [A: $o,B: list_list_nat,A2: $o,B2: list_list_nat] :
      ( ( ( produc8892817347036706481st_nat @ A @ B )
        = ( produc8892817347036706481st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_35_old_Oprod_Oinject,axiom,
    ! [A: $o,B: list_list_list_a,A2: $o,B2: list_list_list_a] :
      ( ( ( produc8548274197880810127list_a @ A @ B )
        = ( produc8548274197880810127list_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_36_prod_Oexpand,axiom,
    ! [Prod: produc2922778490715282969list_a,Prod2: produc2922778490715282969list_a] :
      ( ( ( ( produc3726311466776224373list_a @ Prod )
          = ( produc3726311466776224373list_a @ Prod2 ) )
        & ( ( produc4969414473911996599list_a @ Prod )
          = ( produc4969414473911996599list_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_37_prod_Oexpand,axiom,
    ! [Prod: produc7974263126369132087st_nat,Prod2: produc7974263126369132087st_nat] :
      ( ( ( ( produc3357746125391881157st_nat @ Prod )
          = ( produc3357746125391881157st_nat @ Prod2 ) )
        & ( ( produc4024048262274189571st_nat @ Prod )
          = ( produc4024048262274189571st_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_38_prod_Oexpand,axiom,
    ! [Prod: produc2086622085225315487list_a,Prod2: produc2086622085225315487list_a] :
      ( ( ( ( produc2785537127425143931list_a @ Prod )
          = ( produc2785537127425143931list_a @ Prod2 ) )
        & ( ( produc1153002459082627517list_a @ Prod )
          = ( produc1153002459082627517list_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_39_prod__eqI,axiom,
    ! [P2: produc2922778490715282969list_a,Q2: produc2922778490715282969list_a] :
      ( ( ( produc3726311466776224373list_a @ P2 )
        = ( produc3726311466776224373list_a @ Q2 ) )
     => ( ( ( produc4969414473911996599list_a @ P2 )
          = ( produc4969414473911996599list_a @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_40_prod__eqI,axiom,
    ! [P2: produc7974263126369132087st_nat,Q2: produc7974263126369132087st_nat] :
      ( ( ( produc3357746125391881157st_nat @ P2 )
        = ( produc3357746125391881157st_nat @ Q2 ) )
     => ( ( ( produc4024048262274189571st_nat @ P2 )
          = ( produc4024048262274189571st_nat @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_41_prod__eqI,axiom,
    ! [P2: produc2086622085225315487list_a,Q2: produc2086622085225315487list_a] :
      ( ( ( produc2785537127425143931list_a @ P2 )
        = ( produc2785537127425143931list_a @ Q2 ) )
     => ( ( ( produc1153002459082627517list_a @ P2 )
          = ( produc1153002459082627517list_a @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_42_exE__realizer_H,axiom,
    ! [P: list_list_a > $o > $o,P2: produc2922778490715282969list_a] :
      ( ( P @ ( produc4969414473911996599list_a @ P2 ) @ ( produc3726311466776224373list_a @ P2 ) )
     => ~ ! [X3: $o,Y3: list_list_a] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_43_exE__realizer_H,axiom,
    ! [P: list_list_nat > $o > $o,P2: produc7974263126369132087st_nat] :
      ( ( P @ ( produc4024048262274189571st_nat @ P2 ) @ ( produc3357746125391881157st_nat @ P2 ) )
     => ~ ! [X3: $o,Y3: list_list_nat] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_44_exE__realizer_H,axiom,
    ! [P: list_list_list_a > $o > $o,P2: produc2086622085225315487list_a] :
      ( ( P @ ( produc1153002459082627517list_a @ P2 ) @ ( produc2785537127425143931list_a @ P2 ) )
     => ~ ! [X3: $o,Y3: list_list_list_a] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_45_Pair__inject,axiom,
    ! [A: $o,B: list_list_a,A2: $o,B2: list_list_a] :
      ( ( ( produc7347373609238049417list_a @ A @ B )
        = ( produc7347373609238049417list_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_46_Pair__inject,axiom,
    ! [A: $o,B: list_list_nat,A2: $o,B2: list_list_nat] :
      ( ( ( produc8892817347036706481st_nat @ A @ B )
        = ( produc8892817347036706481st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_47_Pair__inject,axiom,
    ! [A: $o,B: list_list_list_a,A2: $o,B2: list_list_list_a] :
      ( ( ( produc8548274197880810127list_a @ A @ B )
        = ( produc8548274197880810127list_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_48_prod__cases,axiom,
    ! [P: produc2922778490715282969list_a > $o,P2: produc2922778490715282969list_a] :
      ( ! [A3: $o,B3: list_list_a] : ( P @ ( produc7347373609238049417list_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_49_prod__cases,axiom,
    ! [P: produc7974263126369132087st_nat > $o,P2: produc7974263126369132087st_nat] :
      ( ! [A3: $o,B3: list_list_nat] : ( P @ ( produc8892817347036706481st_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_50_prod__cases,axiom,
    ! [P: produc2086622085225315487list_a > $o,P2: produc2086622085225315487list_a] :
      ( ! [A3: $o,B3: list_list_list_a] : ( P @ ( produc8548274197880810127list_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_51_surj__pair,axiom,
    ! [P2: produc2922778490715282969list_a] :
    ? [X3: $o,Y3: list_list_a] :
      ( P2
      = ( produc7347373609238049417list_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_52_surj__pair,axiom,
    ! [P2: produc7974263126369132087st_nat] :
    ? [X3: $o,Y3: list_list_nat] :
      ( P2
      = ( produc8892817347036706481st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_53_surj__pair,axiom,
    ! [P2: produc2086622085225315487list_a] :
    ? [X3: $o,Y3: list_list_list_a] :
      ( P2
      = ( produc8548274197880810127list_a @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_54_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2922778490715282969list_a] :
      ~ ! [A3: $o,B3: list_list_a] :
          ( Y
         != ( produc7347373609238049417list_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_55_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7974263126369132087st_nat] :
      ~ ! [A3: $o,B3: list_list_nat] :
          ( Y
         != ( produc8892817347036706481st_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2086622085225315487list_a] :
      ~ ! [A3: $o,B3: list_list_list_a] :
          ( Y
         != ( produc8548274197880810127list_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_57_not__Cons__self2,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( cons_list_a @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_58_not__Cons__self2,axiom,
    ! [X: a,Xs: list_a] :
      ( ( cons_a @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_59_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_60_not__Cons__self2,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( cons_list_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_61_not__Cons__self2,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( cons_list_list_a @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_62_snd__conv,axiom,
    ! [X1: $o,X2: list_list_a] :
      ( ( produc4969414473911996599list_a @ ( produc7347373609238049417list_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_63_snd__conv,axiom,
    ! [X1: $o,X2: list_list_nat] :
      ( ( produc4024048262274189571st_nat @ ( produc8892817347036706481st_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_64_snd__conv,axiom,
    ! [X1: $o,X2: list_list_list_a] :
      ( ( produc1153002459082627517list_a @ ( produc8548274197880810127list_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_65_snd__eqD,axiom,
    ! [X: $o,Y: list_list_a,A: list_list_a] :
      ( ( ( produc4969414473911996599list_a @ ( produc7347373609238049417list_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_66_snd__eqD,axiom,
    ! [X: $o,Y: list_list_nat,A: list_list_nat] :
      ( ( ( produc4024048262274189571st_nat @ ( produc8892817347036706481st_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_67_snd__eqD,axiom,
    ! [X: $o,Y: list_list_list_a,A: list_list_list_a] :
      ( ( ( produc1153002459082627517list_a @ ( produc8548274197880810127list_a @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_68_fst__conv,axiom,
    ! [X1: $o,X2: list_list_a] :
      ( ( produc3726311466776224373list_a @ ( produc7347373609238049417list_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_69_fst__conv,axiom,
    ! [X1: $o,X2: list_list_nat] :
      ( ( produc3357746125391881157st_nat @ ( produc8892817347036706481st_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_70_fst__conv,axiom,
    ! [X1: $o,X2: list_list_list_a] :
      ( ( produc2785537127425143931list_a @ ( produc8548274197880810127list_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_71_fst__eqD,axiom,
    ! [X: $o,Y: list_list_a,A: $o] :
      ( ( ( produc3726311466776224373list_a @ ( produc7347373609238049417list_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_72_fst__eqD,axiom,
    ! [X: $o,Y: list_list_nat,A: $o] :
      ( ( ( produc3357746125391881157st_nat @ ( produc8892817347036706481st_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_73_fst__eqD,axiom,
    ! [X: $o,Y: list_list_list_a,A: $o] :
      ( ( ( produc2785537127425143931list_a @ ( produc8548274197880810127list_a @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_74_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2922778490715282969list_a,Z: produc2922778490715282969list_a] : ( Y4 = Z ) )
    = ( ^ [S: produc2922778490715282969list_a,T2: produc2922778490715282969list_a] :
          ( ( ( produc3726311466776224373list_a @ S )
            = ( produc3726311466776224373list_a @ T2 ) )
          & ( ( produc4969414473911996599list_a @ S )
            = ( produc4969414473911996599list_a @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_75_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc7974263126369132087st_nat,Z: produc7974263126369132087st_nat] : ( Y4 = Z ) )
    = ( ^ [S: produc7974263126369132087st_nat,T2: produc7974263126369132087st_nat] :
          ( ( ( produc3357746125391881157st_nat @ S )
            = ( produc3357746125391881157st_nat @ T2 ) )
          & ( ( produc4024048262274189571st_nat @ S )
            = ( produc4024048262274189571st_nat @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_76_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc2086622085225315487list_a,Z: produc2086622085225315487list_a] : ( Y4 = Z ) )
    = ( ^ [S: produc2086622085225315487list_a,T2: produc2086622085225315487list_a] :
          ( ( ( produc2785537127425143931list_a @ S )
            = ( produc2785537127425143931list_a @ T2 ) )
          & ( ( produc1153002459082627517list_a @ S )
            = ( produc1153002459082627517list_a @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_77_eq__fst__iff,axiom,
    ! [A: $o,P2: produc2922778490715282969list_a] :
      ( ( A
        = ( produc3726311466776224373list_a @ P2 ) )
      = ( ? [B4: list_list_a] :
            ( P2
            = ( produc7347373609238049417list_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_78_eq__fst__iff,axiom,
    ! [A: $o,P2: produc7974263126369132087st_nat] :
      ( ( A
        = ( produc3357746125391881157st_nat @ P2 ) )
      = ( ? [B4: list_list_nat] :
            ( P2
            = ( produc8892817347036706481st_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_79_eq__fst__iff,axiom,
    ! [A: $o,P2: produc2086622085225315487list_a] :
      ( ( A
        = ( produc2785537127425143931list_a @ P2 ) )
      = ( ? [B4: list_list_list_a] :
            ( P2
            = ( produc8548274197880810127list_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_80_fstI,axiom,
    ! [X: produc2922778490715282969list_a,Y: $o,Z2: list_list_a] :
      ( ( X
        = ( produc7347373609238049417list_a @ Y @ Z2 ) )
     => ( ( produc3726311466776224373list_a @ X )
        = Y ) ) ).

% fstI
thf(fact_81_fstI,axiom,
    ! [X: produc7974263126369132087st_nat,Y: $o,Z2: list_list_nat] :
      ( ( X
        = ( produc8892817347036706481st_nat @ Y @ Z2 ) )
     => ( ( produc3357746125391881157st_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_82_fstI,axiom,
    ! [X: produc2086622085225315487list_a,Y: $o,Z2: list_list_list_a] :
      ( ( X
        = ( produc8548274197880810127list_a @ Y @ Z2 ) )
     => ( ( produc2785537127425143931list_a @ X )
        = Y ) ) ).

% fstI
thf(fact_83_eq__snd__iff,axiom,
    ! [B: list_list_a,P2: produc2922778490715282969list_a] :
      ( ( B
        = ( produc4969414473911996599list_a @ P2 ) )
      = ( ? [A4: $o] :
            ( P2
            = ( produc7347373609238049417list_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_84_eq__snd__iff,axiom,
    ! [B: list_list_nat,P2: produc7974263126369132087st_nat] :
      ( ( B
        = ( produc4024048262274189571st_nat @ P2 ) )
      = ( ? [A4: $o] :
            ( P2
            = ( produc8892817347036706481st_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_85_eq__snd__iff,axiom,
    ! [B: list_list_list_a,P2: produc2086622085225315487list_a] :
      ( ( B
        = ( produc1153002459082627517list_a @ P2 ) )
      = ( ? [A4: $o] :
            ( P2
            = ( produc8548274197880810127list_a @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_86_sndI,axiom,
    ! [X: produc2922778490715282969list_a,Y: $o,Z2: list_list_a] :
      ( ( X
        = ( produc7347373609238049417list_a @ Y @ Z2 ) )
     => ( ( produc4969414473911996599list_a @ X )
        = Z2 ) ) ).

% sndI
thf(fact_87_sndI,axiom,
    ! [X: produc7974263126369132087st_nat,Y: $o,Z2: list_list_nat] :
      ( ( X
        = ( produc8892817347036706481st_nat @ Y @ Z2 ) )
     => ( ( produc4024048262274189571st_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_88_sndI,axiom,
    ! [X: produc2086622085225315487list_a,Y: $o,Z2: list_list_list_a] :
      ( ( X
        = ( produc8548274197880810127list_a @ Y @ Z2 ) )
     => ( ( produc1153002459082627517list_a @ X )
        = Z2 ) ) ).

% sndI
thf(fact_89_prefixes__in__list__helper_Osimps_I2_J,axiom,
    ! [X: a,Yss: list_list_a,Res: produc2922778490715282969list_a] :
      ( ( simple2673724192819300657lper_a @ X @ ( cons_list_a @ nil_a @ Yss ) @ Res )
      = ( simple2673724192819300657lper_a @ X @ Yss @ ( produc7347373609238049417list_a @ $true @ ( produc4969414473911996599list_a @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(2)
thf(fact_90_prefixes__in__list__helper_Osimps_I2_J,axiom,
    ! [X: nat,Yss: list_list_nat,Res: produc7974263126369132087st_nat] :
      ( ( simple1541803723984754397er_nat @ X @ ( cons_list_nat @ nil_nat @ Yss ) @ Res )
      = ( simple1541803723984754397er_nat @ X @ Yss @ ( produc8892817347036706481st_nat @ $true @ ( produc4024048262274189571st_nat @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(2)
thf(fact_91_prefixes__in__list__helper_Osimps_I2_J,axiom,
    ! [X: list_a,Yss: list_list_list_a,Res: produc2086622085225315487list_a] :
      ( ( simple5839407689009508407list_a @ X @ ( cons_list_list_a @ nil_list_a @ Yss ) @ Res )
      = ( simple5839407689009508407list_a @ X @ Yss @ ( produc8548274197880810127list_a @ $true @ ( produc1153002459082627517list_a @ Res ) ) ) ) ).

% prefixes_in_list_helper.simps(2)
thf(fact_92_prod_Oswap__def,axiom,
    ( produc2143262938492138053st_a_o
    = ( ^ [P3: produc3899788845695434549st_a_o] : ( produc7347373609238049417list_a @ ( produc4427084236405564115st_a_o @ P3 ) @ ( produc3183981229269791889st_a_o @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_93_prod_Oswap__def,axiom,
    ( produc6223120681554541487_nat_o
    = ( ^ [P3: produc3539853629763079581_nat_o] : ( produc8892817347036706481st_nat @ ( produc6463515711230485409_nat_o @ P3 ) @ ( produc5797213574348176995_nat_o @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_94_prod_Oswap__def,axiom,
    ( produc701336489830518079st_a_o
    = ( ^ [P3: produc5183940224784037295st_a_o] : ( produc8548274197880810127list_a @ ( produc7187467912298323661st_a_o @ P3 ) @ ( produc8820002580640840075st_a_o @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_95_prod_Oswap__def,axiom,
    ( produc2685593175998570537list_a
    = ( ^ [P3: produc2922778490715282969list_a] : ( produc6805043371731616933st_a_o @ ( produc4969414473911996599list_a @ P3 ) @ ( produc3726311466776224373list_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_96_prod_Oswap__def,axiom,
    ( produc3783653232598245649st_nat
    = ( ^ [P3: produc7974263126369132087st_nat] : ( produc2108912759138226511_nat_o @ ( produc4024048262274189571st_nat @ P3 ) @ ( produc3357746125391881157st_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_97_prod_Oswap__def,axiom,
    ( produc3890243073469597743list_a
    = ( ^ [P3: produc2086622085225315487list_a] : ( produc5359367614241730463st_a_o @ ( produc1153002459082627517list_a @ P3 ) @ ( produc2785537127425143931list_a @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_98_mem__Collect__eq,axiom,
    ! [A: list_a,P: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
    ! [A5: set_list_a] :
      ( ( collect_list_a
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_100_Collect__cong,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X3: list_a] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_list_a @ P )
        = ( collect_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_101_snd__swap,axiom,
    ! [X: produc2922778490715282969list_a] :
      ( ( produc4427084236405564115st_a_o @ ( produc2685593175998570537list_a @ X ) )
      = ( produc3726311466776224373list_a @ X ) ) ).

% snd_swap
thf(fact_102_snd__swap,axiom,
    ! [X: produc7974263126369132087st_nat] :
      ( ( produc6463515711230485409_nat_o @ ( produc3783653232598245649st_nat @ X ) )
      = ( produc3357746125391881157st_nat @ X ) ) ).

% snd_swap
thf(fact_103_snd__swap,axiom,
    ! [X: produc2086622085225315487list_a] :
      ( ( produc7187467912298323661st_a_o @ ( produc3890243073469597743list_a @ X ) )
      = ( produc2785537127425143931list_a @ X ) ) ).

% snd_swap
thf(fact_104_snd__swap,axiom,
    ! [X: produc3899788845695434549st_a_o] :
      ( ( produc4969414473911996599list_a @ ( produc2143262938492138053st_a_o @ X ) )
      = ( produc3183981229269791889st_a_o @ X ) ) ).

% snd_swap
thf(fact_105_snd__swap,axiom,
    ! [X: produc3539853629763079581_nat_o] :
      ( ( produc4024048262274189571st_nat @ ( produc6223120681554541487_nat_o @ X ) )
      = ( produc5797213574348176995_nat_o @ X ) ) ).

% snd_swap
thf(fact_106_snd__swap,axiom,
    ! [X: produc5183940224784037295st_a_o] :
      ( ( produc1153002459082627517list_a @ ( produc701336489830518079st_a_o @ X ) )
      = ( produc8820002580640840075st_a_o @ X ) ) ).

% snd_swap
thf(fact_107_fst__swap,axiom,
    ! [X: produc3899788845695434549st_a_o] :
      ( ( produc3726311466776224373list_a @ ( produc2143262938492138053st_a_o @ X ) )
      = ( produc4427084236405564115st_a_o @ X ) ) ).

% fst_swap
thf(fact_108_fst__swap,axiom,
    ! [X: produc3539853629763079581_nat_o] :
      ( ( produc3357746125391881157st_nat @ ( produc6223120681554541487_nat_o @ X ) )
      = ( produc6463515711230485409_nat_o @ X ) ) ).

% fst_swap
thf(fact_109_fst__swap,axiom,
    ! [X: produc5183940224784037295st_a_o] :
      ( ( produc2785537127425143931list_a @ ( produc701336489830518079st_a_o @ X ) )
      = ( produc7187467912298323661st_a_o @ X ) ) ).

% fst_swap
thf(fact_110_fst__swap,axiom,
    ! [X: produc2922778490715282969list_a] :
      ( ( produc3183981229269791889st_a_o @ ( produc2685593175998570537list_a @ X ) )
      = ( produc4969414473911996599list_a @ X ) ) ).

% fst_swap
thf(fact_111_fst__swap,axiom,
    ! [X: produc7974263126369132087st_nat] :
      ( ( produc5797213574348176995_nat_o @ ( produc3783653232598245649st_nat @ X ) )
      = ( produc4024048262274189571st_nat @ X ) ) ).

% fst_swap
thf(fact_112_fst__swap,axiom,
    ! [X: produc2086622085225315487list_a] :
      ( ( produc8820002580640840075st_a_o @ ( produc3890243073469597743list_a @ X ) )
      = ( produc1153002459082627517list_a @ X ) ) ).

% fst_swap
thf(fact_113_prefixes__in__list__helper_Oelims,axiom,
    ! [X: list_nat,Xa: list_list_list_nat,Xb: produc3585458782675957319st_nat,Y: produc3585458782675957319st_nat] :
      ( ( ( simple6281051842103134061st_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_list_list_nat )
         => ( Y != Xb ) )
       => ( ! [Yss2: list_list_list_nat] :
              ( ( Xa
                = ( cons_list_list_nat @ nil_list_nat @ Yss2 ) )
             => ( Y
               != ( simple6281051842103134061st_nat @ X @ Yss2 @ ( produc3734841530647658561st_nat @ $true @ ( produc5745704060670869907st_nat @ Xb ) ) ) ) )
         => ~ ! [Y3: list_nat,Ys2: list_list_nat,Yss2: list_list_list_nat] :
                ( ( Xa
                  = ( cons_list_list_nat @ ( cons_list_nat @ Y3 @ Ys2 ) @ Yss2 ) )
               => ~ ( ( ( X = Y3 )
                     => ( Y
                        = ( simple6281051842103134061st_nat @ X @ Yss2 @ ( produc3734841530647658561st_nat @ ( produc5989645662155510613st_nat @ Xb ) @ ( cons_list_list_nat @ Ys2 @ ( produc5745704060670869907st_nat @ Xb ) ) ) ) ) )
                    & ( ( X != Y3 )
                     => ( Y
                        = ( simple6281051842103134061st_nat @ X @ Yss2 @ Xb ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.elims
thf(fact_114_prefixes__in__list__helper_Oelims,axiom,
    ! [X: list_list_a,Xa: list_l7815035709764258326list_a,Xb: produc2144844930413989541list_a,Y: produc2144844930413989541list_a] :
      ( ( ( simple7980934897880681149list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_list_list_list_a )
         => ( Y != Xb ) )
       => ( ! [Yss2: list_l7815035709764258326list_a] :
              ( ( Xa
                = ( cons_l8577867181958743494list_a @ nil_list_list_a @ Yss2 ) )
             => ( Y
               != ( simple7980934897880681149list_a @ X @ Yss2 @ ( produc3597484984460465685list_a @ $true @ ( produc8615863343057196611list_a @ Xb ) ) ) ) )
         => ~ ! [Y3: list_list_a,Ys2: list_list_list_a,Yss2: list_l7815035709764258326list_a] :
                ( ( Xa
                  = ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ Y3 @ Ys2 ) @ Yss2 ) )
               => ~ ( ( ( X = Y3 )
                     => ( Y
                        = ( simple7980934897880681149list_a @ X @ Yss2 @ ( produc3597484984460465685list_a @ ( produc7804899884738361857list_a @ Xb ) @ ( cons_l8577867181958743494list_a @ Ys2 @ ( produc8615863343057196611list_a @ Xb ) ) ) ) ) )
                    & ( ( X != Y3 )
                     => ( Y
                        = ( simple7980934897880681149list_a @ X @ Yss2 @ Xb ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.elims
thf(fact_115_prefixes__in__list__helper_Oelims,axiom,
    ! [X: a,Xa: list_list_a,Xb: produc2922778490715282969list_a,Y: produc2922778490715282969list_a] :
      ( ( ( simple2673724192819300657lper_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_list_a )
         => ( Y != Xb ) )
       => ( ! [Yss2: list_list_a] :
              ( ( Xa
                = ( cons_list_a @ nil_a @ Yss2 ) )
             => ( Y
               != ( simple2673724192819300657lper_a @ X @ Yss2 @ ( produc7347373609238049417list_a @ $true @ ( produc4969414473911996599list_a @ Xb ) ) ) ) )
         => ~ ! [Y3: a,Ys2: list_a,Yss2: list_list_a] :
                ( ( Xa
                  = ( cons_list_a @ ( cons_a @ Y3 @ Ys2 ) @ Yss2 ) )
               => ~ ( ( ( X = Y3 )
                     => ( Y
                        = ( simple2673724192819300657lper_a @ X @ Yss2 @ ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ Xb ) @ ( cons_list_a @ Ys2 @ ( produc4969414473911996599list_a @ Xb ) ) ) ) ) )
                    & ( ( X != Y3 )
                     => ( Y
                        = ( simple2673724192819300657lper_a @ X @ Yss2 @ Xb ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.elims
thf(fact_116_prefixes__in__list__helper_Oelims,axiom,
    ! [X: nat,Xa: list_list_nat,Xb: produc7974263126369132087st_nat,Y: produc7974263126369132087st_nat] :
      ( ( ( simple1541803723984754397er_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_list_nat )
         => ( Y != Xb ) )
       => ( ! [Yss2: list_list_nat] :
              ( ( Xa
                = ( cons_list_nat @ nil_nat @ Yss2 ) )
             => ( Y
               != ( simple1541803723984754397er_nat @ X @ Yss2 @ ( produc8892817347036706481st_nat @ $true @ ( produc4024048262274189571st_nat @ Xb ) ) ) ) )
         => ~ ! [Y3: nat,Ys2: list_nat,Yss2: list_list_nat] :
                ( ( Xa
                  = ( cons_list_nat @ ( cons_nat @ Y3 @ Ys2 ) @ Yss2 ) )
               => ~ ( ( ( X = Y3 )
                     => ( Y
                        = ( simple1541803723984754397er_nat @ X @ Yss2 @ ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ Xb ) @ ( cons_list_nat @ Ys2 @ ( produc4024048262274189571st_nat @ Xb ) ) ) ) ) )
                    & ( ( X != Y3 )
                     => ( Y
                        = ( simple1541803723984754397er_nat @ X @ Yss2 @ Xb ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.elims
thf(fact_117_prefixes__in__list__helper_Oelims,axiom,
    ! [X: list_a,Xa: list_list_list_a,Xb: produc2086622085225315487list_a,Y: produc2086622085225315487list_a] :
      ( ( ( simple5839407689009508407list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_list_list_a )
         => ( Y != Xb ) )
       => ( ! [Yss2: list_list_list_a] :
              ( ( Xa
                = ( cons_list_list_a @ nil_list_a @ Yss2 ) )
             => ( Y
               != ( simple5839407689009508407list_a @ X @ Yss2 @ ( produc8548274197880810127list_a @ $true @ ( produc1153002459082627517list_a @ Xb ) ) ) ) )
         => ~ ! [Y3: list_a,Ys2: list_list_a,Yss2: list_list_list_a] :
                ( ( Xa
                  = ( cons_list_list_a @ ( cons_list_a @ Y3 @ Ys2 ) @ Yss2 ) )
               => ~ ( ( ( X = Y3 )
                     => ( Y
                        = ( simple5839407689009508407list_a @ X @ Yss2 @ ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ Xb ) @ ( cons_list_list_a @ Ys2 @ ( produc1153002459082627517list_a @ Xb ) ) ) ) ) )
                    & ( ( X != Y3 )
                     => ( Y
                        = ( simple5839407689009508407list_a @ X @ Yss2 @ Xb ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.elims
thf(fact_118_swap__simp,axiom,
    ! [X: list_list_a,Y: $o] :
      ( ( produc2143262938492138053st_a_o @ ( produc6805043371731616933st_a_o @ X @ Y ) )
      = ( produc7347373609238049417list_a @ Y @ X ) ) ).

% swap_simp
thf(fact_119_swap__simp,axiom,
    ! [X: list_list_nat,Y: $o] :
      ( ( produc6223120681554541487_nat_o @ ( produc2108912759138226511_nat_o @ X @ Y ) )
      = ( produc8892817347036706481st_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_120_swap__simp,axiom,
    ! [X: list_list_list_a,Y: $o] :
      ( ( produc701336489830518079st_a_o @ ( produc5359367614241730463st_a_o @ X @ Y ) )
      = ( produc8548274197880810127list_a @ Y @ X ) ) ).

% swap_simp
thf(fact_121_swap__simp,axiom,
    ! [X: $o,Y: list_list_a] :
      ( ( produc2685593175998570537list_a @ ( produc7347373609238049417list_a @ X @ Y ) )
      = ( produc6805043371731616933st_a_o @ Y @ X ) ) ).

% swap_simp
thf(fact_122_swap__simp,axiom,
    ! [X: $o,Y: list_list_nat] :
      ( ( produc3783653232598245649st_nat @ ( produc8892817347036706481st_nat @ X @ Y ) )
      = ( produc2108912759138226511_nat_o @ Y @ X ) ) ).

% swap_simp
thf(fact_123_swap__simp,axiom,
    ! [X: $o,Y: list_list_list_a] :
      ( ( produc3890243073469597743list_a @ ( produc8548274197880810127list_a @ X @ Y ) )
      = ( produc5359367614241730463st_a_o @ Y @ X ) ) ).

% swap_simp
thf(fact_124_transpose_Ocases,axiom,
    ! [X: list_list_list_nat] :
      ( ( X != nil_list_list_nat )
     => ( ! [Xss: list_list_list_nat] :
            ( X
           != ( cons_list_list_nat @ nil_list_nat @ Xss ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
              ( X
             != ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_125_transpose_Ocases,axiom,
    ! [X: list_l7815035709764258326list_a] :
      ( ( X != nil_list_list_list_a )
     => ( ! [Xss: list_l7815035709764258326list_a] :
            ( X
           != ( cons_l8577867181958743494list_a @ nil_list_list_a @ Xss ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Xss: list_l7815035709764258326list_a] :
              ( X
             != ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_126_transpose_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
              ( X
             != ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_127_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_128_transpose_Ocases,axiom,
    ! [X: list_list_list_a] :
      ( ( X != nil_list_list_a )
     => ( ! [Xss: list_list_list_a] :
            ( X
           != ( cons_list_list_a @ nil_list_a @ Xss ) )
       => ~ ! [X3: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
              ( X
             != ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_129_prefixes__in__list__helper_Ocases,axiom,
    ! [X: produc8039914070420196669st_nat] :
      ( ! [X3: list_nat,Res2: produc3585458782675957319st_nat] :
          ( X
         != ( produc5135401712870579951st_nat @ X3 @ ( produc5677073168774101088st_nat @ nil_list_list_nat @ Res2 ) ) )
     => ( ! [X3: list_nat,Yss2: list_list_list_nat,Res2: produc3585458782675957319st_nat] :
            ( X
           != ( produc5135401712870579951st_nat @ X3 @ ( produc5677073168774101088st_nat @ ( cons_list_list_nat @ nil_list_nat @ Yss2 ) @ Res2 ) ) )
       => ~ ! [X3: list_nat,Y3: list_nat,Ys2: list_list_nat,Yss2: list_list_list_nat,Res2: produc3585458782675957319st_nat] :
              ( X
             != ( produc5135401712870579951st_nat @ X3 @ ( produc5677073168774101088st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ Y3 @ Ys2 ) @ Yss2 ) @ Res2 ) ) ) ) ) ).

% prefixes_in_list_helper.cases
thf(fact_130_prefixes__in__list__helper_Ocases,axiom,
    ! [X: produc3785893746454133715list_a] :
      ( ! [X3: list_list_a,Res2: produc2144844930413989541list_a] :
          ( X
         != ( produc3043550568142011971list_a @ X3 @ ( produc3003139637434810150list_a @ nil_list_list_list_a @ Res2 ) ) )
     => ( ! [X3: list_list_a,Yss2: list_l7815035709764258326list_a,Res2: produc2144844930413989541list_a] :
            ( X
           != ( produc3043550568142011971list_a @ X3 @ ( produc3003139637434810150list_a @ ( cons_l8577867181958743494list_a @ nil_list_list_a @ Yss2 ) @ Res2 ) ) )
       => ~ ! [X3: list_list_a,Y3: list_list_a,Ys2: list_list_list_a,Yss2: list_l7815035709764258326list_a,Res2: produc2144844930413989541list_a] :
              ( X
             != ( produc3043550568142011971list_a @ X3 @ ( produc3003139637434810150list_a @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ Y3 @ Ys2 ) @ Yss2 ) @ Res2 ) ) ) ) ) ).

% prefixes_in_list_helper.cases
thf(fact_131_prefixes__in__list__helper_Ocases,axiom,
    ! [X: produc3276350626183733855list_a] :
      ( ! [X3: a,Res2: produc2922778490715282969list_a] :
          ( X
         != ( produc5627178164694999503list_a @ X3 @ ( produc6979364057698131238list_a @ nil_list_a @ Res2 ) ) )
     => ( ! [X3: a,Yss2: list_list_a,Res2: produc2922778490715282969list_a] :
            ( X
           != ( produc5627178164694999503list_a @ X3 @ ( produc6979364057698131238list_a @ ( cons_list_a @ nil_a @ Yss2 ) @ Res2 ) ) )
       => ~ ! [X3: a,Y3: a,Ys2: list_a,Yss2: list_list_a,Res2: produc2922778490715282969list_a] :
              ( X
             != ( produc5627178164694999503list_a @ X3 @ ( produc6979364057698131238list_a @ ( cons_list_a @ ( cons_a @ Y3 @ Ys2 ) @ Yss2 ) @ Res2 ) ) ) ) ) ).

% prefixes_in_list_helper.cases
thf(fact_132_prefixes__in__list__helper_Ocases,axiom,
    ! [X: produc19496163836001069st_nat] :
      ( ! [X3: nat,Res2: produc7974263126369132087st_nat] :
          ( X
         != ( produc3115374921362880095st_nat @ X3 @ ( produc3419769971367040704st_nat @ nil_list_nat @ Res2 ) ) )
     => ( ! [X3: nat,Yss2: list_list_nat,Res2: produc7974263126369132087st_nat] :
            ( X
           != ( produc3115374921362880095st_nat @ X3 @ ( produc3419769971367040704st_nat @ ( cons_list_nat @ nil_nat @ Yss2 ) @ Res2 ) ) )
       => ~ ! [X3: nat,Y3: nat,Ys2: list_nat,Yss2: list_list_nat,Res2: produc7974263126369132087st_nat] :
              ( X
             != ( produc3115374921362880095st_nat @ X3 @ ( produc3419769971367040704st_nat @ ( cons_list_nat @ ( cons_nat @ Y3 @ Ys2 ) @ Yss2 ) @ Res2 ) ) ) ) ) ).

% prefixes_in_list_helper.cases
thf(fact_133_prefixes__in__list__helper_Ocases,axiom,
    ! [X: produc1773973501398603737list_a] :
      ( ! [X3: list_a,Res2: produc2086622085225315487list_a] :
          ( X
         != ( produc5745923727215425737list_a @ X3 @ ( produc7568776524366471974list_a @ nil_list_list_a @ Res2 ) ) )
     => ( ! [X3: list_a,Yss2: list_list_list_a,Res2: produc2086622085225315487list_a] :
            ( X
           != ( produc5745923727215425737list_a @ X3 @ ( produc7568776524366471974list_a @ ( cons_list_list_a @ nil_list_a @ Yss2 ) @ Res2 ) ) )
       => ~ ! [X3: list_a,Y3: list_a,Ys2: list_list_a,Yss2: list_list_list_a,Res2: produc2086622085225315487list_a] :
              ( X
             != ( produc5745923727215425737list_a @ X3 @ ( produc7568776524366471974list_a @ ( cons_list_list_a @ ( cons_list_a @ Y3 @ Ys2 ) @ Yss2 ) @ Res2 ) ) ) ) ) ).

% prefixes_in_list_helper.cases
thf(fact_134_list_Odistinct_I1_J,axiom,
    ! [X21: list_a,X22: list_list_a] :
      ( nil_list_a
     != ( cons_list_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_135_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_136_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_137_list_Odistinct_I1_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] :
      ( nil_list_nat
     != ( cons_list_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_138_list_Odistinct_I1_J,axiom,
    ! [X21: list_list_a,X22: list_list_list_a] :
      ( nil_list_list_a
     != ( cons_list_list_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_139_list_OdiscI,axiom,
    ! [List: list_list_a,X21: list_a,X22: list_list_a] :
      ( ( List
        = ( cons_list_a @ X21 @ X22 ) )
     => ( List != nil_list_a ) ) ).

% list.discI
thf(fact_140_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_141_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_142_list_OdiscI,axiom,
    ! [List: list_list_nat,X21: list_nat,X22: list_list_nat] :
      ( ( List
        = ( cons_list_nat @ X21 @ X22 ) )
     => ( List != nil_list_nat ) ) ).

% list.discI
thf(fact_143_list_OdiscI,axiom,
    ! [List: list_list_list_a,X21: list_list_a,X22: list_list_list_a] :
      ( ( List
        = ( cons_list_list_a @ X21 @ X22 ) )
     => ( List != nil_list_list_a ) ) ).

% list.discI
thf(fact_144_list_Oexhaust,axiom,
    ! [Y: list_list_a] :
      ( ( Y != nil_list_a )
     => ~ ! [X212: list_a,X222: list_list_a] :
            ( Y
           != ( cons_list_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_145_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_146_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_147_list_Oexhaust,axiom,
    ! [Y: list_list_nat] :
      ( ( Y != nil_list_nat )
     => ~ ! [X212: list_nat,X222: list_list_nat] :
            ( Y
           != ( cons_list_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_148_list_Oexhaust,axiom,
    ! [Y: list_list_list_a] :
      ( ( Y != nil_list_list_a )
     => ~ ! [X212: list_list_a,X222: list_list_list_a] :
            ( Y
           != ( cons_list_list_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_149_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat,Xs2: list_nat] :
          ( X
         != ( cons_nat @ X3 @ Xs2 ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_150_min__list_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ! [X3: list_nat,Xs2: list_list_nat] :
          ( X
         != ( cons_list_nat @ X3 @ Xs2 ) )
     => ( X = nil_list_nat ) ) ).

% min_list.cases
thf(fact_151_remdups__adj_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ( ! [X3: list_a] :
            ( X
           != ( cons_list_a @ X3 @ nil_list_a ) )
       => ~ ! [X3: list_a,Y3: list_a,Xs2: list_list_a] :
              ( X
             != ( cons_list_a @ X3 @ ( cons_list_a @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_152_remdups__adj_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ( ! [X3: a] :
            ( X
           != ( cons_a @ X3 @ nil_a ) )
       => ~ ! [X3: a,Y3: a,Xs2: list_a] :
              ( X
             != ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_153_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X3: nat] :
            ( X
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
              ( X
             != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_154_remdups__adj_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [X3: list_nat] :
            ( X
           != ( cons_list_nat @ X3 @ nil_list_nat ) )
       => ~ ! [X3: list_nat,Y3: list_nat,Xs2: list_list_nat] :
              ( X
             != ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_155_remdups__adj_Ocases,axiom,
    ! [X: list_list_list_a] :
      ( ( X != nil_list_list_a )
     => ( ! [X3: list_list_a] :
            ( X
           != ( cons_list_list_a @ X3 @ nil_list_list_a ) )
       => ~ ! [X3: list_list_a,Y3: list_list_a,Xs2: list_list_list_a] :
              ( X
             != ( cons_list_list_a @ X3 @ ( cons_list_list_a @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_156_neq__Nil__conv,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
      = ( ? [Y5: list_a,Ys3: list_list_a] :
            ( Xs
            = ( cons_list_a @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_157_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y5: a,Ys3: list_a] :
            ( Xs
            = ( cons_a @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_158_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y5: nat,Ys3: list_nat] :
            ( Xs
            = ( cons_nat @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_159_neq__Nil__conv,axiom,
    ! [Xs: list_list_nat] :
      ( ( Xs != nil_list_nat )
      = ( ? [Y5: list_nat,Ys3: list_list_nat] :
            ( Xs
            = ( cons_list_nat @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_160_neq__Nil__conv,axiom,
    ! [Xs: list_list_list_a] :
      ( ( Xs != nil_list_list_a )
      = ( ? [Y5: list_list_a,Ys3: list_list_list_a] :
            ( Xs
            = ( cons_list_list_a @ Y5 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_161_list__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
       => ( ! [Y3: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys2 ) )
         => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_162_list__induct2_H,axiom,
    ! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
      ( ( P @ nil_a @ nil_nat )
     => ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_a @ ( cons_nat @ Y3 @ Ys2 ) )
         => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_163_list__induct2_H,axiom,
    ! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
      ( ( P @ nil_nat @ nil_a )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_a )
       => ( ! [Y3: a,Ys2: list_a] : ( P @ nil_nat @ ( cons_a @ Y3 @ Ys2 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_164_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_165_list__induct2_H,axiom,
    ! [P: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
      ( ( P @ nil_list_a @ nil_a )
     => ( ! [X3: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X3 @ Xs2 ) @ nil_a )
       => ( ! [Y3: a,Ys2: list_a] : ( P @ nil_list_a @ ( cons_a @ Y3 @ Ys2 ) )
         => ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_166_list__induct2_H,axiom,
    ! [P: list_list_a > list_nat > $o,Xs: list_list_a,Ys: list_nat] :
      ( ( P @ nil_list_a @ nil_nat )
     => ( ! [X3: list_a,Xs2: list_list_a] : ( P @ ( cons_list_a @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_list_a @ ( cons_nat @ Y3 @ Ys2 ) )
         => ( ! [X3: list_a,Xs2: list_list_a,Y3: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_167_list__induct2_H,axiom,
    ! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
      ( ( P @ nil_a @ nil_list_a )
     => ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_list_a )
       => ( ! [Y3: list_a,Ys2: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y3 @ Ys2 ) )
         => ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_168_list__induct2_H,axiom,
    ! [P: list_a > list_list_nat > $o,Xs: list_a,Ys: list_list_nat] :
      ( ( P @ nil_a @ nil_list_nat )
     => ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_list_nat )
       => ( ! [Y3: list_nat,Ys2: list_list_nat] : ( P @ nil_a @ ( cons_list_nat @ Y3 @ Ys2 ) )
         => ( ! [X3: a,Xs2: list_a,Y3: list_nat,Ys2: list_list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_169_list__induct2_H,axiom,
    ! [P: list_nat > list_list_a > $o,Xs: list_nat,Ys: list_list_a] :
      ( ( P @ nil_nat @ nil_list_a )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_list_a )
       => ( ! [Y3: list_a,Ys2: list_list_a] : ( P @ nil_nat @ ( cons_list_a @ Y3 @ Ys2 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: list_a,Ys2: list_list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_170_list__induct2_H,axiom,
    ! [P: list_nat > list_list_nat > $o,Xs: list_nat,Ys: list_list_nat] :
      ( ( P @ nil_nat @ nil_list_nat )
     => ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_list_nat )
       => ( ! [Y3: list_nat,Ys2: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y3 @ Ys2 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_171_list__nonempty__induct,axiom,
    ! [Xs: list_list_a,P: list_list_a > $o] :
      ( ( Xs != nil_list_a )
     => ( ! [X3: list_a] : ( P @ ( cons_list_a @ X3 @ nil_list_a ) )
       => ( ! [X3: list_a,Xs2: list_list_a] :
              ( ( Xs2 != nil_list_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_list_a @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_172_list__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_173_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_174_list__nonempty__induct,axiom,
    ! [Xs: list_list_nat,P: list_list_nat > $o] :
      ( ( Xs != nil_list_nat )
     => ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
       => ( ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( Xs2 != nil_list_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_list_nat @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_175_list__nonempty__induct,axiom,
    ! [Xs: list_list_list_a,P: list_list_list_a > $o] :
      ( ( Xs != nil_list_list_a )
     => ( ! [X3: list_list_a] : ( P @ ( cons_list_list_a @ X3 @ nil_list_list_a ) )
       => ( ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( Xs2 != nil_list_list_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_176_prefixes__in__list_Ocases,axiom,
    ! [X: produc6284823174688480481list_a] :
      ( ! [Prev: list_list_a,Yss2: list_list_list_a,Res2: list_list_list_a] :
          ( X
         != ( produc6222199892163455185list_a @ nil_list_a @ ( produc2142498889433442228list_a @ Prev @ ( produc1091363791885468951list_a @ Yss2 @ Res2 ) ) ) )
     => ~ ! [X3: list_a,Xs2: list_list_a,Prev: list_list_a,Yss2: list_list_list_a,Res2: list_list_list_a] :
            ( X
           != ( produc6222199892163455185list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( produc2142498889433442228list_a @ Prev @ ( produc1091363791885468951list_a @ Yss2 @ Res2 ) ) ) ) ) ).

% prefixes_in_list.cases
thf(fact_177_prefixes__in__list_Ocases,axiom,
    ! [X: produc6233025874956846573list_a] :
      ( ! [Prev: list_a,Yss2: list_list_a,Res2: list_list_a] :
          ( X
         != ( produc7311868682961280221list_a @ nil_a @ ( produc5592687884163559226list_a @ Prev @ ( produc8696003437204565271list_a @ Yss2 @ Res2 ) ) ) )
     => ~ ! [X3: a,Xs2: list_a,Prev: list_a,Yss2: list_list_a,Res2: list_list_a] :
            ( X
           != ( produc7311868682961280221list_a @ ( cons_a @ X3 @ Xs2 ) @ ( produc5592687884163559226list_a @ Prev @ ( produc8696003437204565271list_a @ Yss2 @ Res2 ) ) ) ) ) ).

% prefixes_in_list.cases
thf(fact_178_prefixes__in__list_Ocases,axiom,
    ! [X: produc2175610488484699543st_nat] :
      ( ! [Prev: list_nat,Yss2: list_list_nat,Res2: list_list_nat] :
          ( X
         != ( produc4774092089535011407st_nat @ nil_nat @ ( produc8807741811528022664st_nat @ Prev @ ( produc7129799990162260089st_nat @ Yss2 @ Res2 ) ) ) )
     => ~ ! [X3: nat,Xs2: list_nat,Prev: list_nat,Yss2: list_list_nat,Res2: list_list_nat] :
            ( X
           != ( produc4774092089535011407st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( produc8807741811528022664st_nat @ Prev @ ( produc7129799990162260089st_nat @ Yss2 @ Res2 ) ) ) ) ) ).

% prefixes_in_list.cases
thf(fact_179_prefixes__in__list_Ocases,axiom,
    ! [X: produc7003464719162312503st_nat] :
      ( ! [Prev: list_list_nat,Yss2: list_list_list_nat,Res2: list_list_list_nat] :
          ( X
         != ( produc1064228693451757551st_nat @ nil_list_nat @ ( produc4063199782333192088st_nat @ Prev @ ( produc7456843779855495193st_nat @ Yss2 @ Res2 ) ) ) )
     => ~ ! [X3: list_nat,Xs2: list_list_nat,Prev: list_list_nat,Yss2: list_list_list_nat,Res2: list_list_list_nat] :
            ( X
           != ( produc1064228693451757551st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ ( produc4063199782333192088st_nat @ Prev @ ( produc7456843779855495193st_nat @ Yss2 @ Res2 ) ) ) ) ) ).

% prefixes_in_list.cases
thf(fact_180_prefixes__in__list_Ocases,axiom,
    ! [X: produc3892838266400288981list_a] :
      ( ! [Prev: list_list_list_a,Yss2: list_l7815035709764258326list_a,Res2: list_l7815035709764258326list_a] :
          ( X
         != ( produc2131702706070630853list_a @ nil_list_list_a @ ( produc2991501225836924590list_a @ Prev @ ( produc1620520244420031255list_a @ Yss2 @ Res2 ) ) ) )
     => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Prev: list_list_list_a,Yss2: list_l7815035709764258326list_a,Res2: list_l7815035709764258326list_a] :
            ( X
           != ( produc2131702706070630853list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ ( produc2991501225836924590list_a @ Prev @ ( produc1620520244420031255list_a @ Yss2 @ Res2 ) ) ) ) ) ).

% prefixes_in_list.cases
thf(fact_181_successively_Ocases,axiom,
    ! [X: produc1303580075620398275list_a] :
      ( ! [P4: list_a > list_a > $o] :
          ( X
         != ( produc3233922274162870707list_a @ P4 @ nil_list_a ) )
     => ( ! [P4: list_a > list_a > $o,X3: list_a] :
            ( X
           != ( produc3233922274162870707list_a @ P4 @ ( cons_list_a @ X3 @ nil_list_a ) ) )
       => ~ ! [P4: list_a > list_a > $o,X3: list_a,Y3: list_a,Xs2: list_list_a] :
              ( X
             != ( produc3233922274162870707list_a @ P4 @ ( cons_list_a @ X3 @ ( cons_list_a @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_182_successively_Ocases,axiom,
    ! [X: produc5032551385658279741list_a] :
      ( ! [P4: a > a > $o] :
          ( X
         != ( produc8111569692950616493list_a @ P4 @ nil_a ) )
     => ( ! [P4: a > a > $o,X3: a] :
            ( X
           != ( produc8111569692950616493list_a @ P4 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [P4: a > a > $o,X3: a,Y3: a,Xs2: list_a] :
              ( X
             != ( produc8111569692950616493list_a @ P4 @ ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_183_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
     => ( ! [P4: nat > nat > $o,X3: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P4: nat > nat > $o,X3: nat,Y3: nat,Xs2: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_184_successively_Ocases,axiom,
    ! [X: produc2068713617857708901st_nat] :
      ( ! [P4: list_nat > list_nat > $o] :
          ( X
         != ( produc6344517630863869919st_nat @ P4 @ nil_list_nat ) )
     => ( ! [P4: list_nat > list_nat > $o,X3: list_nat] :
            ( X
           != ( produc6344517630863869919st_nat @ P4 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) )
       => ~ ! [P4: list_nat > list_nat > $o,X3: list_nat,Y3: list_nat,Xs2: list_list_nat] :
              ( X
             != ( produc6344517630863869919st_nat @ P4 @ ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_185_successively_Ocases,axiom,
    ! [X: produc493231452093230025list_a] :
      ( ! [P4: list_list_a > list_list_a > $o] :
          ( X
         != ( produc4273680062205805369list_a @ P4 @ nil_list_list_a ) )
     => ( ! [P4: list_list_a > list_list_a > $o,X3: list_list_a] :
            ( X
           != ( produc4273680062205805369list_a @ P4 @ ( cons_list_list_a @ X3 @ nil_list_list_a ) ) )
       => ~ ! [P4: list_list_a > list_list_a > $o,X3: list_list_a,Y3: list_list_a,Xs2: list_list_list_a] :
              ( X
             != ( produc4273680062205805369list_a @ P4 @ ( cons_list_list_a @ X3 @ ( cons_list_list_a @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_186_sorted__wrt_Ocases,axiom,
    ! [X: produc1303580075620398275list_a] :
      ( ! [P4: list_a > list_a > $o] :
          ( X
         != ( produc3233922274162870707list_a @ P4 @ nil_list_a ) )
     => ~ ! [P4: list_a > list_a > $o,X3: list_a,Ys2: list_list_a] :
            ( X
           != ( produc3233922274162870707list_a @ P4 @ ( cons_list_a @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_187_sorted__wrt_Ocases,axiom,
    ! [X: produc5032551385658279741list_a] :
      ( ! [P4: a > a > $o] :
          ( X
         != ( produc8111569692950616493list_a @ P4 @ nil_a ) )
     => ~ ! [P4: a > a > $o,X3: a,Ys2: list_a] :
            ( X
           != ( produc8111569692950616493list_a @ P4 @ ( cons_a @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_188_sorted__wrt_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
     => ~ ! [P4: nat > nat > $o,X3: nat,Ys2: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_189_sorted__wrt_Ocases,axiom,
    ! [X: produc2068713617857708901st_nat] :
      ( ! [P4: list_nat > list_nat > $o] :
          ( X
         != ( produc6344517630863869919st_nat @ P4 @ nil_list_nat ) )
     => ~ ! [P4: list_nat > list_nat > $o,X3: list_nat,Ys2: list_list_nat] :
            ( X
           != ( produc6344517630863869919st_nat @ P4 @ ( cons_list_nat @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_190_sorted__wrt_Ocases,axiom,
    ! [X: produc493231452093230025list_a] :
      ( ! [P4: list_list_a > list_list_a > $o] :
          ( X
         != ( produc4273680062205805369list_a @ P4 @ nil_list_list_a ) )
     => ~ ! [P4: list_list_a > list_list_a > $o,X3: list_list_a,Ys2: list_list_list_a] :
            ( X
           != ( produc4273680062205805369list_a @ P4 @ ( cons_list_list_a @ X3 @ Ys2 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_191_shuffles_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [Ys2: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
     => ( ! [Xs2: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) )
       => ~ ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a] :
              ( X
             != ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_192_shuffles_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ( ! [Xs2: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
       => ~ ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_193_shuffles_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys2: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
     => ( ! [Xs2: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_194_shuffles_Ocases,axiom,
    ! [X: produc4326814125627636033st_nat] :
      ( ! [Ys2: list_list_nat] :
          ( X
         != ( produc7129799990162260089st_nat @ nil_list_nat @ Ys2 ) )
     => ( ! [Xs2: list_list_nat] :
            ( X
           != ( produc7129799990162260089st_nat @ Xs2 @ nil_list_nat ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
              ( X
             != ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_195_shuffles_Ocases,axiom,
    ! [X: produc3789376428941379879list_a] :
      ( ! [Ys2: list_list_list_a] :
          ( X
         != ( produc1091363791885468951list_a @ nil_list_list_a @ Ys2 ) )
     => ( ! [Xs2: list_list_list_a] :
            ( X
           != ( produc1091363791885468951list_a @ Xs2 @ nil_list_list_a ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Y3: list_list_a,Ys2: list_list_list_a] :
              ( X
             != ( produc1091363791885468951list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ ( cons_list_list_a @ Y3 @ Ys2 ) ) ) ) ) ).

% shuffles.cases
thf(fact_196_splice_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [Ys2: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
     => ~ ! [X3: list_a,Xs2: list_list_a,Ys2: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_197_splice_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
     => ~ ! [X3: a,Xs2: list_a,Ys2: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_198_splice_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys2: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
     => ~ ! [X3: nat,Xs2: list_nat,Ys2: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_199_splice_Ocases,axiom,
    ! [X: produc4326814125627636033st_nat] :
      ( ! [Ys2: list_list_nat] :
          ( X
         != ( produc7129799990162260089st_nat @ nil_list_nat @ Ys2 ) )
     => ~ ! [X3: list_nat,Xs2: list_list_nat,Ys2: list_list_nat] :
            ( X
           != ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_200_splice_Ocases,axiom,
    ! [X: produc3789376428941379879list_a] :
      ( ! [Ys2: list_list_list_a] :
          ( X
         != ( produc1091363791885468951list_a @ nil_list_list_a @ Ys2 ) )
     => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Ys2: list_list_list_a] :
            ( X
           != ( produc1091363791885468951list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).

% splice.cases
thf(fact_201_prefixes__in__list__helper_Osimps_I1_J,axiom,
    ! [X: a,Res: produc2922778490715282969list_a] :
      ( ( simple2673724192819300657lper_a @ X @ nil_list_a @ Res )
      = Res ) ).

% prefixes_in_list_helper.simps(1)
thf(fact_202_prefixes__in__list__helper_Osimps_I1_J,axiom,
    ! [X: nat,Res: produc7974263126369132087st_nat] :
      ( ( simple1541803723984754397er_nat @ X @ nil_list_nat @ Res )
      = Res ) ).

% prefixes_in_list_helper.simps(1)
thf(fact_203_prefixes__in__list__helper_Osimps_I1_J,axiom,
    ! [X: list_a,Res: produc2086622085225315487list_a] :
      ( ( simple5839407689009508407list_a @ X @ nil_list_list_a @ Res )
      = Res ) ).

% prefixes_in_list_helper.simps(1)
thf(fact_204_prefixes__in__list__helper_Opelims,axiom,
    ! [X: list_nat,Xa: list_list_list_nat,Xb: produc3585458782675957319st_nat,Y: produc3585458782675957319st_nat] :
      ( ( ( simple6281051842103134061st_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7334067637247705716st_nat @ simple1477450101163209230st_nat @ ( produc5135401712870579951st_nat @ X @ ( produc5677073168774101088st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_list_list_nat )
           => ( ( Y = Xb )
             => ~ ( accp_P7334067637247705716st_nat @ simple1477450101163209230st_nat @ ( produc5135401712870579951st_nat @ X @ ( produc5677073168774101088st_nat @ nil_list_list_nat @ Xb ) ) ) ) )
         => ( ! [Yss2: list_list_list_nat] :
                ( ( Xa
                  = ( cons_list_list_nat @ nil_list_nat @ Yss2 ) )
               => ( ( Y
                    = ( simple6281051842103134061st_nat @ X @ Yss2 @ ( produc3734841530647658561st_nat @ $true @ ( produc5745704060670869907st_nat @ Xb ) ) ) )
                 => ~ ( accp_P7334067637247705716st_nat @ simple1477450101163209230st_nat @ ( produc5135401712870579951st_nat @ X @ ( produc5677073168774101088st_nat @ ( cons_list_list_nat @ nil_list_nat @ Yss2 ) @ Xb ) ) ) ) )
           => ~ ! [Y3: list_nat,Ys2: list_list_nat,Yss2: list_list_list_nat] :
                  ( ( Xa
                    = ( cons_list_list_nat @ ( cons_list_nat @ Y3 @ Ys2 ) @ Yss2 ) )
                 => ( ( ( ( X = Y3 )
                       => ( Y
                          = ( simple6281051842103134061st_nat @ X @ Yss2 @ ( produc3734841530647658561st_nat @ ( produc5989645662155510613st_nat @ Xb ) @ ( cons_list_list_nat @ Ys2 @ ( produc5745704060670869907st_nat @ Xb ) ) ) ) ) )
                      & ( ( X != Y3 )
                       => ( Y
                          = ( simple6281051842103134061st_nat @ X @ Yss2 @ Xb ) ) ) )
                   => ~ ( accp_P7334067637247705716st_nat @ simple1477450101163209230st_nat @ ( produc5135401712870579951st_nat @ X @ ( produc5677073168774101088st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ Y3 @ Ys2 ) @ Yss2 ) @ Xb ) ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.pelims
thf(fact_205_prefixes__in__list__helper_Opelims,axiom,
    ! [X: list_list_a,Xa: list_l7815035709764258326list_a,Xb: produc2144844930413989541list_a,Y: produc2144844930413989541list_a] :
      ( ( ( simple7980934897880681149list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6221345546568043612list_a @ simple8282031061603003100list_a @ ( produc3043550568142011971list_a @ X @ ( produc3003139637434810150list_a @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_list_list_list_a )
           => ( ( Y = Xb )
             => ~ ( accp_P6221345546568043612list_a @ simple8282031061603003100list_a @ ( produc3043550568142011971list_a @ X @ ( produc3003139637434810150list_a @ nil_list_list_list_a @ Xb ) ) ) ) )
         => ( ! [Yss2: list_l7815035709764258326list_a] :
                ( ( Xa
                  = ( cons_l8577867181958743494list_a @ nil_list_list_a @ Yss2 ) )
               => ( ( Y
                    = ( simple7980934897880681149list_a @ X @ Yss2 @ ( produc3597484984460465685list_a @ $true @ ( produc8615863343057196611list_a @ Xb ) ) ) )
                 => ~ ( accp_P6221345546568043612list_a @ simple8282031061603003100list_a @ ( produc3043550568142011971list_a @ X @ ( produc3003139637434810150list_a @ ( cons_l8577867181958743494list_a @ nil_list_list_a @ Yss2 ) @ Xb ) ) ) ) )
           => ~ ! [Y3: list_list_a,Ys2: list_list_list_a,Yss2: list_l7815035709764258326list_a] :
                  ( ( Xa
                    = ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ Y3 @ Ys2 ) @ Yss2 ) )
                 => ( ( ( ( X = Y3 )
                       => ( Y
                          = ( simple7980934897880681149list_a @ X @ Yss2 @ ( produc3597484984460465685list_a @ ( produc7804899884738361857list_a @ Xb ) @ ( cons_l8577867181958743494list_a @ Ys2 @ ( produc8615863343057196611list_a @ Xb ) ) ) ) ) )
                      & ( ( X != Y3 )
                       => ( Y
                          = ( simple7980934897880681149list_a @ X @ Yss2 @ Xb ) ) ) )
                   => ~ ( accp_P6221345546568043612list_a @ simple8282031061603003100list_a @ ( produc3043550568142011971list_a @ X @ ( produc3003139637434810150list_a @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ Y3 @ Ys2 ) @ Yss2 ) @ Xb ) ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.pelims
thf(fact_206_prefixes__in__list__helper_Opelims,axiom,
    ! [X: a,Xa: list_list_a,Xb: produc2922778490715282969list_a,Y: produc2922778490715282969list_a] :
      ( ( ( simple2673724192819300657lper_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8215929052579022312list_a @ simple7477122334461730896_rel_a @ ( produc5627178164694999503list_a @ X @ ( produc6979364057698131238list_a @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_list_a )
           => ( ( Y = Xb )
             => ~ ( accp_P8215929052579022312list_a @ simple7477122334461730896_rel_a @ ( produc5627178164694999503list_a @ X @ ( produc6979364057698131238list_a @ nil_list_a @ Xb ) ) ) ) )
         => ( ! [Yss2: list_list_a] :
                ( ( Xa
                  = ( cons_list_a @ nil_a @ Yss2 ) )
               => ( ( Y
                    = ( simple2673724192819300657lper_a @ X @ Yss2 @ ( produc7347373609238049417list_a @ $true @ ( produc4969414473911996599list_a @ Xb ) ) ) )
                 => ~ ( accp_P8215929052579022312list_a @ simple7477122334461730896_rel_a @ ( produc5627178164694999503list_a @ X @ ( produc6979364057698131238list_a @ ( cons_list_a @ nil_a @ Yss2 ) @ Xb ) ) ) ) )
           => ~ ! [Y3: a,Ys2: list_a,Yss2: list_list_a] :
                  ( ( Xa
                    = ( cons_list_a @ ( cons_a @ Y3 @ Ys2 ) @ Yss2 ) )
                 => ( ( ( ( X = Y3 )
                       => ( Y
                          = ( simple2673724192819300657lper_a @ X @ Yss2 @ ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ Xb ) @ ( cons_list_a @ Ys2 @ ( produc4969414473911996599list_a @ Xb ) ) ) ) ) )
                      & ( ( X != Y3 )
                       => ( Y
                          = ( simple2673724192819300657lper_a @ X @ Yss2 @ Xb ) ) ) )
                   => ~ ( accp_P8215929052579022312list_a @ simple7477122334461730896_rel_a @ ( produc5627178164694999503list_a @ X @ ( produc6979364057698131238list_a @ ( cons_list_a @ ( cons_a @ Y3 @ Ys2 ) @ Yss2 ) @ Xb ) ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.pelims
thf(fact_207_prefixes__in__list__helper_Opelims,axiom,
    ! [X: nat,Xa: list_list_nat,Xb: produc7974263126369132087st_nat,Y: produc7974263126369132087st_nat] :
      ( ( ( simple1541803723984754397er_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P4148377174436316900st_nat @ simple2190780988582970878el_nat @ ( produc3115374921362880095st_nat @ X @ ( produc3419769971367040704st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_list_nat )
           => ( ( Y = Xb )
             => ~ ( accp_P4148377174436316900st_nat @ simple2190780988582970878el_nat @ ( produc3115374921362880095st_nat @ X @ ( produc3419769971367040704st_nat @ nil_list_nat @ Xb ) ) ) ) )
         => ( ! [Yss2: list_list_nat] :
                ( ( Xa
                  = ( cons_list_nat @ nil_nat @ Yss2 ) )
               => ( ( Y
                    = ( simple1541803723984754397er_nat @ X @ Yss2 @ ( produc8892817347036706481st_nat @ $true @ ( produc4024048262274189571st_nat @ Xb ) ) ) )
                 => ~ ( accp_P4148377174436316900st_nat @ simple2190780988582970878el_nat @ ( produc3115374921362880095st_nat @ X @ ( produc3419769971367040704st_nat @ ( cons_list_nat @ nil_nat @ Yss2 ) @ Xb ) ) ) ) )
           => ~ ! [Y3: nat,Ys2: list_nat,Yss2: list_list_nat] :
                  ( ( Xa
                    = ( cons_list_nat @ ( cons_nat @ Y3 @ Ys2 ) @ Yss2 ) )
                 => ( ( ( ( X = Y3 )
                       => ( Y
                          = ( simple1541803723984754397er_nat @ X @ Yss2 @ ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ Xb ) @ ( cons_list_nat @ Ys2 @ ( produc4024048262274189571st_nat @ Xb ) ) ) ) ) )
                      & ( ( X != Y3 )
                       => ( Y
                          = ( simple1541803723984754397er_nat @ X @ Yss2 @ Xb ) ) ) )
                   => ~ ( accp_P4148377174436316900st_nat @ simple2190780988582970878el_nat @ ( produc3115374921362880095st_nat @ X @ ( produc3419769971367040704st_nat @ ( cons_list_nat @ ( cons_nat @ Y3 @ Ys2 ) @ Yss2 ) @ Xb ) ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.pelims
thf(fact_208_prefixes__in__list__helper_Opelims,axiom,
    ! [X: list_a,Xa: list_list_list_a,Xb: produc2086622085225315487list_a,Y: produc2086622085225315487list_a] :
      ( ( ( simple5839407689009508407list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6831594742703354338list_a @ simple2326924577853762518list_a @ ( produc5745923727215425737list_a @ X @ ( produc7568776524366471974list_a @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_list_list_a )
           => ( ( Y = Xb )
             => ~ ( accp_P6831594742703354338list_a @ simple2326924577853762518list_a @ ( produc5745923727215425737list_a @ X @ ( produc7568776524366471974list_a @ nil_list_list_a @ Xb ) ) ) ) )
         => ( ! [Yss2: list_list_list_a] :
                ( ( Xa
                  = ( cons_list_list_a @ nil_list_a @ Yss2 ) )
               => ( ( Y
                    = ( simple5839407689009508407list_a @ X @ Yss2 @ ( produc8548274197880810127list_a @ $true @ ( produc1153002459082627517list_a @ Xb ) ) ) )
                 => ~ ( accp_P6831594742703354338list_a @ simple2326924577853762518list_a @ ( produc5745923727215425737list_a @ X @ ( produc7568776524366471974list_a @ ( cons_list_list_a @ nil_list_a @ Yss2 ) @ Xb ) ) ) ) )
           => ~ ! [Y3: list_a,Ys2: list_list_a,Yss2: list_list_list_a] :
                  ( ( Xa
                    = ( cons_list_list_a @ ( cons_list_a @ Y3 @ Ys2 ) @ Yss2 ) )
                 => ( ( ( ( X = Y3 )
                       => ( Y
                          = ( simple5839407689009508407list_a @ X @ Yss2 @ ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ Xb ) @ ( cons_list_list_a @ Ys2 @ ( produc1153002459082627517list_a @ Xb ) ) ) ) ) )
                      & ( ( X != Y3 )
                       => ( Y
                          = ( simple5839407689009508407list_a @ X @ Yss2 @ Xb ) ) ) )
                   => ~ ( accp_P6831594742703354338list_a @ simple2326924577853762518list_a @ ( produc5745923727215425737list_a @ X @ ( produc7568776524366471974list_a @ ( cons_list_list_a @ ( cons_list_a @ Y3 @ Ys2 ) @ Yss2 ) @ Xb ) ) ) ) ) ) ) ) ) ).

% prefixes_in_list_helper.pelims
thf(fact_209_linear__order__from__list__position_H_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ~ ! [X3: list_a,Xs2: list_list_a] :
            ( X
           != ( cons_list_a @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_210_linear__order__from__list__position_H_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ~ ! [X3: a,Xs2: list_a] :
            ( X
           != ( cons_a @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_211_linear__order__from__list__position_H_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( X
           != ( cons_nat @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_212_linear__order__from__list__position_H_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ~ ! [X3: list_nat,Xs2: list_list_nat] :
            ( X
           != ( cons_list_nat @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_213_linear__order__from__list__position_H_Ocases,axiom,
    ! [X: list_list_list_a] :
      ( ( X != nil_list_list_a )
     => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
            ( X
           != ( cons_list_list_a @ X3 @ Xs2 ) ) ) ).

% linear_order_from_list_position'.cases
thf(fact_214_longest__common__prefix_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ nil_list_a @ Uv ) )
       => ~ ! [Uu: list_list_a] :
              ( X
             != ( produc8696003437204565271list_a @ Uu @ nil_list_a ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_215_longest__common__prefix_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ nil_a @ Uv ) )
       => ~ ! [Uu: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ Uu @ nil_a ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_216_longest__common__prefix_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ nil_nat @ Uv ) )
       => ~ ! [Uu: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ Uu @ nil_nat ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_217_longest__common__prefix_Ocases,axiom,
    ! [X: produc4326814125627636033st_nat] :
      ( ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
          ( X
         != ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_list_nat] :
            ( X
           != ( produc7129799990162260089st_nat @ nil_list_nat @ Uv ) )
       => ~ ! [Uu: list_list_nat] :
              ( X
             != ( produc7129799990162260089st_nat @ Uu @ nil_list_nat ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_218_longest__common__prefix_Ocases,axiom,
    ! [X: produc3789376428941379879list_a] :
      ( ! [X3: list_list_a,Xs2: list_list_list_a,Y3: list_list_a,Ys2: list_list_list_a] :
          ( X
         != ( produc1091363791885468951list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ ( cons_list_list_a @ Y3 @ Ys2 ) ) )
     => ( ! [Uv: list_list_list_a] :
            ( X
           != ( produc1091363791885468951list_a @ nil_list_list_a @ Uv ) )
       => ~ ! [Uu: list_list_list_a] :
              ( X
             != ( produc1091363791885468951list_a @ Uu @ nil_list_list_a ) ) ) ) ).

% longest_common_prefix.cases
thf(fact_219_is__prefix_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [Uu: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ nil_list_a @ Uu ) )
     => ( ! [X3: list_a,Xs2: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ nil_list_a ) )
       => ~ ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a] :
              ( X
             != ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_220_is__prefix_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Uu: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Uu ) )
     => ( ! [X3: a,Xs2: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_a ) )
       => ~ ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_221_is__prefix_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Uu: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Uu ) )
     => ( ! [X3: nat,Xs2: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_222_is__prefix_Ocases,axiom,
    ! [X: produc4326814125627636033st_nat] :
      ( ! [Uu: list_list_nat] :
          ( X
         != ( produc7129799990162260089st_nat @ nil_list_nat @ Uu ) )
     => ( ! [X3: list_nat,Xs2: list_list_nat] :
            ( X
           != ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ nil_list_nat ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
              ( X
             != ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_223_is__prefix_Ocases,axiom,
    ! [X: produc3789376428941379879list_a] :
      ( ! [Uu: list_list_list_a] :
          ( X
         != ( produc1091363791885468951list_a @ nil_list_list_a @ Uu ) )
     => ( ! [X3: list_list_a,Xs2: list_list_list_a] :
            ( X
           != ( produc1091363791885468951list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ nil_list_list_a ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Y3: list_list_a,Ys2: list_list_list_a] :
              ( X
             != ( produc1091363791885468951list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ ( cons_list_list_a @ Y3 @ Ys2 ) ) ) ) ) ).

% is_prefix.cases
thf(fact_224_find__index_Ocases,axiom,
    ! [X: produc7690851789404205254list_a] :
      ( ! [F: list_a > $o] :
          ( X
         != ( produc103144377828961528list_a @ F @ nil_list_a ) )
     => ~ ! [F: list_a > $o,X3: list_a,Xs2: list_list_a] :
            ( X
           != ( produc103144377828961528list_a @ F @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_225_find__index_Ocases,axiom,
    ! [X: produc7567695631791103674list_a] :
      ( ! [F: a > $o] :
          ( X
         != ( produc5033538197335023340list_a @ F @ nil_a ) )
     => ~ ! [F: a > $o,X3: a,Xs2: list_a] :
            ( X
           != ( produc5033538197335023340list_a @ F @ ( cons_a @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_226_find__index_Ocases,axiom,
    ! [X: produc4226810134323546766st_nat] :
      ( ! [F: nat > $o] :
          ( X
         != ( produc8587622027977423880st_nat @ F @ nil_nat ) )
     => ~ ! [F: nat > $o,X3: nat,Xs2: list_nat] :
            ( X
           != ( produc8587622027977423880st_nat @ F @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_227_find__index_Ocases,axiom,
    ! [X: produc4248194078104742542st_nat] :
      ( ! [F: list_nat > $o] :
          ( X
         != ( produc8316565932239993096st_nat @ F @ nil_list_nat ) )
     => ~ ! [F: list_nat > $o,X3: list_nat,Xs2: list_list_nat] :
            ( X
           != ( produc8316565932239993096st_nat @ F @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_228_find__index_Ocases,axiom,
    ! [X: produc1111118985262953426list_a] :
      ( ! [F: list_list_a > $o] :
          ( X
         != ( produc791465279206632452list_a @ F @ nil_list_list_a ) )
     => ~ ! [F: list_list_a > $o,X3: list_list_a,Xs2: list_list_list_a] :
            ( X
           != ( produc791465279206632452list_a @ F @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) ) ).

% find_index.cases
thf(fact_229_find__remove_H_Ocases,axiom,
    ! [X: produc6925218617054031843list_a] :
      ( ! [P4: list_a > $o,Uu: list_list_a] :
          ( X
         != ( produc3589492913885683861list_a @ P4 @ ( produc8696003437204565271list_a @ nil_list_a @ Uu ) ) )
     => ~ ! [P4: list_a > $o,X3: list_a,Xs2: list_list_a,Prev: list_list_a] :
            ( X
           != ( produc3589492913885683861list_a @ P4 @ ( produc8696003437204565271list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_230_find__remove_H_Ocases,axiom,
    ! [X: produc3286415118216283229list_a] :
      ( ! [P4: a > $o,Uu: list_a] :
          ( X
         != ( produc8731264218526379663list_a @ P4 @ ( produc6837034575241423639list_a @ nil_a @ Uu ) ) )
     => ~ ! [P4: a > $o,X3: a,Xs2: list_a,Prev: list_a] :
            ( X
           != ( produc8731264218526379663list_a @ P4 @ ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_231_find__remove_H_Ocases,axiom,
    ! [X: produc1950156742809263241st_nat] :
      ( ! [P4: nat > $o,Uu: list_nat] :
          ( X
         != ( produc1851982983375519865st_nat @ P4 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
     => ~ ! [P4: nat > $o,X3: nat,Xs2: list_nat,Prev: list_nat] :
            ( X
           != ( produc1851982983375519865st_nat @ P4 @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_232_find__remove_H_Ocases,axiom,
    ! [X: produc91608529370230841st_nat] :
      ( ! [P4: list_nat > $o,Uu: list_list_nat] :
          ( X
         != ( produc328518199612081577st_nat @ P4 @ ( produc7129799990162260089st_nat @ nil_list_nat @ Uu ) ) )
     => ~ ! [P4: list_nat > $o,X3: list_nat,Xs2: list_list_nat,Prev: list_list_nat] :
            ( X
           != ( produc328518199612081577st_nat @ P4 @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_233_find__remove_H_Ocases,axiom,
    ! [X: produc7425903735044159721list_a] :
      ( ! [P4: list_list_a > $o,Uu: list_list_list_a] :
          ( X
         != ( produc7480101680193318939list_a @ P4 @ ( produc1091363791885468951list_a @ nil_list_list_a @ Uu ) ) )
     => ~ ! [P4: list_list_a > $o,X3: list_list_a,Xs2: list_list_list_a,Prev: list_list_list_a] :
            ( X
           != ( produc7480101680193318939list_a @ P4 @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Prev ) ) ) ) ).

% find_remove'.cases
thf(fact_234_delete__aux_Ocases,axiom,
    ! [X: produc2919922447449601198list_a] :
      ( ! [K: $o] :
          ( X
         != ( produc1700513956625826974list_a @ K @ nil_Pr1488683542974317855list_a ) )
     => ~ ! [K: $o,K2: $o,V: list_list_a,Xs2: list_P369347992569427103list_a] :
            ( X
           != ( produc1700513956625826974list_a @ K @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_235_delete__aux_Ocases,axiom,
    ! [X: produc5234193878020974990st_nat] :
      ( ! [K: $o] :
          ( X
         != ( produc3028385585695882376st_nat @ K @ nil_Pr1938078369830740401st_nat ) )
     => ~ ! [K: $o,K2: $o,V: list_list_nat,Xs2: list_P3026376796534914631st_nat] :
            ( X
           != ( produc3028385585695882376st_nat @ K @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_236_delete__aux_Ocases,axiom,
    ! [X: produc756509542915563956list_a] :
      ( ! [K: $o] :
          ( X
         != ( produc4612989676349016868list_a @ K @ nil_Pr7514062926313711781list_a ) )
     => ~ ! [K: $o,K2: $o,V: list_list_list_a,Xs2: list_P1915399164375025317list_a] :
            ( X
           != ( produc4612989676349016868list_a @ K @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_237_sublists_Osimps_I1_J,axiom,
    ( ( sublists_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% sublists.simps(1)
thf(fact_238_sublists_Osimps_I1_J,axiom,
    ( ( sublists_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% sublists.simps(1)
thf(fact_239_sublists_Osimps_I1_J,axiom,
    ( ( sublists_list_a @ nil_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% sublists.simps(1)
thf(fact_240_Util_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffixes_a2 @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% Util.suffixes.simps(1)
thf(fact_241_Util_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffixes_nat2 @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% Util.suffixes.simps(1)
thf(fact_242_Util_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffixes_list_a2 @ nil_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% Util.suffixes.simps(1)
thf(fact_243_pow__list_Osimps_I1_J,axiom,
    ( ( pow_list_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% pow_list.simps(1)
thf(fact_244_pow__list_Osimps_I1_J,axiom,
    ( ( pow_list_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% pow_list.simps(1)
thf(fact_245_pow__list_Osimps_I1_J,axiom,
    ( ( pow_list_list_a @ nil_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% pow_list.simps(1)
thf(fact_246_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_a @ nil_list_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% product_lists.simps(1)
thf(fact_247_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_248_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_list_a @ nil_list_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% product_lists.simps(1)
thf(fact_249_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% subseqs.simps(1)
thf(fact_250_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_251_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_list_a @ nil_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% subseqs.simps(1)
thf(fact_252_insert__Nil,axiom,
    ! [X: list_a] :
      ( ( insert_list_a @ X @ nil_list_a )
      = ( cons_list_a @ X @ nil_list_a ) ) ).

% insert_Nil
thf(fact_253_insert__Nil,axiom,
    ! [X: a] :
      ( ( insert_a @ X @ nil_a )
      = ( cons_a @ X @ nil_a ) ) ).

% insert_Nil
thf(fact_254_insert__Nil,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ nil_nat )
      = ( cons_nat @ X @ nil_nat ) ) ).

% insert_Nil
thf(fact_255_insert__Nil,axiom,
    ! [X: list_nat] :
      ( ( insert_list_nat @ X @ nil_list_nat )
      = ( cons_list_nat @ X @ nil_list_nat ) ) ).

% insert_Nil
thf(fact_256_insert__Nil,axiom,
    ! [X: list_list_a] :
      ( ( insert_list_list_a @ X @ nil_list_list_a )
      = ( cons_list_list_a @ X @ nil_list_list_a ) ) ).

% insert_Nil
thf(fact_257_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C )
      = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_258_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_259_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_260_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_261_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_262_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_263_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_264_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_265_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_266_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_267_self__append__conv,axiom,
    ! [Y: list_a,Ys: list_a] :
      ( ( Y
        = ( append_a @ Y @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_268_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_269_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_270_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_271_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y
        = ( append_a @ Xs @ Y ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_272_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_273_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_274_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_275_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_276_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_277_append1__eq__conv,axiom,
    ! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a] :
      ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
        = ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_278_append1__eq__conv,axiom,
    ! [Xs: list_a,X: a,Ys: list_a,Y: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
        = ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_279_append1__eq__conv,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_280_append1__eq__conv,axiom,
    ! [Xs: list_list_nat,X: list_nat,Ys: list_list_nat,Y: list_nat] :
      ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) )
        = ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_281_append1__eq__conv,axiom,
    ! [Xs: list_list_list_a,X: list_list_a,Ys: list_list_list_a,Y: list_list_a] :
      ( ( ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) )
        = ( append_list_list_a @ Ys @ ( cons_list_list_a @ Y @ nil_list_list_a ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_282_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_283_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_284_append__Cons,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
      ( ( append_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
      = ( cons_list_a @ X @ ( append_list_a @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_285_append__Cons,axiom,
    ! [X: a,Xs: list_a,Ys: list_a] :
      ( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
      = ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_286_append__Cons,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
      = ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_287_append__Cons,axiom,
    ! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( append_list_nat @ ( cons_list_nat @ X @ Xs ) @ Ys )
      = ( cons_list_nat @ X @ ( append_list_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_288_append__Cons,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a] :
      ( ( append_list_list_a @ ( cons_list_list_a @ X @ Xs ) @ Ys )
      = ( cons_list_list_a @ X @ ( append_list_list_a @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_289_Cons__eq__appendI,axiom,
    ! [X: list_a,Xs1: list_list_a,Ys: list_list_a,Xs: list_list_a,Zs: list_list_a] :
      ( ( ( cons_list_a @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_list_a @ Xs1 @ Zs ) )
       => ( ( cons_list_a @ X @ Xs )
          = ( append_list_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_290_Cons__eq__appendI,axiom,
    ! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( cons_a @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_a @ Xs1 @ Zs ) )
       => ( ( cons_a @ X @ Xs )
          = ( append_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_291_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X @ Xs )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_292_Cons__eq__appendI,axiom,
    ! [X: list_nat,Xs1: list_list_nat,Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
      ( ( ( cons_list_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_list_nat @ Xs1 @ Zs ) )
       => ( ( cons_list_nat @ X @ Xs )
          = ( append_list_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_293_Cons__eq__appendI,axiom,
    ! [X: list_list_a,Xs1: list_list_list_a,Ys: list_list_list_a,Xs: list_list_list_a,Zs: list_list_list_a] :
      ( ( ( cons_list_list_a @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_list_list_a @ Xs1 @ Zs ) )
       => ( ( cons_list_list_a @ X @ Xs )
          = ( append_list_list_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_294_append__Nil,axiom,
    ! [Ys: list_a] :
      ( ( append_a @ nil_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_295_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_296_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_297_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_298_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_a @ nil_a @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_299_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_300_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( suffixes_list_a2 @ ( cons_list_a @ X @ Xs ) )
      = ( append_list_list_a @ ( suffixes_list_a2 @ Xs ) @ ( cons_list_list_a @ ( cons_list_a @ X @ Xs ) @ nil_list_list_a ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_301_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( suffixes_a2 @ ( cons_a @ X @ Xs ) )
      = ( append_list_a @ ( suffixes_a2 @ Xs ) @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ nil_list_a ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_302_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( suffixes_nat2 @ ( cons_nat @ X @ Xs ) )
      = ( append_list_nat @ ( suffixes_nat2 @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_303_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( suffixes_list_nat2 @ ( cons_list_nat @ X @ Xs ) )
      = ( append_list_list_nat @ ( suffixes_list_nat2 @ Xs ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ nil_list_list_nat ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_304_Util_Osuffixes_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( suffixes_list_list_a2 @ ( cons_list_list_a @ X @ Xs ) )
      = ( append6004555433696164139list_a @ ( suffixes_list_list_a2 @ Xs ) @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X @ Xs ) @ nil_list_list_list_a ) ) ) ).

% Util.suffixes.simps(2)
thf(fact_305_rev__nonempty__induct,axiom,
    ! [Xs: list_list_a,P: list_list_a > $o] :
      ( ( Xs != nil_list_a )
     => ( ! [X3: list_a] : ( P @ ( cons_list_a @ X3 @ nil_list_a ) )
       => ( ! [X3: list_a,Xs2: list_list_a] :
              ( ( Xs2 != nil_list_a )
             => ( ( P @ Xs2 )
               => ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X3 @ nil_list_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_306_rev__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_307_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_308_rev__nonempty__induct,axiom,
    ! [Xs: list_list_nat,P: list_list_nat > $o] :
      ( ( Xs != nil_list_nat )
     => ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
       => ( ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( Xs2 != nil_list_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_309_rev__nonempty__induct,axiom,
    ! [Xs: list_list_list_a,P: list_list_list_a > $o] :
      ( ( Xs != nil_list_list_a )
     => ( ! [X3: list_list_a] : ( P @ ( cons_list_list_a @ X3 @ nil_list_list_a ) )
       => ( ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( Xs2 != nil_list_list_a )
             => ( ( P @ Xs2 )
               => ( P @ ( append_list_list_a @ Xs2 @ ( cons_list_list_a @ X3 @ nil_list_list_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_310_append__eq__Cons__conv,axiom,
    ! [Ys: list_list_a,Zs: list_list_a,X: list_a,Xs: list_list_a] :
      ( ( ( append_list_a @ Ys @ Zs )
        = ( cons_list_a @ X @ Xs ) )
      = ( ( ( Ys = nil_list_a )
          & ( Zs
            = ( cons_list_a @ X @ Xs ) ) )
        | ? [Ys4: list_list_a] :
            ( ( Ys
              = ( cons_list_a @ X @ Ys4 ) )
            & ( ( append_list_a @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_311_append__eq__Cons__conv,axiom,
    ! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs )
        = ( cons_a @ X @ Xs ) )
      = ( ( ( Ys = nil_a )
          & ( Zs
            = ( cons_a @ X @ Xs ) ) )
        | ? [Ys4: list_a] :
            ( ( Ys
              = ( cons_a @ X @ Ys4 ) )
            & ( ( append_a @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_312_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs
            = ( cons_nat @ X @ Xs ) ) )
        | ? [Ys4: list_nat] :
            ( ( Ys
              = ( cons_nat @ X @ Ys4 ) )
            & ( ( append_nat @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_313_append__eq__Cons__conv,axiom,
    ! [Ys: list_list_nat,Zs: list_list_nat,X: list_nat,Xs: list_list_nat] :
      ( ( ( append_list_nat @ Ys @ Zs )
        = ( cons_list_nat @ X @ Xs ) )
      = ( ( ( Ys = nil_list_nat )
          & ( Zs
            = ( cons_list_nat @ X @ Xs ) ) )
        | ? [Ys4: list_list_nat] :
            ( ( Ys
              = ( cons_list_nat @ X @ Ys4 ) )
            & ( ( append_list_nat @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_314_append__eq__Cons__conv,axiom,
    ! [Ys: list_list_list_a,Zs: list_list_list_a,X: list_list_a,Xs: list_list_list_a] :
      ( ( ( append_list_list_a @ Ys @ Zs )
        = ( cons_list_list_a @ X @ Xs ) )
      = ( ( ( Ys = nil_list_list_a )
          & ( Zs
            = ( cons_list_list_a @ X @ Xs ) ) )
        | ? [Ys4: list_list_list_a] :
            ( ( Ys
              = ( cons_list_list_a @ X @ Ys4 ) )
            & ( ( append_list_list_a @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_315_Cons__eq__append__conv,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
      ( ( ( cons_list_a @ X @ Xs )
        = ( append_list_a @ Ys @ Zs ) )
      = ( ( ( Ys = nil_list_a )
          & ( ( cons_list_a @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_list_a] :
            ( ( ( cons_list_a @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_list_a @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_316_Cons__eq__append__conv,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ( ( Ys = nil_a )
          & ( ( cons_a @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_a] :
            ( ( ( cons_a @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_a @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_317_Cons__eq__append__conv,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_nat] :
            ( ( ( cons_nat @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_318_Cons__eq__append__conv,axiom,
    ! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
      ( ( ( cons_list_nat @ X @ Xs )
        = ( append_list_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_list_nat )
          & ( ( cons_list_nat @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_list_nat] :
            ( ( ( cons_list_nat @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_list_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_319_Cons__eq__append__conv,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,Zs: list_list_list_a] :
      ( ( ( cons_list_list_a @ X @ Xs )
        = ( append_list_list_a @ Ys @ Zs ) )
      = ( ( ( Ys = nil_list_list_a )
          & ( ( cons_list_list_a @ X @ Xs )
            = Zs ) )
        | ? [Ys4: list_list_list_a] :
            ( ( ( cons_list_list_a @ X @ Ys4 )
              = Ys )
            & ( Xs
              = ( append_list_list_a @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_320_rev__exhaust,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
     => ~ ! [Ys2: list_list_a,Y3: list_a] :
            ( Xs
           != ( append_list_a @ Ys2 @ ( cons_list_a @ Y3 @ nil_list_a ) ) ) ) ).

% rev_exhaust
thf(fact_321_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys2: list_a,Y3: a] :
            ( Xs
           != ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).

% rev_exhaust
thf(fact_322_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys2: list_nat,Y3: nat] :
            ( Xs
           != ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_323_rev__exhaust,axiom,
    ! [Xs: list_list_nat] :
      ( ( Xs != nil_list_nat )
     => ~ ! [Ys2: list_list_nat,Y3: list_nat] :
            ( Xs
           != ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) ).

% rev_exhaust
thf(fact_324_rev__exhaust,axiom,
    ! [Xs: list_list_list_a] :
      ( ( Xs != nil_list_list_a )
     => ~ ! [Ys2: list_list_list_a,Y3: list_list_a] :
            ( Xs
           != ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ Y3 @ nil_list_list_a ) ) ) ) ).

% rev_exhaust
thf(fact_325_rev__induct,axiom,
    ! [P: list_list_a > $o,Xs: list_list_a] :
      ( ( P @ nil_list_a )
     => ( ! [X3: list_a,Xs2: list_list_a] :
            ( ( P @ Xs2 )
           => ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X3 @ nil_list_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_326_rev__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ( P @ nil_a )
     => ( ! [X3: a,Xs2: list_a] :
            ( ( P @ Xs2 )
           => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_327_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X3: nat,Xs2: list_nat] :
            ( ( P @ Xs2 )
           => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_328_rev__induct,axiom,
    ! [P: list_list_nat > $o,Xs: list_list_nat] :
      ( ( P @ nil_list_nat )
     => ( ! [X3: list_nat,Xs2: list_list_nat] :
            ( ( P @ Xs2 )
           => ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_329_rev__induct,axiom,
    ! [P: list_list_list_a > $o,Xs: list_list_list_a] :
      ( ( P @ nil_list_list_a )
     => ( ! [X3: list_list_a,Xs2: list_list_list_a] :
            ( ( P @ Xs2 )
           => ( P @ ( append_list_list_a @ Xs2 @ ( cons_list_list_a @ X3 @ nil_list_list_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_330_listrel_ONil,axiom,
    ! [R: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R ) ) ).

% listrel.Nil
thf(fact_331_listrel_ONil,axiom,
    ! [R: set_Pr4934435412358123699_a_nat] : ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ nil_a @ nil_nat ) @ ( listrel_a_nat @ R ) ) ).

% listrel.Nil
thf(fact_332_listrel_ONil,axiom,
    ! [R: set_Pr4193341848836149977_nat_a] : ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ nil_nat @ nil_a ) @ ( listrel_nat_a @ R ) ) ).

% listrel.Nil
thf(fact_333_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_334_listrel__Nil1,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R ) )
     => ( Xs = nil_a ) ) ).

% listrel_Nil1
thf(fact_335_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ nil_a @ Xs ) @ ( listrel_a_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_336_listrel__Nil1,axiom,
    ! [Xs: list_a,R: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ nil_nat @ Xs ) @ ( listrel_nat_a @ R ) )
     => ( Xs = nil_a ) ) ).

% listrel_Nil1
thf(fact_337_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_338_listrel__Nil2,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R ) )
     => ( Xs = nil_a ) ) ).

% listrel_Nil2
thf(fact_339_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ nil_a ) @ ( listrel_nat_a @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_340_listrel__Nil2,axiom,
    ! [Xs: list_a,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ nil_nat ) @ ( listrel_a_nat @ R ) )
     => ( Xs = nil_a ) ) ).

% listrel_Nil2
thf(fact_341_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_342_suffixes_Oelims,axiom,
    ! [X: list_list_nat,Y: list_list_list_nat] :
      ( ( ( suffixes_list_nat2 @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y
           != ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_list_nat @ ( suffixes_list_nat2 @ Xs2 ) @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ nil_list_list_nat ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_343_suffixes_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_l7815035709764258326list_a] :
      ( ( ( suffixes_list_list_a2 @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y
           != ( cons_l8577867181958743494list_a @ nil_list_list_a @ nil_list_list_list_a ) ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append6004555433696164139list_a @ ( suffixes_list_list_a2 @ Xs2 ) @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ nil_list_list_list_a ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_344_suffixes_Oelims,axiom,
    ! [X: list_a,Y: list_list_a] :
      ( ( ( suffixes_a2 @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y
           != ( cons_list_a @ nil_a @ nil_list_a ) ) )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( X
                = ( cons_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_a @ ( suffixes_a2 @ Xs2 ) @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_list_a ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_345_suffixes_Oelims,axiom,
    ! [X: list_nat,Y: list_list_nat] :
      ( ( ( suffixes_nat2 @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y
           != ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_nat @ ( suffixes_nat2 @ Xs2 ) @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ nil_list_nat ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_346_suffixes_Oelims,axiom,
    ! [X: list_list_a,Y: list_list_list_a] :
      ( ( ( suffixes_list_a2 @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y
           != ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) )
       => ~ ! [X3: list_a,Xs2: list_list_a] :
              ( ( X
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_list_a @ ( suffixes_list_a2 @ Xs2 ) @ ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ nil_list_list_a ) ) ) ) ) ) ).

% suffixes.elims
thf(fact_347_listrel_OCons,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
     => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R ) ) ) ) ).

% listrel.Cons
thf(fact_348_listrel_OCons,axiom,
    ! [X: a,Y: nat,R: set_Pr4934435412358123699_a_nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R )
     => ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R ) )
       => ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_349_listrel_OCons,axiom,
    ! [X: nat,Y: a,R: set_Pr4193341848836149977_nat_a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y ) @ R )
     => ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ Ys ) @ ( listrel_nat_a @ R ) )
       => ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ ( cons_nat @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_nat_a @ R ) ) ) ) ).

% listrel.Cons
thf(fact_350_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_351_listrel_OCons,axiom,
    ! [X: list_a,Y: a,R: set_Pr8962057229576493569st_a_a,Xs: list_list_a,Ys: list_a] :
      ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X @ Y ) @ R )
     => ( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs @ Ys ) @ ( listrel_list_a_a @ R ) )
       => ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_list_a_a @ R ) ) ) ) ).

% listrel.Cons
thf(fact_352_listrel_OCons,axiom,
    ! [X: list_a,Y: nat,R: set_Pr6248017591093387577_a_nat,Xs: list_list_a,Ys: list_nat] :
      ( ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ X @ Y ) @ R )
     => ( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs @ Ys ) @ ( listrel_list_a_nat @ R ) )
       => ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ ( cons_list_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_list_a_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_353_listrel_OCons,axiom,
    ! [X: a,Y: list_a,R: set_Pr2070066670564046349list_a,Xs: list_a,Ys: list_list_a] :
      ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X @ Y ) @ R )
     => ( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs @ Ys ) @ ( listrel_a_list_a @ R ) )
       => ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ ( cons_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel_a_list_a @ R ) ) ) ) ).

% listrel.Cons
thf(fact_354_listrel_OCons,axiom,
    ! [X: a,Y: list_nat,R: set_Pr6062262858518114627st_nat,Xs: list_a,Ys: list_list_nat] :
      ( ( member9064520187905290916st_nat @ ( produc6928161557333929735st_nat @ X @ Y ) @ R )
     => ( ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ Xs @ Ys ) @ ( listrel_a_list_nat @ R ) )
       => ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ ( cons_a @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( listrel_a_list_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_355_listrel_OCons,axiom,
    ! [X: nat,Y: list_a,R: set_Pr3870785875473343199list_a,Xs: list_nat,Ys: list_list_a] :
      ( ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ X @ Y ) @ R )
     => ( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs @ Ys ) @ ( listrel_nat_list_a @ R ) )
       => ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ ( cons_nat @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel_nat_list_a @ R ) ) ) ) ).

% listrel.Cons
thf(fact_356_listrel_OCons,axiom,
    ! [X: nat,Y: list_nat,R: set_Pr698032948827214833st_nat,Xs: list_nat,Ys: list_list_nat] :
      ( ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X @ Y ) @ R )
     => ( ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ Xs @ Ys ) @ ( listrel_nat_list_nat @ R ) )
       => ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( listrel_nat_list_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_357_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_a @ R ) )
     => ~ ! [Y3: a,Ys2: list_a] :
            ( ( Xs
              = ( cons_a @ Y3 @ Ys2 ) )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y3 ) @ R )
             => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_358_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_nat @ R ) )
     => ~ ! [Y3: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys2 ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Y3 ) @ R )
             => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Ys @ Ys2 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_359_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_a,R: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_a @ R ) )
     => ~ ! [Y3: a,Ys2: list_a] :
            ( ( Xs
              = ( cons_a @ Y3 @ Ys2 ) )
           => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ Y @ Y3 ) @ R )
             => ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Ys @ Ys2 ) @ ( listrel_nat_a @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_360_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [Y3: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_361_listrel__Cons1,axiom,
    ! [Y: list_a,Ys: list_list_a,Xs: list_a,R: set_Pr8962057229576493569st_a_a] :
      ( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ ( cons_list_a @ Y @ Ys ) @ Xs ) @ ( listrel_list_a_a @ R ) )
     => ~ ! [Y3: a,Ys2: list_a] :
            ( ( Xs
              = ( cons_a @ Y3 @ Ys2 ) )
           => ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ Y @ Y3 ) @ R )
             => ~ ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Ys @ Ys2 ) @ ( listrel_list_a_a @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_362_listrel__Cons1,axiom,
    ! [Y: list_a,Ys: list_list_a,Xs: list_nat,R: set_Pr6248017591093387577_a_nat] :
      ( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ ( cons_list_a @ Y @ Ys ) @ Xs ) @ ( listrel_list_a_nat @ R ) )
     => ~ ! [Y3: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys2 ) )
           => ( ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ Y @ Y3 ) @ R )
             => ~ ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Ys @ Ys2 ) @ ( listrel_list_a_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_363_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs: list_list_a,R: set_Pr2070066670564046349list_a] :
      ( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_list_a @ R ) )
     => ~ ! [Y3: list_a,Ys2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Y3 @ Ys2 ) )
           => ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ Y @ Y3 ) @ R )
             => ~ ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Ys @ Ys2 ) @ ( listrel_a_list_a @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_364_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs: list_list_nat,R: set_Pr6062262858518114627st_nat] :
      ( ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_list_nat @ R ) )
     => ~ ! [Y3: list_nat,Ys2: list_list_nat] :
            ( ( Xs
              = ( cons_list_nat @ Y3 @ Ys2 ) )
           => ( ( member9064520187905290916st_nat @ ( produc6928161557333929735st_nat @ Y @ Y3 ) @ R )
             => ~ ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ Ys @ Ys2 ) @ ( listrel_a_list_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_365_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_list_a,R: set_Pr3870785875473343199list_a] :
      ( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_list_a @ R ) )
     => ~ ! [Y3: list_a,Ys2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Y3 @ Ys2 ) )
           => ( ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ Y @ Y3 ) @ R )
             => ~ ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Ys @ Ys2 ) @ ( listrel_nat_list_a @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_366_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_list_nat,R: set_Pr698032948827214833st_nat] :
      ( ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_list_nat @ R ) )
     => ~ ! [Y3: list_nat,Ys2: list_list_nat] :
            ( ( Xs
              = ( cons_list_nat @ Y3 @ Ys2 ) )
           => ( ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ Y @ Y3 ) @ R )
             => ~ ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ Ys @ Ys2 ) @ ( listrel_nat_list_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_367_listrel__Cons2,axiom,
    ! [Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R ) )
     => ~ ! [X3: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X3 @ Xs2 ) )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
             => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_368_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: a,Ys: list_a,R: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_nat_a @ R ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X3 @ Y ) @ R )
             => ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs2 @ Ys ) @ ( listrel_nat_a @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_369_listrel__Cons2,axiom,
    ! [Xs: list_a,Y: nat,Ys: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R ) )
     => ~ ! [X3: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X3 @ Xs2 ) )
           => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ R )
             => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys ) @ ( listrel_a_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_370_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_371_listrel__Cons2,axiom,
    ! [Xs: list_a,Y: list_a,Ys: list_list_a,R: set_Pr2070066670564046349list_a] :
      ( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel_a_list_a @ R ) )
     => ~ ! [X3: a,Xs2: list_a] :
            ( ( Xs
              = ( cons_a @ X3 @ Xs2 ) )
           => ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X3 @ Y ) @ R )
             => ~ ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs2 @ Ys ) @ ( listrel_a_list_a @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_372_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: list_a,Ys: list_list_a,R: set_Pr3870785875473343199list_a] :
      ( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel_nat_list_a @ R ) )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs2 ) )
           => ( ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ X3 @ Y ) @ R )
             => ~ ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs2 @ Ys ) @ ( listrel_nat_list_a @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_373_listrel__Cons2,axiom,
    ! [Xs: list_list_a,Y: a,Ys: list_a,R: set_Pr8962057229576493569st_a_a] :
      ( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_list_a_a @ R ) )
     => ~ ! [X3: list_a,Xs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ X3 @ Xs2 ) )
           => ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X3 @ Y ) @ R )
             => ~ ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs2 @ Ys ) @ ( listrel_list_a_a @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_374_listrel__Cons2,axiom,
    ! [Xs: list_list_nat,Y: a,Ys: list_a,R: set_Pr8540246825521615561_nat_a] :
      ( ( member2562985910519063712list_a @ ( produc1825125547596988987list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_list_nat_a @ R ) )
     => ~ ! [X3: list_nat,Xs2: list_list_nat] :
            ( ( Xs
              = ( cons_list_nat @ X3 @ Xs2 ) )
           => ( ( member6908590373215729450_nat_a @ ( produc1615968307924265413_nat_a @ X3 @ Y ) @ R )
             => ~ ( member2562985910519063712list_a @ ( produc1825125547596988987list_a @ Xs2 @ Ys ) @ ( listrel_list_nat_a @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_375_listrel__Cons2,axiom,
    ! [Xs: list_list_a,Y: nat,Ys: list_nat,R: set_Pr6248017591093387577_a_nat] :
      ( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_list_a_nat @ R ) )
     => ~ ! [X3: list_a,Xs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ X3 @ Xs2 ) )
           => ( ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ X3 @ Y ) @ R )
             => ~ ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs2 @ Ys ) @ ( listrel_list_a_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_376_listrel__Cons2,axiom,
    ! [Xs: list_list_nat,Y: nat,Ys: list_nat,R: set_Pr7549493975000385137at_nat] :
      ( ( member2583041412010425562st_nat @ ( produc8303078633012690025st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_list_nat_nat @ R ) )
     => ~ ! [X3: list_nat,Xs2: list_list_nat] :
            ( ( Xs
              = ( cons_list_nat @ X3 @ Xs2 ) )
           => ( ( member7820552544826472890at_nat @ ( produc1298395424260782409at_nat @ X3 @ Y ) @ R )
             => ~ ( member2583041412010425562st_nat @ ( produc8303078633012690025st_nat @ Xs2 @ Ys ) @ ( listrel_list_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_377_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_a ) )
       => ~ ! [X3: a,Y3: a,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs2 ) )
             => ! [Ys2: list_a] :
                  ( ( A22
                    = ( cons_a @ Y3 @ Ys2 ) )
                 => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
                   => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_378_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_nat ) )
       => ~ ! [X3: a,Y3: nat,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys2 ) )
                 => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys2 ) @ ( listrel_a_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_379_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_a,R: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ A1 @ A22 ) @ ( listrel_nat_a @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_a ) )
       => ~ ! [X3: nat,Y3: a,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X3 @ Xs2 ) )
             => ! [Ys2: list_a] :
                  ( ( A22
                    = ( cons_a @ Y3 @ Ys2 ) )
                 => ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X3 @ Y3 ) @ R )
                   => ~ ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs2 @ Ys2 ) @ ( listrel_nat_a @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_380_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X3 @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys2 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_381_listrel_Ocases,axiom,
    ! [A1: list_list_a,A22: list_a,R: set_Pr8962057229576493569st_a_a] :
      ( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ A1 @ A22 ) @ ( listrel_list_a_a @ R ) )
     => ( ( ( A1 = nil_list_a )
         => ( A22 != nil_a ) )
       => ~ ! [X3: list_a,Y3: a,Xs2: list_list_a] :
              ( ( A1
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ! [Ys2: list_a] :
                  ( ( A22
                    = ( cons_a @ Y3 @ Ys2 ) )
                 => ( ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X3 @ Y3 ) @ R )
                   => ~ ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs2 @ Ys2 ) @ ( listrel_list_a_a @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_382_listrel_Ocases,axiom,
    ! [A1: list_list_a,A22: list_nat,R: set_Pr6248017591093387577_a_nat] :
      ( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ A1 @ A22 ) @ ( listrel_list_a_nat @ R ) )
     => ( ( ( A1 = nil_list_a )
         => ( A22 != nil_nat ) )
       => ~ ! [X3: list_a,Y3: nat,Xs2: list_list_a] :
              ( ( A1
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys2 ) )
                 => ( ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs2 @ Ys2 ) @ ( listrel_list_a_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_383_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_list_a,R: set_Pr2070066670564046349list_a] :
      ( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ A1 @ A22 ) @ ( listrel_a_list_a @ R ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_list_a ) )
       => ~ ! [X3: a,Y3: list_a,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs2 ) )
             => ! [Ys2: list_list_a] :
                  ( ( A22
                    = ( cons_list_a @ Y3 @ Ys2 ) )
                 => ( ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X3 @ Y3 ) @ R )
                   => ~ ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs2 @ Ys2 ) @ ( listrel_a_list_a @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_384_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_list_nat,R: set_Pr6062262858518114627st_nat] :
      ( ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ A1 @ A22 ) @ ( listrel_a_list_nat @ R ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_list_nat ) )
       => ~ ! [X3: a,Y3: list_nat,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs2 ) )
             => ! [Ys2: list_list_nat] :
                  ( ( A22
                    = ( cons_list_nat @ Y3 @ Ys2 ) )
                 => ( ( member9064520187905290916st_nat @ ( produc6928161557333929735st_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ Xs2 @ Ys2 ) @ ( listrel_a_list_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_385_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_list_a,R: set_Pr3870785875473343199list_a] :
      ( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ A1 @ A22 ) @ ( listrel_nat_list_a @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_list_a ) )
       => ~ ! [X3: nat,Y3: list_a,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X3 @ Xs2 ) )
             => ! [Ys2: list_list_a] :
                  ( ( A22
                    = ( cons_list_a @ Y3 @ Ys2 ) )
                 => ( ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ X3 @ Y3 ) @ R )
                   => ~ ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs2 @ Ys2 ) @ ( listrel_nat_list_a @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_386_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_list_nat,R: set_Pr698032948827214833st_nat] :
      ( ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ A1 @ A22 ) @ ( listrel_nat_list_nat @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_list_nat ) )
       => ~ ! [X3: nat,Y3: list_nat,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X3 @ Xs2 ) )
             => ! [Ys2: list_list_nat] :
                  ( ( A22
                    = ( cons_list_nat @ Y3 @ Ys2 ) )
                 => ( ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_list_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_387_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_a ) )
        | ? [X4: a,Y5: a,Xs3: list_a,Ys3: list_a] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_a @ Y5 @ Ys3 ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y5 ) @ R )
            & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_388_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_nat,R: set_Pr4934435412358123699_a_nat] :
      ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_nat ) )
        | ? [X4: a,Y5: nat,Xs3: list_a,Ys3: list_nat] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y5 ) @ R )
            & ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs3 @ Ys3 ) @ ( listrel_a_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_389_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_a,R: set_Pr4193341848836149977_nat_a] :
      ( ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ A1 @ A22 ) @ ( listrel_nat_a @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_a ) )
        | ? [X4: nat,Y5: a,Xs3: list_nat,Ys3: list_a] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_a @ Y5 @ Ys3 ) )
            & ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X4 @ Y5 ) @ R )
            & ( member5932150393272073264list_a @ ( produc7723716010052024011list_a @ Xs3 @ Ys3 ) @ ( listrel_nat_a @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_390_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X4: nat,Y5: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_391_listrel_Osimps,axiom,
    ! [A1: list_list_a,A22: list_a,R: set_Pr8962057229576493569st_a_a] :
      ( ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ A1 @ A22 ) @ ( listrel_list_a_a @ R ) )
      = ( ( ( A1 = nil_list_a )
          & ( A22 = nil_a ) )
        | ? [X4: list_a,Y5: a,Xs3: list_list_a,Ys3: list_a] :
            ( ( A1
              = ( cons_list_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_a @ Y5 @ Ys3 ) )
            & ( member8006451231845903178st_a_a @ ( produc4781227316648555537st_a_a @ X4 @ Y5 ) @ R )
            & ( member4371779931761811402list_a @ ( produc1599761694186162065list_a @ Xs3 @ Ys3 ) @ ( listrel_list_a_a @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_392_listrel_Osimps,axiom,
    ! [A1: list_list_a,A22: list_nat,R: set_Pr6248017591093387577_a_nat] :
      ( ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ A1 @ A22 ) @ ( listrel_list_a_nat @ R ) )
      = ( ( ( A1 = nil_list_a )
          & ( A22 = nil_nat ) )
        | ? [X4: list_a,Y5: nat,Xs3: list_list_a,Ys3: list_nat] :
            ( ( A1
              = ( cons_list_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys3 ) )
            & ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ X4 @ Y5 ) @ R )
            & ( member6137020595371076400st_nat @ ( produc3549405625035910547st_nat @ Xs3 @ Ys3 ) @ ( listrel_list_a_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_393_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_list_a,R: set_Pr2070066670564046349list_a] :
      ( ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ A1 @ A22 ) @ ( listrel_a_list_a @ R ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_list_a ) )
        | ? [X4: a,Y5: list_a,Xs3: list_a,Ys3: list_list_a] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_list_a @ Y5 @ Ys3 ) )
            & ( member4889668945541975382list_a @ ( produc6670463072477821725list_a @ X4 @ Y5 ) @ R )
            & ( member3917598494194944214list_a @ ( produc5682643643425543581list_a @ Xs3 @ Ys3 ) @ ( listrel_a_list_a @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_394_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_list_nat,R: set_Pr6062262858518114627st_nat] :
      ( ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ A1 @ A22 ) @ ( listrel_a_list_nat @ R ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_list_nat ) )
        | ? [X4: a,Y5: list_nat,Xs3: list_a,Ys3: list_list_nat] :
            ( ( A1
              = ( cons_a @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_list_nat @ Y5 @ Ys3 ) )
            & ( member9064520187905290916st_nat @ ( produc6928161557333929735st_nat @ X4 @ Y5 ) @ R )
            & ( member4047528884340026938st_nat @ ( produc8676886407030044317st_nat @ Xs3 @ Ys3 ) @ ( listrel_a_list_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_395_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_list_a,R: set_Pr3870785875473343199list_a] :
      ( ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ A1 @ A22 ) @ ( listrel_nat_list_a @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_list_a ) )
        | ? [X4: nat,Y5: list_a,Xs3: list_nat,Ys3: list_list_a] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_list_a @ Y5 @ Ys3 ) )
            & ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ X4 @ Y5 ) @ R )
            & ( member7437635676025943222list_a @ ( produc2013466663055341393list_a @ Xs3 @ Ys3 ) @ ( listrel_nat_list_a @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_396_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_list_nat,R: set_Pr698032948827214833st_nat] :
      ( ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ A1 @ A22 ) @ ( listrel_nat_list_nat @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_list_nat ) )
        | ? [X4: nat,Y5: list_nat,Xs3: list_nat,Ys3: list_list_nat] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs3 ) )
            & ( A22
              = ( cons_list_nat @ Y5 @ Ys3 ) )
            & ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X4 @ Y5 ) @ R )
            & ( member3963112742161418842st_nat @ ( produc7676750419334090729st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_list_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_397_map__entry_Oelims,axiom,
    ! [X: $o,Xa: list_list_a > list_list_a,Xb: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( ( map_en7508539461263903803list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Pr1488683542974317855list_a )
         => ( Y != nil_Pr1488683542974317855list_a ) )
       => ~ ! [P5: produc2922778490715282969list_a,Ps: list_P369347992569427103list_a] :
              ( ( Xb
                = ( cons_P3051609457517836495list_a @ P5 @ Ps ) )
             => ~ ( ( ( ( produc3726311466776224373list_a @ P5 )
                      = X )
                   => ( Y
                      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ X @ ( Xa @ ( produc4969414473911996599list_a @ P5 ) ) ) @ Ps ) ) )
                  & ( ( ( produc3726311466776224373list_a @ P5 )
                      = ~ X )
                   => ( Y
                      = ( cons_P3051609457517836495list_a @ P5 @ ( map_en7508539461263903803list_a @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_398_map__entry_Oelims,axiom,
    ! [X: $o,Xa: list_list_nat > list_list_nat,Xb: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( ( map_en1378505499878274175st_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Pr1938078369830740401st_nat )
         => ( Y != nil_Pr1938078369830740401st_nat ) )
       => ~ ! [P5: produc7974263126369132087st_nat,Ps: list_P3026376796534914631st_nat] :
              ( ( Xb
                = ( cons_P5527778705151500801st_nat @ P5 @ Ps ) )
             => ~ ( ( ( ( produc3357746125391881157st_nat @ P5 )
                      = X )
                   => ( Y
                      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ X @ ( Xa @ ( produc4024048262274189571st_nat @ P5 ) ) ) @ Ps ) ) )
                  & ( ( ( produc3357746125391881157st_nat @ P5 )
                      = ~ X )
                   => ( Y
                      = ( cons_P5527778705151500801st_nat @ P5 @ ( map_en1378505499878274175st_nat @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_399_map__entry_Oelims,axiom,
    ! [X: $o,Xa: list_list_list_a > list_list_list_a,Xb: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( ( map_en7631213772203213633list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xb = nil_Pr7514062926313711781list_a )
         => ( Y != nil_Pr7514062926313711781list_a ) )
       => ~ ! [P5: produc2086622085225315487list_a,Ps: list_P1915399164375025317list_a] :
              ( ( Xb
                = ( cons_P2654393883304477781list_a @ P5 @ Ps ) )
             => ~ ( ( ( ( produc2785537127425143931list_a @ P5 )
                      = X )
                   => ( Y
                      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ X @ ( Xa @ ( produc1153002459082627517list_a @ P5 ) ) ) @ Ps ) ) )
                  & ( ( ( produc2785537127425143931list_a @ P5 )
                      = ~ X )
                   => ( Y
                      = ( cons_P2654393883304477781list_a @ P5 @ ( map_en7631213772203213633list_a @ X @ Xa @ Ps ) ) ) ) ) ) ) ) ).

% map_entry.elims
thf(fact_400_map__default_Oelims,axiom,
    ! [X: $o,Xa: list_list_a,Xb: list_list_a > list_list_a,Xc: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( ( map_de7478545310729120074list_a @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( Xc = nil_Pr1488683542974317855list_a )
         => ( Y
           != ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ X @ Xa ) @ nil_Pr1488683542974317855list_a ) ) )
       => ~ ! [P5: produc2922778490715282969list_a,Ps: list_P369347992569427103list_a] :
              ( ( Xc
                = ( cons_P3051609457517836495list_a @ P5 @ Ps ) )
             => ~ ( ( ( ( produc3726311466776224373list_a @ P5 )
                      = X )
                   => ( Y
                      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ X @ ( Xb @ ( produc4969414473911996599list_a @ P5 ) ) ) @ Ps ) ) )
                  & ( ( ( produc3726311466776224373list_a @ P5 )
                      = ~ X )
                   => ( Y
                      = ( cons_P3051609457517836495list_a @ P5 @ ( map_de7478545310729120074list_a @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_401_map__default_Oelims,axiom,
    ! [X: $o,Xa: list_list_nat,Xb: list_list_nat > list_list_nat,Xc: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( ( map_de5183444947954434480st_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( Xc = nil_Pr1938078369830740401st_nat )
         => ( Y
           != ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ X @ Xa ) @ nil_Pr1938078369830740401st_nat ) ) )
       => ~ ! [P5: produc7974263126369132087st_nat,Ps: list_P3026376796534914631st_nat] :
              ( ( Xc
                = ( cons_P5527778705151500801st_nat @ P5 @ Ps ) )
             => ~ ( ( ( ( produc3357746125391881157st_nat @ P5 )
                      = X )
                   => ( Y
                      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ X @ ( Xb @ ( produc4024048262274189571st_nat @ P5 ) ) ) @ Ps ) ) )
                  & ( ( ( produc3357746125391881157st_nat @ P5 )
                      = ~ X )
                   => ( Y
                      = ( cons_P5527778705151500801st_nat @ P5 @ ( map_de5183444947954434480st_nat @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_402_map__default_Oelims,axiom,
    ! [X: $o,Xa: list_list_list_a,Xb: list_list_list_a > list_list_list_a,Xc: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( ( map_de2428310187333209296list_a @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( Xc = nil_Pr7514062926313711781list_a )
         => ( Y
           != ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ X @ Xa ) @ nil_Pr7514062926313711781list_a ) ) )
       => ~ ! [P5: produc2086622085225315487list_a,Ps: list_P1915399164375025317list_a] :
              ( ( Xc
                = ( cons_P2654393883304477781list_a @ P5 @ Ps ) )
             => ~ ( ( ( ( produc2785537127425143931list_a @ P5 )
                      = X )
                   => ( Y
                      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ X @ ( Xb @ ( produc1153002459082627517list_a @ P5 ) ) ) @ Ps ) ) )
                  & ( ( ( produc2785537127425143931list_a @ P5 )
                      = ~ X )
                   => ( Y
                      = ( cons_P2654393883304477781list_a @ P5 @ ( map_de2428310187333209296list_a @ X @ Xa @ Xb @ Ps ) ) ) ) ) ) ) ) ).

% map_default.elims
thf(fact_403_update__with__aux_Osimps_I2_J,axiom,
    ! [P2: produc2922778490715282969list_a,K3: $o,V2: list_list_a,F2: list_list_a > list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( ( ( produc3726311466776224373list_a @ P2 )
          = K3 )
       => ( ( update3143123273235729040st_a_o @ V2 @ K3 @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ ( produc4969414473911996599list_a @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc3726311466776224373list_a @ P2 )
          = ~ K3 )
       => ( ( update3143123273235729040st_a_o @ V2 @ K3 @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ P2 @ ( update3143123273235729040st_a_o @ V2 @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_404_update__with__aux_Osimps_I2_J,axiom,
    ! [P2: produc7974263126369132087st_nat,K3: $o,V2: list_list_nat,F2: list_list_nat > list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( ( ( produc3357746125391881157st_nat @ P2 )
          = K3 )
       => ( ( update2517193581708834468_nat_o @ V2 @ K3 @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ ( produc4024048262274189571st_nat @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc3357746125391881157st_nat @ P2 )
          = ~ K3 )
       => ( ( update2517193581708834468_nat_o @ V2 @ K3 @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ P2 @ ( update2517193581708834468_nat_o @ V2 @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_405_update__with__aux_Osimps_I2_J,axiom,
    ! [P2: produc2086622085225315487list_a,K3: $o,V2: list_list_list_a,F2: list_list_list_a > list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( ( ( produc2785537127425143931list_a @ P2 )
          = K3 )
       => ( ( update1976731422980637706st_a_o @ V2 @ K3 @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ ( produc1153002459082627517list_a @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc2785537127425143931list_a @ P2 )
          = ~ K3 )
       => ( ( update1976731422980637706st_a_o @ V2 @ K3 @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ P2 @ ( update1976731422980637706st_a_o @ V2 @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% update_with_aux.simps(2)
thf(fact_406_map__entry_Osimps_I2_J,axiom,
    ! [P2: produc2922778490715282969list_a,K3: $o,F2: list_list_a > list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( ( ( produc3726311466776224373list_a @ P2 )
          = K3 )
       => ( ( map_en7508539461263903803list_a @ K3 @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ ( produc4969414473911996599list_a @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc3726311466776224373list_a @ P2 )
          = ~ K3 )
       => ( ( map_en7508539461263903803list_a @ K3 @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ P2 @ ( map_en7508539461263903803list_a @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_407_map__entry_Osimps_I2_J,axiom,
    ! [P2: produc7974263126369132087st_nat,K3: $o,F2: list_list_nat > list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( ( ( produc3357746125391881157st_nat @ P2 )
          = K3 )
       => ( ( map_en1378505499878274175st_nat @ K3 @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ ( produc4024048262274189571st_nat @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc3357746125391881157st_nat @ P2 )
          = ~ K3 )
       => ( ( map_en1378505499878274175st_nat @ K3 @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ P2 @ ( map_en1378505499878274175st_nat @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_408_map__entry_Osimps_I2_J,axiom,
    ! [P2: produc2086622085225315487list_a,K3: $o,F2: list_list_list_a > list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( ( ( produc2785537127425143931list_a @ P2 )
          = K3 )
       => ( ( map_en7631213772203213633list_a @ K3 @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ ( produc1153002459082627517list_a @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc2785537127425143931list_a @ P2 )
          = ~ K3 )
       => ( ( map_en7631213772203213633list_a @ K3 @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ P2 @ ( map_en7631213772203213633list_a @ K3 @ F2 @ Ps2 ) ) ) ) ) ).

% map_entry.simps(2)
thf(fact_409_map__default_Osimps_I2_J,axiom,
    ! [P2: produc2922778490715282969list_a,K3: $o,V2: list_list_a,F2: list_list_a > list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( ( ( produc3726311466776224373list_a @ P2 )
          = K3 )
       => ( ( map_de7478545310729120074list_a @ K3 @ V2 @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ ( produc4969414473911996599list_a @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc3726311466776224373list_a @ P2 )
          = ~ K3 )
       => ( ( map_de7478545310729120074list_a @ K3 @ V2 @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ P2 @ ( map_de7478545310729120074list_a @ K3 @ V2 @ F2 @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_410_map__default_Osimps_I2_J,axiom,
    ! [P2: produc7974263126369132087st_nat,K3: $o,V2: list_list_nat,F2: list_list_nat > list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( ( ( produc3357746125391881157st_nat @ P2 )
          = K3 )
       => ( ( map_de5183444947954434480st_nat @ K3 @ V2 @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ ( produc4024048262274189571st_nat @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc3357746125391881157st_nat @ P2 )
          = ~ K3 )
       => ( ( map_de5183444947954434480st_nat @ K3 @ V2 @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ P2 @ ( map_de5183444947954434480st_nat @ K3 @ V2 @ F2 @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_411_map__default_Osimps_I2_J,axiom,
    ! [P2: produc2086622085225315487list_a,K3: $o,V2: list_list_list_a,F2: list_list_list_a > list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( ( ( produc2785537127425143931list_a @ P2 )
          = K3 )
       => ( ( map_de2428310187333209296list_a @ K3 @ V2 @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ ( produc1153002459082627517list_a @ P2 ) ) ) @ Ps2 ) ) )
      & ( ( ( produc2785537127425143931list_a @ P2 )
          = ~ K3 )
       => ( ( map_de2428310187333209296list_a @ K3 @ V2 @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ P2 @ ( map_de2428310187333209296list_a @ K3 @ V2 @ F2 @ Ps2 ) ) ) ) ) ).

% map_default.simps(2)
thf(fact_412_SuccD,axiom,
    ! [K3: list_a,Kl: set_list_list_a,Kl2: list_list_a] :
      ( ( member_list_a @ K3 @ ( bNF_Gr4634511371912843295list_a @ Kl @ Kl2 ) )
     => ( member_list_list_a @ ( append_list_a @ Kl2 @ ( cons_list_a @ K3 @ nil_list_a ) ) @ Kl ) ) ).

% SuccD
thf(fact_413_SuccD,axiom,
    ! [K3: a,Kl: set_list_a,Kl2: list_a] :
      ( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
     => ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K3 @ nil_a ) ) @ Kl ) ) ).

% SuccD
thf(fact_414_SuccD,axiom,
    ! [K3: nat,Kl: set_list_nat,Kl2: list_nat] :
      ( ( member_nat @ K3 @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
     => ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_415_SuccD,axiom,
    ! [K3: list_nat,Kl: set_list_list_nat,Kl2: list_list_nat] :
      ( ( member_list_nat @ K3 @ ( bNF_Gr3053708287304744325st_nat @ Kl @ Kl2 ) )
     => ( member_list_list_nat @ ( append_list_nat @ Kl2 @ ( cons_list_nat @ K3 @ nil_list_nat ) ) @ Kl ) ) ).

% SuccD
thf(fact_416_SuccD,axiom,
    ! [K3: list_list_a,Kl: set_list_list_list_a,Kl2: list_list_list_a] :
      ( ( member_list_list_a @ K3 @ ( bNF_Gr6039074496463509157list_a @ Kl @ Kl2 ) )
     => ( member5342144027231129785list_a @ ( append_list_list_a @ Kl2 @ ( cons_list_list_a @ K3 @ nil_list_list_a ) ) @ Kl ) ) ).

% SuccD
thf(fact_417_SuccI,axiom,
    ! [Kl2: list_list_a,K3: list_a,Kl: set_list_list_a] :
      ( ( member_list_list_a @ ( append_list_a @ Kl2 @ ( cons_list_a @ K3 @ nil_list_a ) ) @ Kl )
     => ( member_list_a @ K3 @ ( bNF_Gr4634511371912843295list_a @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_418_SuccI,axiom,
    ! [Kl2: list_a,K3: a,Kl: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K3 @ nil_a ) ) @ Kl )
     => ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_419_SuccI,axiom,
    ! [Kl2: list_nat,K3: nat,Kl: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl )
     => ( member_nat @ K3 @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_420_SuccI,axiom,
    ! [Kl2: list_list_nat,K3: list_nat,Kl: set_list_list_nat] :
      ( ( member_list_list_nat @ ( append_list_nat @ Kl2 @ ( cons_list_nat @ K3 @ nil_list_nat ) ) @ Kl )
     => ( member_list_nat @ K3 @ ( bNF_Gr3053708287304744325st_nat @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_421_SuccI,axiom,
    ! [Kl2: list_list_list_a,K3: list_list_a,Kl: set_list_list_list_a] :
      ( ( member5342144027231129785list_a @ ( append_list_list_a @ Kl2 @ ( cons_list_list_a @ K3 @ nil_list_list_a ) ) @ Kl )
     => ( member_list_list_a @ K3 @ ( bNF_Gr6039074496463509157list_a @ Kl @ Kl2 ) ) ) ).

% SuccI
thf(fact_422_bind__simps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a,F2: list_a > list_nat] :
      ( ( bind_list_a_nat @ ( cons_list_a @ X @ Xs ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_list_a_nat @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_423_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F2: a > list_nat] :
      ( ( bind_a_nat @ ( cons_a @ X @ Xs ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_a_nat @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_424_bind__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,F2: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_nat_nat @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_425_bind__simps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat,F2: list_nat > list_nat] :
      ( ( bind_list_nat_nat @ ( cons_list_nat @ X @ Xs ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_list_nat_nat @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_426_bind__simps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,F2: list_list_a > list_nat] :
      ( ( bind_list_list_a_nat @ ( cons_list_list_a @ X @ Xs ) @ F2 )
      = ( append_nat @ ( F2 @ X ) @ ( bind_list_list_a_nat @ Xs @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_427_prefixes__snoc,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( prefixes_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
      = ( append_list_list_a @ ( prefixes_list_a @ Xs ) @ ( cons_list_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) @ nil_list_list_a ) ) ) ).

% prefixes_snoc
thf(fact_428_prefixes__snoc,axiom,
    ! [Xs: list_a,X: a] :
      ( ( prefixes_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = ( append_list_a @ ( prefixes_a @ Xs ) @ ( cons_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ nil_list_a ) ) ) ).

% prefixes_snoc
thf(fact_429_prefixes__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).

% prefixes_snoc
thf(fact_430_prefixes__snoc,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( prefixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( append_list_list_nat @ ( prefixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).

% prefixes_snoc
thf(fact_431_prefixes__snoc,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( prefixes_list_list_a @ ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = ( append6004555433696164139list_a @ ( prefixes_list_list_a @ Xs ) @ ( cons_l8577867181958743494list_a @ ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) @ nil_list_list_list_a ) ) ) ).

% prefixes_snoc
thf(fact_432_bind__simps_I1_J,axiom,
    ! [F2: a > list_a] :
      ( ( bind_a_a @ nil_a @ F2 )
      = nil_a ) ).

% bind_simps(1)
thf(fact_433_bind__simps_I1_J,axiom,
    ! [F2: a > list_nat] :
      ( ( bind_a_nat @ nil_a @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_434_bind__simps_I1_J,axiom,
    ! [F2: nat > list_a] :
      ( ( bind_nat_a @ nil_nat @ F2 )
      = nil_a ) ).

% bind_simps(1)
thf(fact_435_bind__simps_I1_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_436_prefixes_Osimps_I1_J,axiom,
    ( ( prefixes_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% prefixes.simps(1)
thf(fact_437_prefixes_Osimps_I1_J,axiom,
    ( ( prefixes_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% prefixes.simps(1)
thf(fact_438_prefixes_Osimps_I1_J,axiom,
    ( ( prefixes_list_a @ nil_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% prefixes.simps(1)
thf(fact_439_prefixes__eq__snoc,axiom,
    ! [Ys: list_list_nat,Xs: list_list_list_nat,X: list_list_nat] :
      ( ( ( prefixes_list_nat @ Ys )
        = ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
      = ( ( ( ( Ys = nil_list_nat )
            & ( Xs = nil_list_list_nat ) )
          | ? [Z3: list_nat,Zs2: list_list_nat] :
              ( ( Ys
                = ( append_list_nat @ Zs2 @ ( cons_list_nat @ Z3 @ nil_list_nat ) ) )
              & ( Xs
                = ( prefixes_list_nat @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_440_prefixes__eq__snoc,axiom,
    ! [Ys: list_list_list_a,Xs: list_l7815035709764258326list_a,X: list_list_list_a] :
      ( ( ( prefixes_list_list_a @ Ys )
        = ( append6004555433696164139list_a @ Xs @ ( cons_l8577867181958743494list_a @ X @ nil_list_list_list_a ) ) )
      = ( ( ( ( Ys = nil_list_list_a )
            & ( Xs = nil_list_list_list_a ) )
          | ? [Z3: list_list_a,Zs2: list_list_list_a] :
              ( ( Ys
                = ( append_list_list_a @ Zs2 @ ( cons_list_list_a @ Z3 @ nil_list_list_a ) ) )
              & ( Xs
                = ( prefixes_list_list_a @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_441_prefixes__eq__snoc,axiom,
    ! [Ys: list_a,Xs: list_list_a,X: list_a] :
      ( ( ( prefixes_a @ Ys )
        = ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
      = ( ( ( ( Ys = nil_a )
            & ( Xs = nil_list_a ) )
          | ? [Z3: a,Zs2: list_a] :
              ( ( Ys
                = ( append_a @ Zs2 @ ( cons_a @ Z3 @ nil_a ) ) )
              & ( Xs
                = ( prefixes_a @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_442_prefixes__eq__snoc,axiom,
    ! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
      ( ( ( prefixes_nat @ Ys )
        = ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( ( ( ( Ys = nil_nat )
            & ( Xs = nil_list_nat ) )
          | ? [Z3: nat,Zs2: list_nat] :
              ( ( Ys
                = ( append_nat @ Zs2 @ ( cons_nat @ Z3 @ nil_nat ) ) )
              & ( Xs
                = ( prefixes_nat @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_443_prefixes__eq__snoc,axiom,
    ! [Ys: list_list_a,Xs: list_list_list_a,X: list_list_a] :
      ( ( ( prefixes_list_a @ Ys )
        = ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = ( ( ( ( Ys = nil_list_a )
            & ( Xs = nil_list_list_a ) )
          | ? [Z3: list_a,Zs2: list_list_a] :
              ( ( Ys
                = ( append_list_a @ Zs2 @ ( cons_list_a @ Z3 @ nil_list_a ) ) )
              & ( Xs
                = ( prefixes_list_a @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% prefixes_eq_snoc
thf(fact_444_map__default_Osimps_I1_J,axiom,
    ! [K3: $o,V2: list_list_a,F2: list_list_a > list_list_a] :
      ( ( map_de7478545310729120074list_a @ K3 @ V2 @ F2 @ nil_Pr1488683542974317855list_a )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V2 ) @ nil_Pr1488683542974317855list_a ) ) ).

% map_default.simps(1)
thf(fact_445_map__default_Osimps_I1_J,axiom,
    ! [K3: $o,V2: list_list_nat,F2: list_list_nat > list_list_nat] :
      ( ( map_de5183444947954434480st_nat @ K3 @ V2 @ F2 @ nil_Pr1938078369830740401st_nat )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V2 ) @ nil_Pr1938078369830740401st_nat ) ) ).

% map_default.simps(1)
thf(fact_446_map__default_Osimps_I1_J,axiom,
    ! [K3: $o,V2: list_list_list_a,F2: list_list_list_a > list_list_list_a] :
      ( ( map_de2428310187333209296list_a @ K3 @ V2 @ F2 @ nil_Pr7514062926313711781list_a )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V2 ) @ nil_Pr7514062926313711781list_a ) ) ).

% map_default.simps(1)
thf(fact_447_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: list_list_a,K3: $o,F2: list_list_a > list_list_a] :
      ( ( update3143123273235729040st_a_o @ V2 @ K3 @ F2 @ nil_Pr1488683542974317855list_a )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ V2 ) ) @ nil_Pr1488683542974317855list_a ) ) ).

% update_with_aux.simps(1)
thf(fact_448_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: list_list_nat,K3: $o,F2: list_list_nat > list_list_nat] :
      ( ( update2517193581708834468_nat_o @ V2 @ K3 @ F2 @ nil_Pr1938078369830740401st_nat )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ V2 ) ) @ nil_Pr1938078369830740401st_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_449_update__with__aux_Osimps_I1_J,axiom,
    ! [V2: list_list_list_a,K3: $o,F2: list_list_list_a > list_list_list_a] :
      ( ( update1976731422980637706st_a_o @ V2 @ K3 @ F2 @ nil_Pr7514062926313711781list_a )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ V2 ) ) @ nil_Pr7514062926313711781list_a ) ) ).

% update_with_aux.simps(1)
thf(fact_450_map__default_Opelims,axiom,
    ! [X: $o,Xa: list_list_a,Xb: list_list_a > list_list_a,Xc: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( ( map_de7478545310729120074list_a @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6723639556475776038list_a @ map_de8605239412455127315list_a @ ( produc4163342915187748713list_a @ X @ ( produc8851461697476795554list_a @ Xa @ ( produc1851355790630076705list_a @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr1488683542974317855list_a )
           => ( ( Y
                = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ X @ Xa ) @ nil_Pr1488683542974317855list_a ) )
             => ~ ( accp_P6723639556475776038list_a @ map_de8605239412455127315list_a @ ( produc4163342915187748713list_a @ X @ ( produc8851461697476795554list_a @ Xa @ ( produc1851355790630076705list_a @ Xb @ nil_Pr1488683542974317855list_a ) ) ) ) ) )
         => ~ ! [P5: produc2922778490715282969list_a,Ps: list_P369347992569427103list_a] :
                ( ( Xc
                  = ( cons_P3051609457517836495list_a @ P5 @ Ps ) )
               => ( ( ( ( ( produc3726311466776224373list_a @ P5 )
                        = X )
                     => ( Y
                        = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ X @ ( Xb @ ( produc4969414473911996599list_a @ P5 ) ) ) @ Ps ) ) )
                    & ( ( ( produc3726311466776224373list_a @ P5 )
                        = ~ X )
                     => ( Y
                        = ( cons_P3051609457517836495list_a @ P5 @ ( map_de7478545310729120074list_a @ X @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P6723639556475776038list_a @ map_de8605239412455127315list_a @ ( produc4163342915187748713list_a @ X @ ( produc8851461697476795554list_a @ Xa @ ( produc1851355790630076705list_a @ Xb @ ( cons_P3051609457517836495list_a @ P5 @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_451_map__default_Opelims,axiom,
    ! [X: $o,Xa: list_list_nat,Xb: list_list_nat > list_list_nat,Xc: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( ( map_de5183444947954434480st_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6219854476835865258st_nat @ map_de7349754367479955879st_nat @ ( produc6828981150761895405st_nat @ X @ ( produc8146549173532289118st_nat @ Xa @ ( produc4663047625710595679st_nat @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr1938078369830740401st_nat )
           => ( ( Y
                = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ X @ Xa ) @ nil_Pr1938078369830740401st_nat ) )
             => ~ ( accp_P6219854476835865258st_nat @ map_de7349754367479955879st_nat @ ( produc6828981150761895405st_nat @ X @ ( produc8146549173532289118st_nat @ Xa @ ( produc4663047625710595679st_nat @ Xb @ nil_Pr1938078369830740401st_nat ) ) ) ) ) )
         => ~ ! [P5: produc7974263126369132087st_nat,Ps: list_P3026376796534914631st_nat] :
                ( ( Xc
                  = ( cons_P5527778705151500801st_nat @ P5 @ Ps ) )
               => ( ( ( ( ( produc3357746125391881157st_nat @ P5 )
                        = X )
                     => ( Y
                        = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ X @ ( Xb @ ( produc4024048262274189571st_nat @ P5 ) ) ) @ Ps ) ) )
                    & ( ( ( produc3357746125391881157st_nat @ P5 )
                        = ~ X )
                     => ( Y
                        = ( cons_P5527778705151500801st_nat @ P5 @ ( map_de5183444947954434480st_nat @ X @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P6219854476835865258st_nat @ map_de7349754367479955879st_nat @ ( produc6828981150761895405st_nat @ X @ ( produc8146549173532289118st_nat @ Xa @ ( produc4663047625710595679st_nat @ Xb @ ( cons_P5527778705151500801st_nat @ P5 @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_452_map__default_Opelims,axiom,
    ! [X: $o,Xa: list_list_list_a,Xb: list_list_list_a > list_list_list_a,Xc: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( ( map_de2428310187333209296list_a @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P4204554831414503090list_a @ map_de3558955406781211673list_a @ ( produc291085998517269493list_a @ X @ ( produc8995079174706216494list_a @ Xa @ ( produc6935208604740912679list_a @ Xb @ Xc ) ) ) )
       => ( ( ( Xc = nil_Pr7514062926313711781list_a )
           => ( ( Y
                = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ X @ Xa ) @ nil_Pr7514062926313711781list_a ) )
             => ~ ( accp_P4204554831414503090list_a @ map_de3558955406781211673list_a @ ( produc291085998517269493list_a @ X @ ( produc8995079174706216494list_a @ Xa @ ( produc6935208604740912679list_a @ Xb @ nil_Pr7514062926313711781list_a ) ) ) ) ) )
         => ~ ! [P5: produc2086622085225315487list_a,Ps: list_P1915399164375025317list_a] :
                ( ( Xc
                  = ( cons_P2654393883304477781list_a @ P5 @ Ps ) )
               => ( ( ( ( ( produc2785537127425143931list_a @ P5 )
                        = X )
                     => ( Y
                        = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ X @ ( Xb @ ( produc1153002459082627517list_a @ P5 ) ) ) @ Ps ) ) )
                    & ( ( ( produc2785537127425143931list_a @ P5 )
                        = ~ X )
                     => ( Y
                        = ( cons_P2654393883304477781list_a @ P5 @ ( map_de2428310187333209296list_a @ X @ Xa @ Xb @ Ps ) ) ) ) )
                 => ~ ( accp_P4204554831414503090list_a @ map_de3558955406781211673list_a @ ( produc291085998517269493list_a @ X @ ( produc8995079174706216494list_a @ Xa @ ( produc6935208604740912679list_a @ Xb @ ( cons_P2654393883304477781list_a @ P5 @ Ps ) ) ) ) ) ) ) ) ) ) ).

% map_default.pelims
thf(fact_453_empty__Shift,axiom,
    ! [Kl: set_list_list_a,K3: list_a] :
      ( ( member_list_list_a @ nil_list_a @ Kl )
     => ( ( member_list_a @ K3 @ ( bNF_Gr4634511371912843295list_a @ Kl @ nil_list_a ) )
       => ( member_list_list_a @ nil_list_a @ ( bNF_Gr7042794125918077091list_a @ Kl @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_454_empty__Shift,axiom,
    ! [Kl: set_list_a,K3: a] :
      ( ( member_list_a @ nil_a @ Kl )
     => ( ( member_a @ K3 @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
       => ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_455_empty__Shift,axiom,
    ! [Kl: set_list_nat,K3: nat] :
      ( ( member_list_nat @ nil_nat @ Kl )
     => ( ( member_nat @ K3 @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_456_Succ__Shift,axiom,
    ! [Kl: set_list_list_a,K3: list_a,Kl2: list_list_a] :
      ( ( bNF_Gr4634511371912843295list_a @ ( bNF_Gr7042794125918077091list_a @ Kl @ K3 ) @ Kl2 )
      = ( bNF_Gr4634511371912843295list_a @ Kl @ ( cons_list_a @ K3 @ Kl2 ) ) ) ).

% Succ_Shift
thf(fact_457_Succ__Shift,axiom,
    ! [Kl: set_list_a,K3: a,Kl2: list_a] :
      ( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl @ K3 ) @ Kl2 )
      = ( bNF_Greatest_Succ_a @ Kl @ ( cons_a @ K3 @ Kl2 ) ) ) ).

% Succ_Shift
thf(fact_458_Succ__Shift,axiom,
    ! [Kl: set_list_nat,K3: nat,Kl2: list_nat] :
      ( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K3 ) @ Kl2 )
      = ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K3 @ Kl2 ) ) ) ).

% Succ_Shift
thf(fact_459_Succ__Shift,axiom,
    ! [Kl: set_list_list_nat,K3: list_nat,Kl2: list_list_nat] :
      ( ( bNF_Gr3053708287304744325st_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl @ K3 ) @ Kl2 )
      = ( bNF_Gr3053708287304744325st_nat @ Kl @ ( cons_list_nat @ K3 @ Kl2 ) ) ) ).

% Succ_Shift
thf(fact_460_Succ__Shift,axiom,
    ! [Kl: set_list_list_list_a,K3: list_list_a,Kl2: list_list_list_a] :
      ( ( bNF_Gr6039074496463509157list_a @ ( bNF_Gr1120979937343206697list_a @ Kl @ K3 ) @ Kl2 )
      = ( bNF_Gr6039074496463509157list_a @ Kl @ ( cons_list_list_a @ K3 @ Kl2 ) ) ) ).

% Succ_Shift
thf(fact_461_map__entry_Opelims,axiom,
    ! [X: $o,Xa: list_list_a > list_list_a,Xb: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( ( map_en7508539461263903803list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6298570550759538669list_a @ map_en288630569058640738list_a @ ( produc2365957476491948720list_a @ X @ ( produc1851355790630076705list_a @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr1488683542974317855list_a )
           => ( ( Y = nil_Pr1488683542974317855list_a )
             => ~ ( accp_P6298570550759538669list_a @ map_en288630569058640738list_a @ ( produc2365957476491948720list_a @ X @ ( produc1851355790630076705list_a @ Xa @ nil_Pr1488683542974317855list_a ) ) ) ) )
         => ~ ! [P5: produc2922778490715282969list_a,Ps: list_P369347992569427103list_a] :
                ( ( Xb
                  = ( cons_P3051609457517836495list_a @ P5 @ Ps ) )
               => ( ( ( ( ( produc3726311466776224373list_a @ P5 )
                        = X )
                     => ( Y
                        = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ X @ ( Xa @ ( produc4969414473911996599list_a @ P5 ) ) ) @ Ps ) ) )
                    & ( ( ( produc3726311466776224373list_a @ P5 )
                        = ~ X )
                     => ( Y
                        = ( cons_P3051609457517836495list_a @ P5 @ ( map_en7508539461263903803list_a @ X @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P6298570550759538669list_a @ map_en288630569058640738list_a @ ( produc2365957476491948720list_a @ X @ ( produc1851355790630076705list_a @ Xa @ ( cons_P3051609457517836495list_a @ P5 @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_462_map__entry_Opelims,axiom,
    ! [X: $o,Xa: list_list_nat > list_list_nat,Xb: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( ( map_en1378505499878274175st_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P283175222203424191st_nat @ map_en5395453035494846872st_nat @ ( produc5157861394213783590st_nat @ X @ ( produc4663047625710595679st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr1938078369830740401st_nat )
           => ( ( Y = nil_Pr1938078369830740401st_nat )
             => ~ ( accp_P283175222203424191st_nat @ map_en5395453035494846872st_nat @ ( produc5157861394213783590st_nat @ X @ ( produc4663047625710595679st_nat @ Xa @ nil_Pr1938078369830740401st_nat ) ) ) ) )
         => ~ ! [P5: produc7974263126369132087st_nat,Ps: list_P3026376796534914631st_nat] :
                ( ( Xb
                  = ( cons_P5527778705151500801st_nat @ P5 @ Ps ) )
               => ( ( ( ( ( produc3357746125391881157st_nat @ P5 )
                        = X )
                     => ( Y
                        = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ X @ ( Xa @ ( produc4024048262274189571st_nat @ P5 ) ) ) @ Ps ) ) )
                    & ( ( ( produc3357746125391881157st_nat @ P5 )
                        = ~ X )
                     => ( Y
                        = ( cons_P5527778705151500801st_nat @ P5 @ ( map_en1378505499878274175st_nat @ X @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P283175222203424191st_nat @ map_en5395453035494846872st_nat @ ( produc5157861394213783590st_nat @ X @ ( produc4663047625710595679st_nat @ Xa @ ( cons_P5527778705151500801st_nat @ P5 @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_463_map__entry_Opelims,axiom,
    ! [X: $o,Xa: list_list_list_a > list_list_list_a,Xb: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( ( map_en7631213772203213633list_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P9132993267457354227list_a @ map_en7930330521816029928list_a @ ( produc4469486809467490998list_a @ X @ ( produc6935208604740912679list_a @ Xa @ Xb ) ) )
       => ( ( ( Xb = nil_Pr7514062926313711781list_a )
           => ( ( Y = nil_Pr7514062926313711781list_a )
             => ~ ( accp_P9132993267457354227list_a @ map_en7930330521816029928list_a @ ( produc4469486809467490998list_a @ X @ ( produc6935208604740912679list_a @ Xa @ nil_Pr7514062926313711781list_a ) ) ) ) )
         => ~ ! [P5: produc2086622085225315487list_a,Ps: list_P1915399164375025317list_a] :
                ( ( Xb
                  = ( cons_P2654393883304477781list_a @ P5 @ Ps ) )
               => ( ( ( ( ( produc2785537127425143931list_a @ P5 )
                        = X )
                     => ( Y
                        = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ X @ ( Xa @ ( produc1153002459082627517list_a @ P5 ) ) ) @ Ps ) ) )
                    & ( ( ( produc2785537127425143931list_a @ P5 )
                        = ~ X )
                     => ( Y
                        = ( cons_P2654393883304477781list_a @ P5 @ ( map_en7631213772203213633list_a @ X @ Xa @ Ps ) ) ) ) )
                 => ~ ( accp_P9132993267457354227list_a @ map_en7930330521816029928list_a @ ( produc4469486809467490998list_a @ X @ ( produc6935208604740912679list_a @ Xa @ ( cons_P2654393883304477781list_a @ P5 @ Ps ) ) ) ) ) ) ) ) ) ).

% map_entry.pelims
thf(fact_464_suffixes__eq__snoc,axiom,
    ! [Ys: list_list_nat,Xs: list_list_list_nat,X: list_list_nat] :
      ( ( ( suffixes_list_nat @ Ys )
        = ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
      = ( ( ( ( Ys = nil_list_nat )
            & ( Xs = nil_list_list_nat ) )
          | ? [Z3: list_nat,Zs2: list_list_nat] :
              ( ( Ys
                = ( cons_list_nat @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffixes_list_nat @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_465_suffixes__eq__snoc,axiom,
    ! [Ys: list_list_list_a,Xs: list_l7815035709764258326list_a,X: list_list_list_a] :
      ( ( ( suffixes_list_list_a @ Ys )
        = ( append6004555433696164139list_a @ Xs @ ( cons_l8577867181958743494list_a @ X @ nil_list_list_list_a ) ) )
      = ( ( ( ( Ys = nil_list_list_a )
            & ( Xs = nil_list_list_list_a ) )
          | ? [Z3: list_list_a,Zs2: list_list_list_a] :
              ( ( Ys
                = ( cons_list_list_a @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffixes_list_list_a @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_466_suffixes__eq__snoc,axiom,
    ! [Ys: list_a,Xs: list_list_a,X: list_a] :
      ( ( ( suffixes_a @ Ys )
        = ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
      = ( ( ( ( Ys = nil_a )
            & ( Xs = nil_list_a ) )
          | ? [Z3: a,Zs2: list_a] :
              ( ( Ys
                = ( cons_a @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffixes_a @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_467_suffixes__eq__snoc,axiom,
    ! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
      ( ( ( suffixes_nat @ Ys )
        = ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( ( ( ( Ys = nil_nat )
            & ( Xs = nil_list_nat ) )
          | ? [Z3: nat,Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffixes_nat @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_468_suffixes__eq__snoc,axiom,
    ! [Ys: list_list_a,Xs: list_list_list_a,X: list_list_a] :
      ( ( ( suffixes_list_a @ Ys )
        = ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = ( ( ( ( Ys = nil_list_a )
            & ( Xs = nil_list_list_a ) )
          | ? [Z3: list_a,Zs2: list_list_a] :
              ( ( Ys
                = ( cons_list_a @ Z3 @ Zs2 ) )
              & ( Xs
                = ( suffixes_list_a @ Zs2 ) ) ) )
        & ( X = Ys ) ) ) ).

% suffixes_eq_snoc
thf(fact_469_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_a > list_list_a,P2: produc2922778490715282969list_a,Ps2: list_P369347992569427103list_a] :
      ( ( map_ra3570251311891529249list_a @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ P2 ) @ ( F2 @ ( produc3726311466776224373list_a @ P2 ) @ ( produc4969414473911996599list_a @ P2 ) ) ) @ ( map_ra3570251311891529249list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_470_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_a > list_list_nat,P2: produc2922778490715282969list_a,Ps2: list_P369347992569427103list_a] :
      ( ( map_ra2045802175598081305st_nat @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ ( produc3726311466776224373list_a @ P2 ) @ ( F2 @ ( produc3726311466776224373list_a @ P2 ) @ ( produc4969414473911996599list_a @ P2 ) ) ) @ ( map_ra2045802175598081305st_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_471_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_a > list_list_list_a,P2: produc2922778490715282969list_a,Ps2: list_P369347992569427103list_a] :
      ( ( map_ra8396789647597898279list_a @ F2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ ( produc3726311466776224373list_a @ P2 ) @ ( F2 @ ( produc3726311466776224373list_a @ P2 ) @ ( produc4969414473911996599list_a @ P2 ) ) ) @ ( map_ra8396789647597898279list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_472_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_nat > list_list_a,P2: produc7974263126369132087st_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( map_ra8120288265412832055list_a @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( produc3357746125391881157st_nat @ P2 ) @ ( F2 @ ( produc3357746125391881157st_nat @ P2 ) @ ( produc4024048262274189571st_nat @ P2 ) ) ) @ ( map_ra8120288265412832055list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_473_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_nat > list_list_nat,P2: produc7974263126369132087st_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( map_ra5394953335731719811st_nat @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ P2 ) @ ( F2 @ ( produc3357746125391881157st_nat @ P2 ) @ ( produc4024048262274189571st_nat @ P2 ) ) ) @ ( map_ra5394953335731719811st_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_474_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_nat > list_list_list_a,P2: produc7974263126369132087st_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( map_ra9120075116080084029list_a @ F2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ ( produc3357746125391881157st_nat @ P2 ) @ ( F2 @ ( produc3357746125391881157st_nat @ P2 ) @ ( produc4024048262274189571st_nat @ P2 ) ) ) @ ( map_ra9120075116080084029list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_475_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_list_a > list_list_a,P2: produc2086622085225315487list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( map_ra6112735818830951707list_a @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( produc2785537127425143931list_a @ P2 ) @ ( F2 @ ( produc2785537127425143931list_a @ P2 ) @ ( produc1153002459082627517list_a @ P2 ) ) ) @ ( map_ra6112735818830951707list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_476_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_list_a > list_list_nat,P2: produc2086622085225315487list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( map_ra3239984559097762719st_nat @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ ( produc2785537127425143931list_a @ P2 ) @ ( F2 @ ( produc2785537127425143931list_a @ P2 ) @ ( produc1153002459082627517list_a @ P2 ) ) ) @ ( map_ra3239984559097762719st_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_477_map__ran__Cons__sel,axiom,
    ! [F2: $o > list_list_list_a > list_list_list_a,P2: produc2086622085225315487list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( map_ra4487667581546282017list_a @ F2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ ( produc2785537127425143931list_a @ P2 ) @ ( F2 @ ( produc2785537127425143931list_a @ P2 ) @ ( produc1153002459082627517list_a @ P2 ) ) ) @ ( map_ra4487667581546282017list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_Cons_sel
thf(fact_478_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_list_a )
         => ( ( Ys = nil_a )
            & ( Zs = nil_a ) ) )
        & ( ( Xss2 != nil_list_a )
         => ? [Xss1: list_list_a,Xs3: list_a,Xs4: list_a,Xss22: list_list_a] :
              ( ( Xss2
                = ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_a @ ( concat_a @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append_a @ Xs4 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_479_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_list_nat )
         => ( ( Ys = nil_nat )
            & ( Zs = nil_nat ) ) )
        & ( ( Xss2 != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs3: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
              ( ( Xss2
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_480_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_list_a,Ys: list_list_a,Zs: list_list_a] :
      ( ( ( concat_list_a @ Xss2 )
        = ( append_list_a @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_list_list_a )
         => ( ( Ys = nil_list_a )
            & ( Zs = nil_list_a ) ) )
        & ( ( Xss2 != nil_list_list_a )
         => ? [Xss1: list_list_list_a,Xs3: list_list_a,Xs4: list_list_a,Xss22: list_list_list_a] :
              ( ( Xss2
                = ( append_list_list_a @ Xss1 @ ( cons_list_list_a @ ( append_list_a @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_list_a @ ( concat_list_a @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append_list_a @ Xs4 @ ( concat_list_a @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_481_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) @ ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) ) @ ( listrel1_list_a @ R ) )
      = ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_482_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_a,X: a,Ys: list_a,Y: a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
      = ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_483_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_484_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_list_nat,X: list_nat,Ys: list_list_nat,Y: list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) ) @ ( listrel1_list_nat @ R ) )
      = ( ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_485_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_list_list_a,X: list_list_a,Ys: list_list_list_a,Y: list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) @ ( append_list_list_a @ Ys @ ( cons_list_list_a @ Y @ nil_list_list_a ) ) ) @ ( listrel1_list_list_a @ R ) )
      = ( ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R ) )
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_486_concat__append,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
      = ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).

% concat_append
thf(fact_487_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_a > list_list_a,K3: $o,V2: list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( map_ra3570251311891529249list_a @ F2 @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra3570251311891529249list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_488_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_a > list_list_nat,K3: $o,V2: list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( map_ra2045802175598081305st_nat @ F2 @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra2045802175598081305st_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_489_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_a > list_list_list_a,K3: $o,V2: list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( map_ra8396789647597898279list_a @ F2 @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra8396789647597898279list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_490_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_nat > list_list_a,K3: $o,V2: list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( map_ra8120288265412832055list_a @ F2 @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra8120288265412832055list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_491_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_nat > list_list_nat,K3: $o,V2: list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( map_ra5394953335731719811st_nat @ F2 @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra5394953335731719811st_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_492_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_nat > list_list_list_a,K3: $o,V2: list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( map_ra9120075116080084029list_a @ F2 @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra9120075116080084029list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_493_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_list_a > list_list_a,K3: $o,V2: list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( map_ra6112735818830951707list_a @ F2 @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra6112735818830951707list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_494_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_list_a > list_list_nat,K3: $o,V2: list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( map_ra3239984559097762719st_nat @ F2 @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra3239984559097762719st_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_495_map__ran__simps_I2_J,axiom,
    ! [F2: $o > list_list_list_a > list_list_list_a,K3: $o,V2: list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( map_ra4487667581546282017list_a @ F2 @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V2 ) @ Ps2 ) )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ ( F2 @ K3 @ V2 ) ) @ ( map_ra4487667581546282017list_a @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_496_Cons__listrel1__Cons,axiom,
    ! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R ) )
      = ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_497_Cons__listrel1__Cons,axiom,
    ! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
      = ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_498_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_499_Cons__listrel1__Cons,axiom,
    ! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( listrel1_list_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_500_Cons__listrel1__Cons,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Ys ) ) @ ( listrel1_list_list_a @ R ) )
      = ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_501_concat_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( concat_a @ ( cons_list_a @ X @ Xs ) )
      = ( append_a @ X @ ( concat_a @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_502_concat_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_503_concat_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( concat_list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append_list_a @ X @ ( concat_list_a @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_504_concat_Osimps_I1_J,axiom,
    ( ( concat_a @ nil_list_a )
    = nil_a ) ).

% concat.simps(1)
thf(fact_505_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_506_listrel1I2,axiom,
    ! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a,X: list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
     => ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ X @ Ys ) ) @ ( listrel1_list_a @ R ) ) ) ).

% listrel1I2
thf(fact_507_listrel1I2,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,X: a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys ) ) @ ( listrel1_a @ R ) ) ) ).

% listrel1I2
thf(fact_508_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ X @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I2
thf(fact_509_listrel1I2,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat,X: list_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R ) )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ X @ Ys ) ) @ ( listrel1_list_nat @ R ) ) ) ).

% listrel1I2
thf(fact_510_listrel1I2,axiom,
    ! [Xs: list_list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a,X: list_list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R ) )
     => ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ X @ Ys ) ) @ ( listrel1_list_list_a @ R ) ) ) ).

% listrel1I2
thf(fact_511_not__Nil__listrel1,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R ) ) ).

% not_Nil_listrel1
thf(fact_512_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_513_not__listrel1__Nil,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R ) ) ).

% not_listrel1_Nil
thf(fact_514_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_515_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_516_ShiftD,axiom,
    ! [Kl2: list_list_a,Kl: set_list_list_a,K3: list_a] :
      ( ( member_list_list_a @ Kl2 @ ( bNF_Gr7042794125918077091list_a @ Kl @ K3 ) )
     => ( member_list_list_a @ ( cons_list_a @ K3 @ Kl2 ) @ Kl ) ) ).

% ShiftD
thf(fact_517_ShiftD,axiom,
    ! [Kl2: list_a,Kl: set_list_a,K3: a] :
      ( ( member_list_a @ Kl2 @ ( bNF_Greatest_Shift_a @ Kl @ K3 ) )
     => ( member_list_a @ ( cons_a @ K3 @ Kl2 ) @ Kl ) ) ).

% ShiftD
thf(fact_518_ShiftD,axiom,
    ! [Kl2: list_nat,Kl: set_list_nat,K3: nat] :
      ( ( member_list_nat @ Kl2 @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K3 ) )
     => ( member_list_nat @ ( cons_nat @ K3 @ Kl2 ) @ Kl ) ) ).

% ShiftD
thf(fact_519_ShiftD,axiom,
    ! [Kl2: list_list_nat,Kl: set_list_list_nat,K3: list_nat] :
      ( ( member_list_list_nat @ Kl2 @ ( bNF_Gr9051742241863529473st_nat @ Kl @ K3 ) )
     => ( member_list_list_nat @ ( cons_list_nat @ K3 @ Kl2 ) @ Kl ) ) ).

% ShiftD
thf(fact_520_ShiftD,axiom,
    ! [Kl2: list_list_list_a,Kl: set_list_list_list_a,K3: list_list_a] :
      ( ( member5342144027231129785list_a @ Kl2 @ ( bNF_Gr1120979937343206697list_a @ Kl @ K3 ) )
     => ( member5342144027231129785list_a @ ( cons_list_list_a @ K3 @ Kl2 ) @ Kl ) ) ).

% ShiftD
thf(fact_521_listrel1I1,axiom,
    ! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
     => ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Xs ) ) @ ( listrel1_list_a @ R ) ) ) ).

% listrel1I1
thf(fact_522_listrel1I1,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R ) ) ) ).

% listrel1I1
thf(fact_523_listrel1I1,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_524_listrel1I1,axiom,
    ! [X: list_nat,Y: list_nat,R: set_Pr3451248702717554689st_nat,Xs: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Xs ) ) @ ( listrel1_list_nat @ R ) ) ) ).

% listrel1I1
thf(fact_525_listrel1I1,axiom,
    ! [X: list_list_a,Y: list_list_a,R: set_Pr5382606609415531783list_a,Xs: list_list_list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R )
     => ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Xs ) ) @ ( listrel1_list_list_a @ R ) ) ) ).

% listrel1I1
thf(fact_526_Cons__listrel1E1,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ Ys ) @ ( listrel1_list_a @ R ) )
     => ( ! [Y3: list_a] :
            ( ( Ys
              = ( cons_list_a @ Y3 @ Xs ) )
           => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y3 ) @ R ) )
       => ~ ! [Zs3: list_list_a] :
              ( ( Ys
                = ( cons_list_a @ X @ Zs3 ) )
             => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs3 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_527_Cons__listrel1E1,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ Ys ) @ ( listrel1_a @ R ) )
     => ( ! [Y3: a] :
            ( ( Ys
              = ( cons_a @ Y3 @ Xs ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R ) )
       => ~ ! [Zs3: list_a] :
              ( ( Ys
                = ( cons_a @ X @ Zs3 ) )
             => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs3 ) @ ( listrel1_a @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_528_Cons__listrel1E1,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y3: nat] :
            ( ( Ys
              = ( cons_nat @ Y3 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
       => ~ ! [Zs3: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Zs3 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs3 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_529_Cons__listrel1E1,axiom,
    ! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ Ys ) @ ( listrel1_list_nat @ R ) )
     => ( ! [Y3: list_nat] :
            ( ( Ys
              = ( cons_list_nat @ Y3 @ Xs ) )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ R ) )
       => ~ ! [Zs3: list_list_nat] :
              ( ( Ys
                = ( cons_list_nat @ X @ Zs3 ) )
             => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Zs3 ) @ ( listrel1_list_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_530_Cons__listrel1E1,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ Ys ) @ ( listrel1_list_list_a @ R ) )
     => ( ! [Y3: list_list_a] :
            ( ( Ys
              = ( cons_list_list_a @ Y3 @ Xs ) )
           => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y3 ) @ R ) )
       => ~ ! [Zs3: list_list_list_a] :
              ( ( Ys
                = ( cons_list_list_a @ X @ Zs3 ) )
             => ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Zs3 ) @ ( listrel1_list_list_a @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_531_Cons__listrel1E2,axiom,
    ! [Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R ) )
     => ( ! [X3: list_a] :
            ( ( Xs
              = ( cons_list_a @ X3 @ Ys ) )
           => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y ) @ R ) )
       => ~ ! [Zs3: list_list_a] :
              ( ( Xs
                = ( cons_list_a @ Y @ Zs3 ) )
             => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Zs3 @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_532_Cons__listrel1E2,axiom,
    ! [Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
     => ( ! [X3: a] :
            ( ( Xs
              = ( cons_a @ X3 @ Ys ) )
           => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R ) )
       => ~ ! [Zs3: list_a] :
              ( ( Xs
                = ( cons_a @ Y @ Zs3 ) )
             => ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs3 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_533_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X3: nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs3: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs3 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs3 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_534_Cons__listrel1E2,axiom,
    ! [Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ ( cons_list_nat @ Y @ Ys ) ) @ ( listrel1_list_nat @ R ) )
     => ( ! [X3: list_nat] :
            ( ( Xs
              = ( cons_list_nat @ X3 @ Ys ) )
           => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs3: list_list_nat] :
              ( ( Xs
                = ( cons_list_nat @ Y @ Zs3 ) )
             => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Zs3 @ Ys ) @ ( listrel1_list_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_535_Cons__listrel1E2,axiom,
    ! [Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ ( cons_list_list_a @ Y @ Ys ) ) @ ( listrel1_list_list_a @ R ) )
     => ( ! [X3: list_list_a] :
            ( ( Xs
              = ( cons_list_list_a @ X3 @ Ys ) )
           => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y ) @ R ) )
       => ~ ! [Zs3: list_list_list_a] :
              ( ( Xs
                = ( cons_list_list_a @ Y @ Zs3 ) )
             => ~ ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Zs3 @ Ys ) @ ( listrel1_list_list_a @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_536_Sublist_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffixes_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% Sublist.suffixes.simps(1)
thf(fact_537_Sublist_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffixes_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% Sublist.suffixes.simps(1)
thf(fact_538_Sublist_Osuffixes_Osimps_I1_J,axiom,
    ( ( suffixes_list_a @ nil_list_a )
    = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).

% Sublist.suffixes.simps(1)
thf(fact_539_listrel1E,axiom,
    ! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
     => ~ ! [X3: list_a,Y3: list_a] :
            ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y3 ) @ R )
           => ! [Us3: list_list_a,Vs2: list_list_a] :
                ( ( Xs
                  = ( append_list_a @ Us3 @ ( cons_list_a @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_list_a @ Us3 @ ( cons_list_a @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_540_listrel1E,axiom,
    ! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
     => ~ ! [X3: a,Y3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y3 ) @ R )
           => ! [Us3: list_a,Vs2: list_a] :
                ( ( Xs
                  = ( append_a @ Us3 @ ( cons_a @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_a @ Us3 @ ( cons_a @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_541_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ! [Us3: list_nat,Vs2: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us3 @ ( cons_nat @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_542_listrel1E,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R ) )
     => ~ ! [X3: list_nat,Y3: list_nat] :
            ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y3 ) @ R )
           => ! [Us3: list_list_nat,Vs2: list_list_nat] :
                ( ( Xs
                  = ( append_list_nat @ Us3 @ ( cons_list_nat @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_list_nat @ Us3 @ ( cons_list_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_543_listrel1E,axiom,
    ! [Xs: list_list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R ) )
     => ~ ! [X3: list_list_a,Y3: list_list_a] :
            ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X3 @ Y3 ) @ R )
           => ! [Us3: list_list_list_a,Vs2: list_list_list_a] :
                ( ( Xs
                  = ( append_list_list_a @ Us3 @ ( cons_list_list_a @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append_list_list_a @ Us3 @ ( cons_list_list_a @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_544_listrel1I,axiom,
    ! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a,Us: list_list_a,Vs: list_list_a,Ys: list_list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
     => ( ( Xs
          = ( append_list_a @ Us @ ( cons_list_a @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_list_a @ Us @ ( cons_list_a @ Y @ Vs ) ) )
         => ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).

% listrel1I
thf(fact_545_listrel1I,axiom,
    ! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
     => ( ( Xs
          = ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).

% listrel1I
thf(fact_546_listrel1I,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
     => ( ( Xs
          = ( append_nat @ Us @ ( cons_nat @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_nat @ Us @ ( cons_nat @ Y @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_547_listrel1I,axiom,
    ! [X: list_nat,Y: list_nat,R: set_Pr3451248702717554689st_nat,Xs: list_list_nat,Us: list_list_nat,Vs: list_list_nat,Ys: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
     => ( ( Xs
          = ( append_list_nat @ Us @ ( cons_list_nat @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_list_nat @ Us @ ( cons_list_nat @ Y @ Vs ) ) )
         => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( listrel1_list_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_548_listrel1I,axiom,
    ! [X: list_list_a,Y: list_list_a,R: set_Pr5382606609415531783list_a,Xs: list_list_list_a,Us: list_list_list_a,Vs: list_list_list_a,Ys: list_list_list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R )
     => ( ( Xs
          = ( append_list_list_a @ Us @ ( cons_list_list_a @ X @ Vs ) ) )
       => ( ( Ys
            = ( append_list_list_a @ Us @ ( cons_list_list_a @ Y @ Vs ) ) )
         => ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( listrel1_list_list_a @ R ) ) ) ) ) ).

% listrel1I
thf(fact_549_concat__eq__appendD,axiom,
    ! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys @ Zs ) )
     => ( ( Xss2 != nil_list_a )
       => ? [Xss12: list_list_a,Xs2: list_a,Xs5: list_a,Xss23: list_list_a] :
            ( ( Xss2
              = ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs5 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_a @ ( concat_a @ Xss12 ) @ Xs2 ) )
            & ( Zs
              = ( append_a @ Xs5 @ ( concat_a @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_550_concat__eq__appendD,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys @ Zs ) )
     => ( ( Xss2 != nil_list_nat )
       => ? [Xss12: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss23: list_list_nat] :
            ( ( Xss2
              = ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
            & ( Zs
              = ( append_nat @ Xs5 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_551_concat__eq__appendD,axiom,
    ! [Xss2: list_list_list_a,Ys: list_list_a,Zs: list_list_a] :
      ( ( ( concat_list_a @ Xss2 )
        = ( append_list_a @ Ys @ Zs ) )
     => ( ( Xss2 != nil_list_list_a )
       => ? [Xss12: list_list_list_a,Xs2: list_list_a,Xs5: list_list_a,Xss23: list_list_list_a] :
            ( ( Xss2
              = ( append_list_list_a @ Xss12 @ ( cons_list_list_a @ ( append_list_a @ Xs2 @ Xs5 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_list_a @ ( concat_list_a @ Xss12 ) @ Xs2 ) )
            & ( Zs
              = ( append_list_a @ Xs5 @ ( concat_list_a @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_552_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( suffixes_list_a @ ( cons_list_a @ X @ Xs ) )
      = ( append_list_list_a @ ( suffixes_list_a @ Xs ) @ ( cons_list_list_a @ ( cons_list_a @ X @ Xs ) @ nil_list_list_a ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_553_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( suffixes_a @ ( cons_a @ X @ Xs ) )
      = ( append_list_a @ ( suffixes_a @ Xs ) @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ nil_list_a ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_554_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
      = ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_555_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( suffixes_list_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append_list_list_nat @ ( suffixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ nil_list_list_nat ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_556_Sublist_Osuffixes_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( suffixes_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append6004555433696164139list_a @ ( suffixes_list_list_a @ Xs ) @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X @ Xs ) @ nil_list_list_list_a ) ) ) ).

% Sublist.suffixes.simps(2)
thf(fact_557_delete__aux_Oelims,axiom,
    ! [X: $o,Xa: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( ( delete8241016168524078360list_a @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr1488683542974317855list_a )
         => ( Y != nil_Pr1488683542974317855list_a ) )
       => ~ ! [K2: $o,V: list_list_a,Xs2: list_P369347992569427103list_a] :
              ( ( Xa
                = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X = ~ K2 )
                   => ( Y
                      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K2 @ V ) @ ( delete8241016168524078360list_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_558_delete__aux_Oelims,axiom,
    ! [X: $o,Xa: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( ( delete7818344518816601698st_nat @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr1938078369830740401st_nat )
         => ( Y != nil_Pr1938078369830740401st_nat ) )
       => ~ ! [K2: $o,V: list_list_nat,Xs2: list_P3026376796534914631st_nat] :
              ( ( Xa
                = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X = ~ K2 )
                   => ( Y
                      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K2 @ V ) @ ( delete7818344518816601698st_nat @ X @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_559_delete__aux_Oelims,axiom,
    ! [X: $o,Xa: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( ( delete3389553227512332702list_a @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr7514062926313711781list_a )
         => ( Y != nil_Pr7514062926313711781list_a ) )
       => ~ ! [K2: $o,V: list_list_list_a,Xs2: list_P1915399164375025317list_a] :
              ( ( Xa
                = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X = ~ K2 )
                   => ( Y
                      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K2 @ V ) @ ( delete3389553227512332702list_a @ X @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_560_delete__aux__eq__Nil__conv,axiom,
    ! [K3: $o,Ts: list_P369347992569427103list_a] :
      ( ( ( delete8241016168524078360list_a @ K3 @ Ts )
        = nil_Pr1488683542974317855list_a )
      = ( ( Ts = nil_Pr1488683542974317855list_a )
        | ? [V3: list_list_a] :
            ( Ts
            = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V3 ) @ nil_Pr1488683542974317855list_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_561_delete__aux__eq__Nil__conv,axiom,
    ! [K3: $o,Ts: list_P3026376796534914631st_nat] :
      ( ( ( delete7818344518816601698st_nat @ K3 @ Ts )
        = nil_Pr1938078369830740401st_nat )
      = ( ( Ts = nil_Pr1938078369830740401st_nat )
        | ? [V3: list_list_nat] :
            ( Ts
            = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V3 ) @ nil_Pr1938078369830740401st_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_562_delete__aux__eq__Nil__conv,axiom,
    ! [K3: $o,Ts: list_P1915399164375025317list_a] :
      ( ( ( delete3389553227512332702list_a @ K3 @ Ts )
        = nil_Pr7514062926313711781list_a )
      = ( ( Ts = nil_Pr7514062926313711781list_a )
        | ? [V3: list_list_list_a] :
            ( Ts
            = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V3 ) @ nil_Pr7514062926313711781list_a ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_563_maps__simps_I1_J,axiom,
    ! [F2: list_a > list_nat,X: list_a,Xs: list_list_a] :
      ( ( maps_list_a_nat @ F2 @ ( cons_list_a @ X @ Xs ) )
      = ( append_nat @ ( F2 @ X ) @ ( maps_list_a_nat @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_564_maps__simps_I1_J,axiom,
    ! [F2: a > list_nat,X: a,Xs: list_a] :
      ( ( maps_a_nat @ F2 @ ( cons_a @ X @ Xs ) )
      = ( append_nat @ ( F2 @ X ) @ ( maps_a_nat @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_565_maps__simps_I1_J,axiom,
    ! [F2: nat > list_nat,X: nat,Xs: list_nat] :
      ( ( maps_nat_nat @ F2 @ ( cons_nat @ X @ Xs ) )
      = ( append_nat @ ( F2 @ X ) @ ( maps_nat_nat @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_566_maps__simps_I1_J,axiom,
    ! [F2: list_nat > list_nat,X: list_nat,Xs: list_list_nat] :
      ( ( maps_list_nat_nat @ F2 @ ( cons_list_nat @ X @ Xs ) )
      = ( append_nat @ ( F2 @ X ) @ ( maps_list_nat_nat @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_567_maps__simps_I1_J,axiom,
    ! [F2: list_list_a > list_nat,X: list_list_a,Xs: list_list_list_a] :
      ( ( maps_list_list_a_nat @ F2 @ ( cons_list_list_a @ X @ Xs ) )
      = ( append_nat @ ( F2 @ X ) @ ( maps_list_list_a_nat @ F2 @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_568_suffixes_Opelims,axiom,
    ! [X: list_list_nat,Y: list_list_list_nat] :
      ( ( ( suffixes_list_nat2 @ X )
        = Y )
     => ( ( accp_list_list_nat @ suffix9047829294884260173st_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y
                = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) )
             => ~ ( accp_list_list_nat @ suffix9047829294884260173st_nat @ nil_list_nat ) ) )
         => ~ ! [X3: list_nat,Xs2: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_list_nat @ ( suffixes_list_nat2 @ Xs2 ) @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ nil_list_list_nat ) ) )
                 => ~ ( accp_list_list_nat @ suffix9047829294884260173st_nat @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_569_suffixes_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_l7815035709764258326list_a] :
      ( ( ( suffixes_list_list_a2 @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ suffix3201455351484461277list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y
                = ( cons_l8577867181958743494list_a @ nil_list_list_a @ nil_list_list_list_a ) )
             => ~ ( accp_l3666201008454774169list_a @ suffix3201455351484461277list_a @ nil_list_list_a ) ) )
         => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append6004555433696164139list_a @ ( suffixes_list_list_a2 @ Xs2 ) @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ nil_list_list_list_a ) ) )
                 => ~ ( accp_l3666201008454774169list_a @ suffix3201455351484461277list_a @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_570_suffixes_Opelims,axiom,
    ! [X: list_a,Y: list_list_a] :
      ( ( ( suffixes_a2 @ X )
        = Y )
     => ( ( accp_list_a @ suffixes_rel_a @ X )
       => ( ( ( X = nil_a )
           => ( ( Y
                = ( cons_list_a @ nil_a @ nil_list_a ) )
             => ~ ( accp_list_a @ suffixes_rel_a @ nil_a ) ) )
         => ~ ! [X3: a,Xs2: list_a] :
                ( ( X
                  = ( cons_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_a @ ( suffixes_a2 @ Xs2 ) @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ nil_list_a ) ) )
                 => ~ ( accp_list_a @ suffixes_rel_a @ ( cons_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_571_suffixes_Opelims,axiom,
    ! [X: list_nat,Y: list_list_nat] :
      ( ( ( suffixes_nat2 @ X )
        = Y )
     => ( ( accp_list_nat @ suffixes_rel_nat @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y
                = ( cons_list_nat @ nil_nat @ nil_list_nat ) )
             => ~ ( accp_list_nat @ suffixes_rel_nat @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_nat @ ( suffixes_nat2 @ Xs2 ) @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ nil_list_nat ) ) )
                 => ~ ( accp_list_nat @ suffixes_rel_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_572_suffixes_Opelims,axiom,
    ! [X: list_list_a,Y: list_list_list_a] :
      ( ( ( suffixes_list_a2 @ X )
        = Y )
     => ( ( accp_list_list_a @ suffixes_rel_list_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y
                = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) )
             => ~ ( accp_list_list_a @ suffixes_rel_list_a @ nil_list_a ) ) )
         => ~ ! [X3: list_a,Xs2: list_list_a] :
                ( ( X
                  = ( cons_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_list_a @ ( suffixes_list_a2 @ Xs2 ) @ ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ nil_list_list_a ) ) )
                 => ~ ( accp_list_list_a @ suffixes_rel_list_a @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% suffixes.pelims
thf(fact_573_lexord__append__left__rightI,axiom,
    ! [A: list_a,B: list_a,R: set_Pr4048851178543822343list_a,U: list_list_a,X: list_list_a,Y: list_list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B ) @ R )
     => ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ U @ ( cons_list_a @ A @ X ) ) @ ( append_list_a @ U @ ( cons_list_a @ B @ Y ) ) ) @ ( lexord_list_a @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_574_lexord__append__left__rightI,axiom,
    ! [A: a,B: a,R: set_Product_prod_a_a,U: list_a,X: list_a,Y: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ U @ ( cons_a @ A @ X ) ) @ ( append_a @ U @ ( cons_a @ B @ Y ) ) ) @ ( lexord_a @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_575_lexord__append__left__rightI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X: list_nat,Y: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_576_lexord__append__left__rightI,axiom,
    ! [A: list_nat,B: list_nat,R: set_Pr3451248702717554689st_nat,U: list_list_nat,X: list_list_nat,Y: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( append_list_nat @ U @ ( cons_list_nat @ A @ X ) ) @ ( append_list_nat @ U @ ( cons_list_nat @ B @ Y ) ) ) @ ( lexord_list_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_577_lexord__append__left__rightI,axiom,
    ! [A: list_list_a,B: list_list_a,R: set_Pr5382606609415531783list_a,U: list_list_list_a,X: list_list_list_a,Y: list_list_list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A @ B ) @ R )
     => ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( append_list_list_a @ U @ ( cons_list_list_a @ A @ X ) ) @ ( append_list_list_a @ U @ ( cons_list_list_a @ B @ Y ) ) ) @ ( lexord_list_list_a @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_578_list__ex1__simps_I1_J,axiom,
    ! [P: a > $o] :
      ~ ( list_ex1_a @ P @ nil_a ) ).

% list_ex1_simps(1)
thf(fact_579_list__ex1__simps_I1_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex1_nat @ P @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_580_lexord__Nil__left,axiom,
    ! [Y: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Y ) @ ( lexord_list_a @ R ) )
      = ( ? [A4: list_a,X4: list_list_a] :
            ( Y
            = ( cons_list_a @ A4 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_581_lexord__Nil__left,axiom,
    ! [Y: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Y ) @ ( lexord_a @ R ) )
      = ( ? [A4: a,X4: list_a] :
            ( Y
            = ( cons_a @ A4 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_582_lexord__Nil__left,axiom,
    ! [Y: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R ) )
      = ( ? [A4: nat,X4: list_nat] :
            ( Y
            = ( cons_nat @ A4 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_583_lexord__Nil__left,axiom,
    ! [Y: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ nil_list_nat @ Y ) @ ( lexord_list_nat @ R ) )
      = ( ? [A4: list_nat,X4: list_list_nat] :
            ( Y
            = ( cons_list_nat @ A4 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_584_lexord__Nil__left,axiom,
    ! [Y: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ nil_list_list_a @ Y ) @ ( lexord_list_list_a @ R ) )
      = ( ? [A4: list_list_a,X4: list_list_list_a] :
            ( Y
            = ( cons_list_list_a @ A4 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_585_lexord__cons__cons,axiom,
    ! [A: list_a,X: list_list_a,B: list_a,Y: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ A @ X ) @ ( cons_list_a @ B @ Y ) ) @ ( lexord_list_a @ R ) )
      = ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B ) @ R )
        | ( ( A = B )
          & ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ ( lexord_list_a @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_586_lexord__cons__cons,axiom,
    ! [A: a,X: list_a,B: a,Y: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ A @ X ) @ ( cons_a @ B @ Y ) ) @ ( lexord_a @ R ) )
      = ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R )
        | ( ( A = B )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( lexord_a @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_587_lexord__cons__cons,axiom,
    ! [A: nat,X: list_nat,B: nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_588_lexord__cons__cons,axiom,
    ! [A: list_nat,X: list_list_nat,B: list_nat,Y: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ A @ X ) @ ( cons_list_nat @ B @ Y ) ) @ ( lexord_list_nat @ R ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ Y ) @ ( lexord_list_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_589_lexord__cons__cons,axiom,
    ! [A: list_list_a,X: list_list_list_a,B: list_list_a,Y: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ A @ X ) @ ( cons_list_list_a @ B @ Y ) ) @ ( lexord_list_list_a @ R ) )
      = ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A @ B ) @ R )
        | ( ( A = B )
          & ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ X @ Y ) @ ( lexord_list_list_a @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_590_lexord__Nil__right,axiom,
    ! [X: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ nil_a ) @ ( lexord_a @ R ) ) ).

% lexord_Nil_right
thf(fact_591_lexord__Nil__right,axiom,
    ! [X: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ nil_nat ) @ ( lexord_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_592_lexord__append__leftI,axiom,
    ! [U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat,X: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_leftI
thf(fact_593_maps__simps_I2_J,axiom,
    ! [F2: a > list_a] :
      ( ( maps_a_a @ F2 @ nil_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_594_maps__simps_I2_J,axiom,
    ! [F2: a > list_nat] :
      ( ( maps_a_nat @ F2 @ nil_a )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_595_maps__simps_I2_J,axiom,
    ! [F2: nat > list_a] :
      ( ( maps_nat_a @ F2 @ nil_nat )
      = nil_a ) ).

% maps_simps(2)
thf(fact_596_maps__simps_I2_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( maps_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_597_delete__aux_Osimps_I2_J,axiom,
    ! [K3: $o,K4: $o,V2: list_list_a,Xs: list_P369347992569427103list_a] :
      ( ( ( K3 = K4 )
       => ( ( delete8241016168524078360list_a @ K3 @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K4 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 = ~ K4 )
       => ( ( delete8241016168524078360list_a @ K3 @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K4 @ V2 ) @ Xs ) )
          = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K4 @ V2 ) @ ( delete8241016168524078360list_a @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_598_delete__aux_Osimps_I2_J,axiom,
    ! [K3: $o,K4: $o,V2: list_list_nat,Xs: list_P3026376796534914631st_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete7818344518816601698st_nat @ K3 @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K4 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 = ~ K4 )
       => ( ( delete7818344518816601698st_nat @ K3 @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K4 @ V2 ) @ Xs ) )
          = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K4 @ V2 ) @ ( delete7818344518816601698st_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_599_delete__aux_Osimps_I2_J,axiom,
    ! [K3: $o,K4: $o,V2: list_list_list_a,Xs: list_P1915399164375025317list_a] :
      ( ( ( K3 = K4 )
       => ( ( delete3389553227512332702list_a @ K3 @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K4 @ V2 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 = ~ K4 )
       => ( ( delete3389553227512332702list_a @ K3 @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K4 @ V2 ) @ Xs ) )
          = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K4 @ V2 ) @ ( delete3389553227512332702list_a @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_600_lexord__append__leftD,axiom,
    ! [X: list_nat,U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X @ U ) @ ( append_nat @ X @ V2 ) ) @ ( lexord_nat @ R ) )
     => ( ! [A3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_601_lexord__append__rightI,axiom,
    ! [Y: list_list_a,X: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ? [B5: list_a,Z4: list_list_a] :
          ( Y
          = ( cons_list_a @ B5 @ Z4 ) )
     => ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ ( append_list_a @ X @ Y ) ) @ ( lexord_list_a @ R ) ) ) ).

% lexord_append_rightI
thf(fact_602_lexord__append__rightI,axiom,
    ! [Y: list_a,X: list_a,R: set_Product_prod_a_a] :
      ( ? [B5: a,Z4: list_a] :
          ( Y
          = ( cons_a @ B5 @ Z4 ) )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ ( append_a @ X @ Y ) ) @ ( lexord_a @ R ) ) ) ).

% lexord_append_rightI
thf(fact_603_lexord__append__rightI,axiom,
    ! [Y: list_nat,X: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ? [B5: nat,Z4: list_nat] :
          ( Y
          = ( cons_nat @ B5 @ Z4 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ ( append_nat @ X @ Y ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_604_lexord__append__rightI,axiom,
    ! [Y: list_list_nat,X: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ? [B5: list_nat,Z4: list_list_nat] :
          ( Y
          = ( cons_list_nat @ B5 @ Z4 ) )
     => ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ X @ ( append_list_nat @ X @ Y ) ) @ ( lexord_list_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_605_lexord__append__rightI,axiom,
    ! [Y: list_list_list_a,X: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ? [B5: list_list_a,Z4: list_list_list_a] :
          ( Y
          = ( cons_list_list_a @ B5 @ Z4 ) )
     => ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ X @ ( append_list_list_a @ X @ Y ) ) @ ( lexord_list_list_a @ R ) ) ) ).

% lexord_append_rightI
thf(fact_606_delete__aux_Opelims,axiom,
    ! [X: $o,Xa: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( ( delete8241016168524078360list_a @ X @ Xa )
        = Y )
     => ( ( accp_P4373874884700329527list_a @ delete8886510592334333637list_a @ ( produc1700513956625826974list_a @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr1488683542974317855list_a )
           => ( ( Y = nil_Pr1488683542974317855list_a )
             => ~ ( accp_P4373874884700329527list_a @ delete8886510592334333637list_a @ ( produc1700513956625826974list_a @ X @ nil_Pr1488683542974317855list_a ) ) ) )
         => ~ ! [K2: $o,V: list_list_a,Xs2: list_P369347992569427103list_a] :
                ( ( Xa
                  = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X = K2 )
                     => ( Y = Xs2 ) )
                    & ( ( X = ~ K2 )
                     => ( Y
                        = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K2 @ V ) @ ( delete8241016168524078360list_a @ X @ Xs2 ) ) ) ) )
                 => ~ ( accp_P4373874884700329527list_a @ delete8886510592334333637list_a @ ( produc1700513956625826974list_a @ X @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_607_delete__aux_Opelims,axiom,
    ! [X: $o,Xa: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( ( delete7818344518816601698st_nat @ X @ Xa )
        = Y )
     => ( ( accp_P8060215013527529413st_nat @ delete4210044761207720821st_nat @ ( produc3028385585695882376st_nat @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr1938078369830740401st_nat )
           => ( ( Y = nil_Pr1938078369830740401st_nat )
             => ~ ( accp_P8060215013527529413st_nat @ delete4210044761207720821st_nat @ ( produc3028385585695882376st_nat @ X @ nil_Pr1938078369830740401st_nat ) ) ) )
         => ~ ! [K2: $o,V: list_list_nat,Xs2: list_P3026376796534914631st_nat] :
                ( ( Xa
                  = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X = K2 )
                     => ( Y = Xs2 ) )
                    & ( ( X = ~ K2 )
                     => ( Y
                        = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K2 @ V ) @ ( delete7818344518816601698st_nat @ X @ Xs2 ) ) ) ) )
                 => ~ ( accp_P8060215013527529413st_nat @ delete4210044761207720821st_nat @ ( produc3028385585695882376st_nat @ X @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_608_delete__aux_Opelims,axiom,
    ! [X: $o,Xa: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( ( delete3389553227512332702list_a @ X @ Xa )
        = Y )
     => ( ( accp_P8713446083613076157list_a @ delete1279353588773826763list_a @ ( produc4612989676349016868list_a @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr7514062926313711781list_a )
           => ( ( Y = nil_Pr7514062926313711781list_a )
             => ~ ( accp_P8713446083613076157list_a @ delete1279353588773826763list_a @ ( produc4612989676349016868list_a @ X @ nil_Pr7514062926313711781list_a ) ) ) )
         => ~ ! [K2: $o,V: list_list_list_a,Xs2: list_P1915399164375025317list_a] :
                ( ( Xa
                  = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X = K2 )
                     => ( Y = Xs2 ) )
                    & ( ( X = ~ K2 )
                     => ( Y
                        = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K2 @ V ) @ ( delete3389553227512332702list_a @ X @ Xs2 ) ) ) ) )
                 => ~ ( accp_P8713446083613076157list_a @ delete1279353588773826763list_a @ ( produc4612989676349016868list_a @ X @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_609_sublists_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( sublists_list_a @ ( cons_list_a @ X @ Xs ) )
      = ( append_list_list_a @ ( sublists_list_a @ Xs ) @ ( map_li8713736314956022724list_a @ ( cons_list_a @ X ) @ ( prefixes_list_a @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_610_sublists_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( sublists_a @ ( cons_a @ X @ Xs ) )
      = ( append_list_a @ ( sublists_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_611_sublists_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
      = ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_612_sublists_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( sublists_list_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append_list_list_nat @ ( sublists_list_nat @ Xs ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_613_sublists_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( sublists_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append6004555433696164139list_a @ ( sublists_list_list_a @ Xs ) @ ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X ) @ ( prefixes_list_list_a @ Xs ) ) ) ) ).

% sublists.simps(2)
thf(fact_614_lexord__same__pref__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lexord_nat @ R ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R ) )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_615_lexord__same__pref__iff,axiom,
    ! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lexord_list_a @ R ) )
      = ( ? [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
            & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ X4 ) @ R ) )
        | ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lexord_list_a @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_616_butlast__snoc,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( butlast_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_617_butlast__snoc,axiom,
    ! [Xs: list_a,X: a] :
      ( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_618_butlast__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_619_butlast__snoc,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( butlast_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_620_butlast__snoc,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( butlast_list_list_a @ ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_621_lex__append__leftD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_622_lex__append__left__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_623_list_Omap__disc__iff,axiom,
    ! [F2: a > a,A: list_a] :
      ( ( ( map_a_a @ F2 @ A )
        = nil_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_624_list_Omap__disc__iff,axiom,
    ! [F2: nat > a,A: list_nat] :
      ( ( ( map_nat_a @ F2 @ A )
        = nil_a )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_625_list_Omap__disc__iff,axiom,
    ! [F2: a > nat,A: list_a] :
      ( ( ( map_a_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_626_list_Omap__disc__iff,axiom,
    ! [F2: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_627_Nil__is__map__conv,axiom,
    ! [F2: a > a,Xs: list_a] :
      ( ( nil_a
        = ( map_a_a @ F2 @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_628_Nil__is__map__conv,axiom,
    ! [F2: nat > a,Xs: list_nat] :
      ( ( nil_a
        = ( map_nat_a @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_629_Nil__is__map__conv,axiom,
    ! [F2: a > nat,Xs: list_a] :
      ( ( nil_nat
        = ( map_a_nat @ F2 @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_630_Nil__is__map__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_631_map__is__Nil__conv,axiom,
    ! [F2: a > a,Xs: list_a] :
      ( ( ( map_a_a @ F2 @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_632_map__is__Nil__conv,axiom,
    ! [F2: nat > a,Xs: list_nat] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = nil_a )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_633_map__is__Nil__conv,axiom,
    ! [F2: a > nat,Xs: list_a] :
      ( ( ( map_a_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_634_map__is__Nil__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_635_map__eq__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) ) ) ) ).

% map_eq_conv
thf(fact_636_map__append,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F2 @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ F2 @ Ys ) ) ) ).

% map_append
thf(fact_637_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = nil_nat )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
           => ( X4 = nil_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_638_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_list_a] :
      ( ( ( concat_a @ Xss2 )
        = nil_a )
      = ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xss2 ) )
           => ( X4 = nil_a ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_639_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_list_nat] :
      ( ( nil_nat
        = ( concat_nat @ Xss2 ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
           => ( X4 = nil_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_640_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_list_a] :
      ( ( nil_a
        = ( concat_a @ Xss2 ) )
      = ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xss2 ) )
           => ( X4 = nil_a ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_641_in__set__insert,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ( insert_list_a @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_642_not__in__set__insert,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ( insert_list_a @ X @ Xs )
        = ( cons_list_a @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_643_not__in__set__insert,axiom,
    ! [X: a,Xs: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ( insert_a @ X @ Xs )
        = ( cons_a @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_644_not__in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = ( cons_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_645_not__in__set__insert,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ( ( insert_list_nat @ X @ Xs )
        = ( cons_list_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_646_not__in__set__insert,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ( ( insert_list_list_a @ X @ Xs )
        = ( cons_list_list_a @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_647_in__set__butlast__appendI,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
        | ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
     => ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_648_in__set__butlast__appendI,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
      ( ( ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
        | ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Ys ) ) ) )
     => ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ ( append_list_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_649_prefixes__butlast__is__prefix,axiom,
    ! [Xs: list_a] : ( member_list_a @ ( butlast_a @ Xs ) @ ( set_list_a2 @ ( prefixes_a @ Xs ) ) ) ).

% prefixes_butlast_is_prefix
thf(fact_650_map__set,axiom,
    ! [X: nat,Xs: list_nat,F2: nat > nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( member_nat @ ( F2 @ X ) @ ( set_nat2 @ ( map_nat_nat @ F2 @ Xs ) ) ) ) ).

% map_set
thf(fact_651_map__set,axiom,
    ! [X: list_a,Xs: list_list_a,F2: list_a > list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( member_list_a @ ( F2 @ X ) @ ( set_list_a2 @ ( map_list_a_list_a @ F2 @ Xs ) ) ) ) ).

% map_set
thf(fact_652_set__map__elem,axiom,
    ! [Y: nat,F2: nat > nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( map_nat_nat @ F2 @ Xs ) ) )
     => ~ ! [X3: nat] :
            ( ( Y
              = ( F2 @ X3 ) )
           => ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_653_set__map__elem,axiom,
    ! [Y: list_a,F2: list_a > list_a,Xs: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( map_list_a_list_a @ F2 @ Xs ) ) )
     => ~ ! [X3: list_a] :
            ( ( Y
              = ( F2 @ X3 ) )
           => ~ ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_map_elem
thf(fact_654_list__map__set__prop,axiom,
    ! [X: nat,F2: nat > nat,Xs: list_nat,P: nat > $o] :
      ( ( member_nat @ X @ ( set_nat2 @ ( map_nat_nat @ F2 @ Xs ) ) )
     => ( ! [Y3: nat] : ( P @ ( F2 @ Y3 ) )
       => ( P @ X ) ) ) ).

% list_map_set_prop
thf(fact_655_list__map__source__elem,axiom,
    ! [X: nat,F2: nat > nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( map_nat_nat @ F2 @ Xs ) ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
          & ( X
            = ( F2 @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_656_list__map__source__elem,axiom,
    ! [X: list_a,F2: list_a > list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ ( map_list_a_list_a @ F2 @ Xs ) ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
          & ( X
            = ( F2 @ X3 ) ) ) ) ).

% list_map_source_elem
thf(fact_657_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F2: nat > nat,G: nat > nat] :
      ( ( X = Ya )
     => ( ! [Z5: nat] :
            ( ( member_nat @ Z5 @ ( set_nat2 @ Ya ) )
           => ( ( F2 @ Z5 )
              = ( G @ Z5 ) ) )
       => ( ( map_nat_nat @ F2 @ X )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_658_list_Omap__cong0,axiom,
    ! [X: list_nat,F2: nat > nat,G: nat > nat] :
      ( ! [Z5: nat] :
          ( ( member_nat @ Z5 @ ( set_nat2 @ X ) )
         => ( ( F2 @ Z5 )
            = ( G @ Z5 ) ) )
     => ( ( map_nat_nat @ F2 @ X )
        = ( map_nat_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_659_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F2: nat > nat,Fa: nat > nat] :
      ( ! [Z5: nat,Za: nat] :
          ( ( member_nat @ Z5 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F2 @ Z5 )
                = ( Fa @ Za ) )
             => ( Z5 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F2 @ X )
          = ( map_nat_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_660_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F2: nat > nat] :
      ( ! [Z5: nat] :
          ( ( member_nat @ Z5 @ ( set_nat2 @ T ) )
         => ( ( F2 @ Z5 )
            = Z5 ) )
     => ( ( map_nat_nat @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_661_list_Omap__ident__strong,axiom,
    ! [T: list_list_a,F2: list_a > list_a] :
      ( ! [Z5: list_a] :
          ( ( member_list_a @ Z5 @ ( set_list_a2 @ T ) )
         => ( ( F2 @ Z5 )
            = Z5 ) )
     => ( ( map_list_a_list_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_662_map__ext,axiom,
    ! [Xs: list_nat,F2: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F2 @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_663_map__idI,axiom,
    ! [Xs: list_nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F2 @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_664_map__idI,axiom,
    ! [Xs: list_list_a,F2: list_a > list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
         => ( ( F2 @ X3 )
            = X3 ) )
     => ( ( map_list_a_list_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_665_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F2: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F2 @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_nat @ F2 @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_666_ex__map__conv,axiom,
    ! [Ys: list_nat,F2: nat > nat] :
      ( ( ? [Xs3: list_nat] :
            ( Ys
            = ( map_nat_nat @ F2 @ Xs3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ? [Y5: nat] :
                ( X4
                = ( F2 @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_667_map__butlast,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( map_nat_nat @ F2 @ ( butlast_nat @ Xs ) )
      = ( butlast_nat @ ( map_nat_nat @ F2 @ Xs ) ) ) ).

% map_butlast
thf(fact_668_in__set__butlastD,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ ( butlast_list_a @ Xs ) ) )
     => ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_669_map__concat,axiom,
    ! [F2: nat > nat,Xs: list_list_nat] :
      ( ( map_nat_nat @ F2 @ ( concat_nat @ Xs ) )
      = ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F2 ) @ Xs ) ) ) ).

% map_concat
thf(fact_670_set__concat__map__elem,axiom,
    ! [X: list_a,F2: list_a > list_list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ ( concat_list_a @ ( map_li5729356230488778442list_a @ F2 @ Xs ) ) ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
          & ( member_list_a @ X @ ( set_list_a2 @ ( F2 @ X3 ) ) ) ) ) ).

% set_concat_map_elem
thf(fact_671_set__concat__elem,axiom,
    ! [X: a,Xss2: list_list_a] :
      ( ( member_a @ X @ ( set_a2 @ ( concat_a @ Xss2 ) ) )
     => ~ ! [Xs2: list_a] :
            ( ( member_list_a @ Xs2 @ ( set_list_a2 @ Xss2 ) )
           => ~ ( member_a @ X @ ( set_a2 @ Xs2 ) ) ) ) ).

% set_concat_elem
thf(fact_672_set__concat__elem,axiom,
    ! [X: list_a,Xss2: list_list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ ( concat_list_a @ Xss2 ) ) )
     => ~ ! [Xs2: list_list_a] :
            ( ( member_list_list_a @ Xs2 @ ( set_list_list_a2 @ Xss2 ) )
           => ~ ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) ) ) ) ).

% set_concat_elem
thf(fact_673_concat__map__elem,axiom,
    ! [Y: list_a,F2: list_a > list_list_a,Xs: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( concat_list_a @ ( map_li5729356230488778442list_a @ F2 @ Xs ) ) ) )
     => ~ ! [X3: list_a] :
            ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
           => ~ ( member_list_a @ Y @ ( set_list_a2 @ ( F2 @ X3 ) ) ) ) ) ).

% concat_map_elem
thf(fact_674_set__map__subset,axiom,
    ! [X: list_a,Xs: list_list_a,T: list_a,F2: list_a > list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ( member_list_a @ T @ ( set_list_a2 @ ( map_list_a_list_a @ F2 @ ( cons_list_a @ X @ nil_list_a ) ) ) )
       => ( member_list_a @ T @ ( set_list_a2 @ ( map_list_a_list_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_675_set__map__subset,axiom,
    ! [X: a,Xs: list_a,T: list_a,F2: a > list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ( member_list_a @ T @ ( set_list_a2 @ ( map_a_list_a @ F2 @ ( cons_a @ X @ nil_a ) ) ) )
       => ( member_list_a @ T @ ( set_list_a2 @ ( map_a_list_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_676_set__map__subset,axiom,
    ! [X: nat,Xs: list_nat,T: nat,F2: nat > nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( member_nat @ T @ ( set_nat2 @ ( map_nat_nat @ F2 @ ( cons_nat @ X @ nil_nat ) ) ) )
       => ( member_nat @ T @ ( set_nat2 @ ( map_nat_nat @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_677_set__map__subset,axiom,
    ! [X: nat,Xs: list_nat,T: list_a,F2: nat > list_a] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( member_list_a @ T @ ( set_list_a2 @ ( map_nat_list_a @ F2 @ ( cons_nat @ X @ nil_nat ) ) ) )
       => ( member_list_a @ T @ ( set_list_a2 @ ( map_nat_list_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_678_set__map__subset,axiom,
    ! [X: list_nat,Xs: list_list_nat,T: list_a,F2: list_nat > list_a] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ( ( member_list_a @ T @ ( set_list_a2 @ ( map_list_nat_list_a @ F2 @ ( cons_list_nat @ X @ nil_list_nat ) ) ) )
       => ( member_list_a @ T @ ( set_list_a2 @ ( map_list_nat_list_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_679_set__map__subset,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,T: list_a,F2: list_list_a > list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ( ( member_list_a @ T @ ( set_list_a2 @ ( map_li1646474281249396926list_a @ F2 @ ( cons_list_list_a @ X @ nil_list_list_a ) ) ) )
       => ( member_list_a @ T @ ( set_list_a2 @ ( map_li1646474281249396926list_a @ F2 @ Xs ) ) ) ) ) ).

% set_map_subset
thf(fact_680_list_Osimps_I9_J,axiom,
    ! [F2: a > a,X21: a,X22: list_a] :
      ( ( map_a_a @ F2 @ ( cons_a @ X21 @ X22 ) )
      = ( cons_a @ ( F2 @ X21 ) @ ( map_a_a @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_681_list_Osimps_I9_J,axiom,
    ! [F2: a > nat,X21: a,X22: list_a] :
      ( ( map_a_nat @ F2 @ ( cons_a @ X21 @ X22 ) )
      = ( cons_nat @ ( F2 @ X21 ) @ ( map_a_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_682_list_Osimps_I9_J,axiom,
    ! [F2: nat > a,X21: nat,X22: list_nat] :
      ( ( map_nat_a @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_a @ ( F2 @ X21 ) @ ( map_nat_a @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_683_list_Osimps_I9_J,axiom,
    ! [F2: nat > nat,X21: nat,X22: list_nat] :
      ( ( map_nat_nat @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F2 @ X21 ) @ ( map_nat_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_684_list_Osimps_I9_J,axiom,
    ! [F2: list_a > a,X21: list_a,X22: list_list_a] :
      ( ( map_list_a_a @ F2 @ ( cons_list_a @ X21 @ X22 ) )
      = ( cons_a @ ( F2 @ X21 ) @ ( map_list_a_a @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_685_list_Osimps_I9_J,axiom,
    ! [F2: list_a > nat,X21: list_a,X22: list_list_a] :
      ( ( map_list_a_nat @ F2 @ ( cons_list_a @ X21 @ X22 ) )
      = ( cons_nat @ ( F2 @ X21 ) @ ( map_list_a_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_686_list_Osimps_I9_J,axiom,
    ! [F2: a > list_a,X21: a,X22: list_a] :
      ( ( map_a_list_a @ F2 @ ( cons_a @ X21 @ X22 ) )
      = ( cons_list_a @ ( F2 @ X21 ) @ ( map_a_list_a @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_687_list_Osimps_I9_J,axiom,
    ! [F2: a > list_nat,X21: a,X22: list_a] :
      ( ( map_a_list_nat @ F2 @ ( cons_a @ X21 @ X22 ) )
      = ( cons_list_nat @ ( F2 @ X21 ) @ ( map_a_list_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_688_list_Osimps_I9_J,axiom,
    ! [F2: nat > list_a,X21: nat,X22: list_nat] :
      ( ( map_nat_list_a @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_list_a @ ( F2 @ X21 ) @ ( map_nat_list_a @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_689_list_Osimps_I9_J,axiom,
    ! [F2: nat > list_nat,X21: nat,X22: list_nat] :
      ( ( map_nat_list_nat @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_list_nat @ ( F2 @ X21 ) @ ( map_nat_list_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_690_Cons__eq__map__D,axiom,
    ! [X: a,Xs: list_a,F2: a > a,Ys: list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_a_a @ F2 @ Ys ) )
     => ? [Z5: a,Zs3: list_a] :
          ( ( Ys
            = ( cons_a @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_a_a @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_691_Cons__eq__map__D,axiom,
    ! [X: a,Xs: list_a,F2: nat > a,Ys: list_nat] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_nat_a @ F2 @ Ys ) )
     => ? [Z5: nat,Zs3: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_nat_a @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_692_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F2: a > nat,Ys: list_a] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_a_nat @ F2 @ Ys ) )
     => ? [Z5: a,Zs3: list_a] :
          ( ( Ys
            = ( cons_a @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_a_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_693_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F2 @ Ys ) )
     => ? [Z5: nat,Zs3: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_nat_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_694_Cons__eq__map__D,axiom,
    ! [X: list_a,Xs: list_list_a,F2: a > list_a,Ys: list_a] :
      ( ( ( cons_list_a @ X @ Xs )
        = ( map_a_list_a @ F2 @ Ys ) )
     => ? [Z5: a,Zs3: list_a] :
          ( ( Ys
            = ( cons_a @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_a_list_a @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_695_Cons__eq__map__D,axiom,
    ! [X: list_a,Xs: list_list_a,F2: nat > list_a,Ys: list_nat] :
      ( ( ( cons_list_a @ X @ Xs )
        = ( map_nat_list_a @ F2 @ Ys ) )
     => ? [Z5: nat,Zs3: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_nat_list_a @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_696_Cons__eq__map__D,axiom,
    ! [X: a,Xs: list_a,F2: list_a > a,Ys: list_list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_list_a_a @ F2 @ Ys ) )
     => ? [Z5: list_a,Zs3: list_list_a] :
          ( ( Ys
            = ( cons_list_a @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_list_a_a @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_697_Cons__eq__map__D,axiom,
    ! [X: a,Xs: list_a,F2: list_nat > a,Ys: list_list_nat] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_list_nat_a @ F2 @ Ys ) )
     => ? [Z5: list_nat,Zs3: list_list_nat] :
          ( ( Ys
            = ( cons_list_nat @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_list_nat_a @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_698_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F2: list_a > nat,Ys: list_list_a] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_list_a_nat @ F2 @ Ys ) )
     => ? [Z5: list_a,Zs3: list_list_a] :
          ( ( Ys
            = ( cons_list_a @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_list_a_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_699_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F2: list_nat > nat,Ys: list_list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_list_nat_nat @ F2 @ Ys ) )
     => ? [Z5: list_nat,Zs3: list_list_nat] :
          ( ( Ys
            = ( cons_list_nat @ Z5 @ Zs3 ) )
          & ( X
            = ( F2 @ Z5 ) )
          & ( Xs
            = ( map_list_nat_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_700_map__eq__Cons__D,axiom,
    ! [F2: a > a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( ( map_a_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
     => ? [Z5: a,Zs3: list_a] :
          ( ( Xs
            = ( cons_a @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_a_a @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_701_map__eq__Cons__D,axiom,
    ! [F2: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
     => ? [Z5: nat,Zs3: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_nat_a @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_702_map__eq__Cons__D,axiom,
    ! [F2: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z5: a,Zs3: list_a] :
          ( ( Xs
            = ( cons_a @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_a_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_703_map__eq__Cons__D,axiom,
    ! [F2: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z5: nat,Zs3: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_nat_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_704_map__eq__Cons__D,axiom,
    ! [F2: a > list_a,Xs: list_a,Y: list_a,Ys: list_list_a] :
      ( ( ( map_a_list_a @ F2 @ Xs )
        = ( cons_list_a @ Y @ Ys ) )
     => ? [Z5: a,Zs3: list_a] :
          ( ( Xs
            = ( cons_a @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_a_list_a @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_705_map__eq__Cons__D,axiom,
    ! [F2: nat > list_a,Xs: list_nat,Y: list_a,Ys: list_list_a] :
      ( ( ( map_nat_list_a @ F2 @ Xs )
        = ( cons_list_a @ Y @ Ys ) )
     => ? [Z5: nat,Zs3: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_nat_list_a @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_706_map__eq__Cons__D,axiom,
    ! [F2: list_a > a,Xs: list_list_a,Y: a,Ys: list_a] :
      ( ( ( map_list_a_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
     => ? [Z5: list_a,Zs3: list_list_a] :
          ( ( Xs
            = ( cons_list_a @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_list_a_a @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_707_map__eq__Cons__D,axiom,
    ! [F2: list_nat > a,Xs: list_list_nat,Y: a,Ys: list_a] :
      ( ( ( map_list_nat_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
     => ? [Z5: list_nat,Zs3: list_list_nat] :
          ( ( Xs
            = ( cons_list_nat @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_list_nat_a @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_708_map__eq__Cons__D,axiom,
    ! [F2: list_a > nat,Xs: list_list_a,Y: nat,Ys: list_nat] :
      ( ( ( map_list_a_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z5: list_a,Zs3: list_list_a] :
          ( ( Xs
            = ( cons_list_a @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_list_a_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_709_map__eq__Cons__D,axiom,
    ! [F2: list_nat > nat,Xs: list_list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_list_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z5: list_nat,Zs3: list_list_nat] :
          ( ( Xs
            = ( cons_list_nat @ Z5 @ Zs3 ) )
          & ( ( F2 @ Z5 )
            = Y )
          & ( ( map_list_nat_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_710_Cons__eq__map__conv,axiom,
    ! [X: a,Xs: list_a,F2: a > a,Ys: list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_a_a @ F2 @ Ys ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_a_a @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_711_Cons__eq__map__conv,axiom,
    ! [X: a,Xs: list_a,F2: nat > a,Ys: list_nat] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_nat_a @ F2 @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_nat_a @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_712_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F2: a > nat,Ys: list_a] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_a_nat @ F2 @ Ys ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_a_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_713_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F2 @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_nat_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_714_Cons__eq__map__conv,axiom,
    ! [X: list_a,Xs: list_list_a,F2: a > list_a,Ys: list_a] :
      ( ( ( cons_list_a @ X @ Xs )
        = ( map_a_list_a @ F2 @ Ys ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_a_list_a @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_715_Cons__eq__map__conv,axiom,
    ! [X: list_a,Xs: list_list_a,F2: nat > list_a,Ys: list_nat] :
      ( ( ( cons_list_a @ X @ Xs )
        = ( map_nat_list_a @ F2 @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_nat_list_a @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_716_Cons__eq__map__conv,axiom,
    ! [X: a,Xs: list_a,F2: list_a > a,Ys: list_list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_list_a_a @ F2 @ Ys ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Ys
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_list_a_a @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_717_Cons__eq__map__conv,axiom,
    ! [X: a,Xs: list_a,F2: list_nat > a,Ys: list_list_nat] :
      ( ( ( cons_a @ X @ Xs )
        = ( map_list_nat_a @ F2 @ Ys ) )
      = ( ? [Z3: list_nat,Zs2: list_list_nat] :
            ( ( Ys
              = ( cons_list_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_list_nat_a @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_718_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F2: list_a > nat,Ys: list_list_a] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_list_a_nat @ F2 @ Ys ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Ys
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_list_a_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_719_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F2: list_nat > nat,Ys: list_list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_list_nat_nat @ F2 @ Ys ) )
      = ( ? [Z3: list_nat,Zs2: list_list_nat] :
            ( ( Ys
              = ( cons_list_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_list_nat_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_720_map__eq__Cons__conv,axiom,
    ! [F2: a > a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( ( map_a_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_a_a @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_721_map__eq__Cons__conv,axiom,
    ! [F2: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_nat_a @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_722_map__eq__Cons__conv,axiom,
    ! [F2: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_a_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_723_map__eq__Cons__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_nat_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_724_map__eq__Cons__conv,axiom,
    ! [F2: a > list_a,Xs: list_a,Y: list_a,Ys: list_list_a] :
      ( ( ( map_a_list_a @ F2 @ Xs )
        = ( cons_list_a @ Y @ Ys ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_a_list_a @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_725_map__eq__Cons__conv,axiom,
    ! [F2: nat > list_a,Xs: list_nat,Y: list_a,Ys: list_list_a] :
      ( ( ( map_nat_list_a @ F2 @ Xs )
        = ( cons_list_a @ Y @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_nat_list_a @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_726_map__eq__Cons__conv,axiom,
    ! [F2: list_a > a,Xs: list_list_a,Y: a,Ys: list_a] :
      ( ( ( map_list_a_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_list_a_a @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_727_map__eq__Cons__conv,axiom,
    ! [F2: list_nat > a,Xs: list_list_nat,Y: a,Ys: list_a] :
      ( ( ( map_list_nat_a @ F2 @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( ? [Z3: list_nat,Zs2: list_list_nat] :
            ( ( Xs
              = ( cons_list_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_list_nat_a @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_728_map__eq__Cons__conv,axiom,
    ! [F2: list_a > nat,Xs: list_list_a,Y: nat,Ys: list_nat] :
      ( ( ( map_list_a_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_list_a_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_729_map__eq__Cons__conv,axiom,
    ! [F2: list_nat > nat,Xs: list_list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_list_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z3: list_nat,Zs2: list_list_nat] :
            ( ( Xs
              = ( cons_list_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_list_nat_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_730_list_Osimps_I8_J,axiom,
    ! [F2: a > a] :
      ( ( map_a_a @ F2 @ nil_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_731_list_Osimps_I8_J,axiom,
    ! [F2: a > nat] :
      ( ( map_a_nat @ F2 @ nil_a )
      = nil_nat ) ).

% list.simps(8)
thf(fact_732_list_Osimps_I8_J,axiom,
    ! [F2: nat > a] :
      ( ( map_nat_a @ F2 @ nil_nat )
      = nil_a ) ).

% list.simps(8)
thf(fact_733_list_Osimps_I8_J,axiom,
    ! [F2: nat > nat] :
      ( ( map_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_734_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F2: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F2 @ Xs ) )
      = ( ? [Us2: list_nat,Vs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_nat_nat @ F2 @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F2 @ Vs3 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_735_map__eq__append__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_nat_nat @ F2 @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F2 @ Vs3 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_736_list_Oset__intros_I2_J,axiom,
    ! [Y: list_a,X22: list_list_a,X21: list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
     => ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_737_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X22: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X22 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_738_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_739_list_Oset__intros_I2_J,axiom,
    ! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
      ( ( member_list_nat @ Y @ ( set_list_nat2 @ X22 ) )
     => ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_740_list_Oset__intros_I2_J,axiom,
    ! [Y: list_list_a,X22: list_list_list_a,X21: list_list_a] :
      ( ( member_list_list_a @ Y @ ( set_list_list_a2 @ X22 ) )
     => ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_741_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_742_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_743_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_744_list_Oset__intros_I1_J,axiom,
    ! [X21: list_nat,X22: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_745_list_Oset__intros_I1_J,axiom,
    ! [X21: list_list_a,X22: list_list_list_a] : ( member_list_list_a @ X21 @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_746_list_Oset__cases,axiom,
    ! [E: list_a,A: list_list_a] :
      ( ( member_list_a @ E @ ( set_list_a2 @ A ) )
     => ( ! [Z22: list_list_a] :
            ( A
           != ( cons_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_a,Z22: list_list_a] :
              ( ( A
                = ( cons_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_a @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_747_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_748_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_749_list_Oset__cases,axiom,
    ! [E: list_nat,A: list_list_nat] :
      ( ( member_list_nat @ E @ ( set_list_nat2 @ A ) )
     => ( ! [Z22: list_list_nat] :
            ( A
           != ( cons_list_nat @ E @ Z22 ) )
       => ~ ! [Z1: list_nat,Z22: list_list_nat] :
              ( ( A
                = ( cons_list_nat @ Z1 @ Z22 ) )
             => ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_750_list_Oset__cases,axiom,
    ! [E: list_list_a,A: list_list_list_a] :
      ( ( member_list_list_a @ E @ ( set_list_list_a2 @ A ) )
     => ( ! [Z22: list_list_list_a] :
            ( A
           != ( cons_list_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_list_a,Z22: list_list_list_a] :
              ( ( A
                = ( cons_list_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_list_a @ E @ ( set_list_list_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_751_set__ConsD,axiom,
    ! [Y: list_a,X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_752_set__ConsD,axiom,
    ! [Y: a,X: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_753_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_754_set__ConsD,axiom,
    ! [Y: list_nat,X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_list_nat @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_755_set__ConsD,axiom,
    ! [Y: list_list_a,X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_list_list_a @ Y @ ( set_list_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_756_list__set__sym,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( set_nat2 @ ( append_nat @ X @ Y ) )
      = ( set_nat2 @ ( append_nat @ Y @ X ) ) ) ).

% list_set_sym
thf(fact_757_list__set__sym,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( set_list_a2 @ ( append_list_a @ X @ Y ) )
      = ( set_list_a2 @ ( append_list_a @ Y @ X ) ) ) ).

% list_set_sym
thf(fact_758_list__prefix__elem,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( append_nat @ Xs @ Ys ) ) )
     => ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
       => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_759_list__prefix__elem,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ Xs @ Ys ) ) )
     => ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
       => ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ) ).

% list_prefix_elem
thf(fact_760_list__concat__non__elem,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
       => ~ ( member_nat @ X @ ( set_nat2 @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_761_list__concat__non__elem,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
      ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
       => ~ ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ Xs @ Ys ) ) ) ) ) ).

% list_concat_non_elem
thf(fact_762_butlast_Osimps_I1_J,axiom,
    ( ( butlast_a @ nil_a )
    = nil_a ) ).

% butlast.simps(1)
thf(fact_763_butlast_Osimps_I1_J,axiom,
    ( ( butlast_nat @ nil_nat )
    = nil_nat ) ).

% butlast.simps(1)
thf(fact_764_prefixes__prefixes,axiom,
    ! [Ys: list_a,Xs: list_a,Zs: list_a] :
      ( ( member_list_a @ Ys @ ( set_list_a2 @ ( prefixes_a @ Xs ) ) )
     => ( ( member_list_a @ Zs @ ( set_list_a2 @ ( prefixes_a @ Xs ) ) )
       => ( ( member_list_a @ Ys @ ( set_list_a2 @ ( prefixes_a @ Zs ) ) )
          | ( member_list_a @ Zs @ ( set_list_a2 @ ( prefixes_a @ Ys ) ) ) ) ) ) ).

% prefixes_prefixes
thf(fact_765_subseqs__refl,axiom,
    ! [Xs: list_a] : ( member_list_a @ Xs @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ).

% subseqs_refl
thf(fact_766_list__ex1__iff,axiom,
    ( list_ex1_list_a
    = ( ^ [P6: list_a > $o,Xs3: list_list_a] :
        ? [X4: list_a] :
          ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs3 ) )
          & ( P6 @ X4 )
          & ! [Y5: list_a] :
              ( ( ( member_list_a @ Y5 @ ( set_list_a2 @ Xs3 ) )
                & ( P6 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_767_split__list,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ? [Ys2: list_list_a,Zs3: list_list_a] :
          ( Xs
          = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_768_split__list,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs3: list_a] :
          ( Xs
          = ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_769_split__list,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ? [Ys2: list_nat,Zs3: list_nat] :
          ( Xs
          = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_770_split__list,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ? [Ys2: list_list_nat,Zs3: list_list_nat] :
          ( Xs
          = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_771_split__list,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ? [Ys2: list_list_list_a,Zs3: list_list_list_a] :
          ( Xs
          = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_772_split__list__last,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ? [Ys2: list_list_a,Zs3: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs3 ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_773_split__list__last,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs3: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_774_split__list__last,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ? [Ys2: list_nat,Zs3: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
          & ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_775_split__list__last,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ? [Ys2: list_list_nat,Zs3: list_list_nat] :
          ( ( Xs
            = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) )
          & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_776_split__list__last,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ? [Ys2: list_list_list_a,Zs3: list_list_list_a] :
          ( ( Xs
            = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs3 ) ) )
          & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_777_split__list__prop,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ? [X5: list_a] :
          ( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_a,X3: list_a] :
          ( ? [Zs3: list_list_a] :
              ( Xs
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_778_split__list__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_a,X3: a] :
          ( ? [Zs3: list_a] :
              ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_779_split__list__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_nat,X3: nat] :
          ( ? [Zs3: list_nat] :
              ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_780_split__list__prop,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_nat,X3: list_nat] :
          ( ? [Zs3: list_list_nat] :
              ( Xs
              = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_781_split__list__prop,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ? [X5: list_list_a] :
          ( ( member_list_list_a @ X5 @ ( set_list_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_list_a,X3: list_list_a] :
          ( ? [Zs3: list_list_list_a] :
              ( Xs
              = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_782_split__list__first,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ? [Ys2: list_list_a,Zs3: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs3 ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_783_split__list__first,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys2: list_a,Zs3: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs3 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_784_split__list__first,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ? [Ys2: list_nat,Zs3: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
          & ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_785_split__list__first,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ? [Ys2: list_list_nat,Zs3: list_list_nat] :
          ( ( Xs
            = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) )
          & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_786_split__list__first,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ? [Ys2: list_list_list_a,Zs3: list_list_list_a] :
          ( ( Xs
            = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs3 ) ) )
          & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys2 ) ) ) ) ).

% split_list_first
thf(fact_787_split__list__propE,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ? [X5: list_a] :
          ( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_a,X3: list_a] :
            ( ? [Zs3: list_list_a] :
                ( Xs
                = ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_788_split__list__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_a,X3: a] :
            ( ? [Zs3: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_789_split__list__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_nat,X3: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_790_split__list__propE,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_nat,X3: list_nat] :
            ( ? [Zs3: list_list_nat] :
                ( Xs
                = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_791_split__list__propE,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ? [X5: list_list_a] :
          ( ( member_list_list_a @ X5 @ ( set_list_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_list_a,X3: list_list_a] :
            ( ? [Zs3: list_list_list_a] :
                ( Xs
                = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X3 @ Zs3 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_792_append__Cons__eq__iff,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs6: list_list_a,Ys5: list_list_a] :
      ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
       => ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) )
            = ( append_list_a @ Xs6 @ ( cons_list_a @ X @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_793_append__Cons__eq__iff,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Xs6: list_a,Ys5: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
            = ( append_a @ Xs6 @ ( cons_a @ X @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_794_append__Cons__eq__iff,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Xs6: list_nat,Ys5: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
            = ( append_nat @ Xs6 @ ( cons_nat @ X @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_795_append__Cons__eq__iff,axiom,
    ! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs6: list_list_nat,Ys5: list_list_nat] :
      ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
     => ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
       => ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys ) )
            = ( append_list_nat @ Xs6 @ ( cons_list_nat @ X @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_796_append__Cons__eq__iff,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,Xs6: list_list_list_a,Ys5: list_list_list_a] :
      ( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys ) )
       => ( ( ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ Ys ) )
            = ( append_list_list_a @ Xs6 @ ( cons_list_list_a @ X @ Ys5 ) ) )
          = ( ( Xs = Xs6 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_797_in__set__conv__decomp,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys3: list_list_a,Zs2: list_list_a] :
            ( Xs
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_798_in__set__conv__decomp,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( Xs
            = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_799_in__set__conv__decomp,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys3: list_nat,Zs2: list_nat] :
            ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_800_in__set__conv__decomp,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
      = ( ? [Ys3: list_list_nat,Zs2: list_list_nat] :
            ( Xs
            = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_801_in__set__conv__decomp,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
      = ( ? [Ys3: list_list_list_a,Zs2: list_list_list_a] :
            ( Xs
            = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_802_split__list__last__prop,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ? [X5: list_a] :
          ( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_a,X3: list_a,Zs3: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_a] :
              ( ( member_list_a @ Xa2 @ ( set_list_a2 @ Zs3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_803_split__list__last__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_a,X3: a,Zs3: list_a] :
          ( ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: a] :
              ( ( member_a @ Xa2 @ ( set_a2 @ Zs3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_804_split__list__last__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_nat,X3: nat,Zs3: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_805_split__list__last__prop,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
          ( ( Xs
            = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_nat] :
              ( ( member_list_nat @ Xa2 @ ( set_list_nat2 @ Zs3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_806_split__list__last__prop,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ? [X5: list_list_a] :
          ( ( member_list_list_a @ X5 @ ( set_list_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_list_a,X3: list_list_a,Zs3: list_list_list_a] :
          ( ( Xs
            = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_list_a] :
              ( ( member_list_list_a @ Xa2 @ ( set_list_list_a2 @ Zs3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_807_split__list__first__prop,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ? [X5: list_a] :
          ( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_a,X3: list_a] :
          ( ? [Zs3: list_list_a] :
              ( Xs
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_a] :
              ( ( member_list_a @ Xa2 @ ( set_list_a2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_808_split__list__first__prop,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_a,X3: a] :
          ( ? [Zs3: list_a] :
              ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: a] :
              ( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_809_split__list__first__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_nat,X3: nat] :
          ( ? [Zs3: list_nat] :
              ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_810_split__list__first__prop,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_nat,X3: list_nat] :
          ( ? [Zs3: list_list_nat] :
              ( Xs
              = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_nat] :
              ( ( member_list_nat @ Xa2 @ ( set_list_nat2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_811_split__list__first__prop,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ? [X5: list_list_a] :
          ( ( member_list_list_a @ X5 @ ( set_list_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ? [Ys2: list_list_list_a,X3: list_list_a] :
          ( ? [Zs3: list_list_list_a] :
              ( Xs
              = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X3 @ Zs3 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_list_a] :
              ( ( member_list_list_a @ Xa2 @ ( set_list_list_a2 @ Ys2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_812_split__list__last__propE,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ? [X5: list_a] :
          ( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_a,X3: list_a,Zs3: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_a] :
                    ( ( member_list_a @ Xa2 @ ( set_list_a2 @ Zs3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_813_split__list__last__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_a,X3: a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ ( set_a2 @ Zs3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_814_split__list__last__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_nat,X3: nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_815_split__list__last__propE,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
            ( ( Xs
              = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_nat] :
                    ( ( member_list_nat @ Xa2 @ ( set_list_nat2 @ Zs3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_816_split__list__last__propE,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ? [X5: list_list_a] :
          ( ( member_list_list_a @ X5 @ ( set_list_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_list_a,X3: list_list_a,Zs3: list_list_list_a] :
            ( ( Xs
              = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_list_a] :
                    ( ( member_list_list_a @ Xa2 @ ( set_list_list_a2 @ Zs3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_817_split__list__first__propE,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ? [X5: list_a] :
          ( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_a,X3: list_a] :
            ( ? [Zs3: list_list_a] :
                ( Xs
                = ( append_list_a @ Ys2 @ ( cons_list_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_a] :
                    ( ( member_list_a @ Xa2 @ ( set_list_a2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_818_split__list__first__propE,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_a,X3: a] :
            ( ? [Zs3: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_819_split__list__first__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_nat,X3: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_820_split__list__first__propE,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ? [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_nat,X3: list_nat] :
            ( ? [Zs3: list_list_nat] :
                ( Xs
                = ( append_list_nat @ Ys2 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_nat] :
                    ( ( member_list_nat @ Xa2 @ ( set_list_nat2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_821_split__list__first__propE,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ? [X5: list_list_a] :
          ( ( member_list_list_a @ X5 @ ( set_list_list_a2 @ Xs ) )
          & ( P @ X5 ) )
     => ~ ! [Ys2: list_list_list_a,X3: list_list_a] :
            ( ? [Zs3: list_list_list_a] :
                ( Xs
                = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X3 @ Zs3 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_list_a] :
                    ( ( member_list_list_a @ Xa2 @ ( set_list_list_a2 @ Ys2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_822_in__set__conv__decomp__last,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys3: list_list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_823_in__set__conv__decomp__last,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_824_in__set__conv__decomp__last,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys3: list_nat,Zs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_825_in__set__conv__decomp__last,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
      = ( ? [Ys3: list_list_nat,Zs2: list_list_nat] :
            ( ( Xs
              = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs2 ) ) )
            & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_826_in__set__conv__decomp__last,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
      = ( ? [Ys3: list_list_list_a,Zs2: list_list_list_a] :
            ( ( Xs
              = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_827_in__set__conv__decomp__first,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys3: list_list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_828_in__set__conv__decomp__first,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys3: list_a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_829_in__set__conv__decomp__first,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys3: list_nat,Zs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_830_in__set__conv__decomp__first,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
      = ( ? [Ys3: list_list_nat,Zs2: list_list_nat] :
            ( ( Xs
              = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs2 ) ) )
            & ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_831_in__set__conv__decomp__first,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
      = ( ? [Ys3: list_list_list_a,Zs2: list_list_list_a] :
            ( ( Xs
              = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_832_split__list__last__prop__iff,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ( ? [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_list_a,X4: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: list_a] :
                ( ( member_list_a @ Y5 @ ( set_list_a2 @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_833_split__list__last__prop__iff,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_a,X4: a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: a] :
                ( ( member_a @ Y5 @ ( set_a2 @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_834_split__list__last__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_nat,X4: nat,Zs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( set_nat2 @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_835_split__list__last__prop__iff,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ( ? [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_list_nat,X4: list_nat,Zs2: list_list_nat] :
            ( ( Xs
              = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: list_nat] :
                ( ( member_list_nat @ Y5 @ ( set_list_nat2 @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_836_split__list__last__prop__iff,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ( ? [X4: list_list_a] :
            ( ( member_list_list_a @ X4 @ ( set_list_list_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_list_list_a,X4: list_list_a,Zs2: list_list_list_a] :
            ( ( Xs
              = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: list_list_a] :
                ( ( member_list_list_a @ Y5 @ ( set_list_list_a2 @ Zs2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_837_split__list__first__prop__iff,axiom,
    ! [Xs: list_list_a,P: list_a > $o] :
      ( ( ? [X4: list_a] :
            ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_list_a,X4: list_a] :
            ( ? [Zs2: list_list_a] :
                ( Xs
                = ( append_list_a @ Ys3 @ ( cons_list_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: list_a] :
                ( ( member_list_a @ Y5 @ ( set_list_a2 @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_838_split__list__first__prop__iff,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_a,X4: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys3 @ ( cons_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: a] :
                ( ( member_a @ Y5 @ ( set_a2 @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_839_split__list__first__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_nat,X4: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs
                = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( set_nat2 @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_840_split__list__first__prop__iff,axiom,
    ! [Xs: list_list_nat,P: list_nat > $o] :
      ( ( ? [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_list_nat,X4: list_nat] :
            ( ? [Zs2: list_list_nat] :
                ( Xs
                = ( append_list_nat @ Ys3 @ ( cons_list_nat @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: list_nat] :
                ( ( member_list_nat @ Y5 @ ( set_list_nat2 @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_841_split__list__first__prop__iff,axiom,
    ! [Xs: list_list_list_a,P: list_list_a > $o] :
      ( ( ? [X4: list_list_a] :
            ( ( member_list_list_a @ X4 @ ( set_list_list_a2 @ Xs ) )
            & ( P @ X4 ) ) )
      = ( ? [Ys3: list_list_list_a,X4: list_list_a] :
            ( ? [Zs2: list_list_list_a] :
                ( Xs
                = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X4 @ Zs2 ) ) )
            & ( P @ X4 )
            & ! [Y5: list_list_a] :
                ( ( member_list_list_a @ Y5 @ ( set_list_list_a2 @ Ys3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_842_prefixes__Cons_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
      ( ( member_list_list_a @ ( cons_list_a @ X @ Xs ) @ ( set_list_list_a2 @ ( prefixes_list_a @ ( cons_list_a @ Y @ Ys ) ) ) )
     => ( member_list_list_a @ Xs @ ( set_list_list_a2 @ ( prefixes_list_a @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_843_prefixes__Cons_I2_J,axiom,
    ! [X: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( member_list_a @ ( cons_a @ X @ Xs ) @ ( set_list_a2 @ ( prefixes_a @ ( cons_a @ Y @ Ys ) ) ) )
     => ( member_list_a @ Xs @ ( set_list_a2 @ ( prefixes_a @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_844_prefixes__Cons_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ X @ Xs ) @ ( set_list_nat2 @ ( prefixes_nat @ ( cons_nat @ Y @ Ys ) ) ) )
     => ( member_list_nat @ Xs @ ( set_list_nat2 @ ( prefixes_nat @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_845_prefixes__Cons_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ ( set_list_list_nat2 @ ( prefixes_list_nat @ ( cons_list_nat @ Y @ Ys ) ) ) )
     => ( member_list_list_nat @ Xs @ ( set_list_list_nat2 @ ( prefixes_list_nat @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_846_prefixes__Cons_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a] :
      ( ( member5342144027231129785list_a @ ( cons_list_list_a @ X @ Xs ) @ ( set_list_list_list_a2 @ ( prefixes_list_list_a @ ( cons_list_list_a @ Y @ Ys ) ) ) )
     => ( member5342144027231129785list_a @ Xs @ ( set_list_list_list_a2 @ ( prefixes_list_list_a @ Ys ) ) ) ) ).

% prefixes_Cons(2)
thf(fact_847_prefixes__Cons_I1_J,axiom,
    ! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
      ( ( member_list_list_a @ ( cons_list_a @ X @ Xs ) @ ( set_list_list_a2 @ ( prefixes_list_a @ ( cons_list_a @ Y @ Ys ) ) ) )
     => ( X = Y ) ) ).

% prefixes_Cons(1)
thf(fact_848_prefixes__Cons_I1_J,axiom,
    ! [X: a,Xs: list_a,Y: a,Ys: list_a] :
      ( ( member_list_a @ ( cons_a @ X @ Xs ) @ ( set_list_a2 @ ( prefixes_a @ ( cons_a @ Y @ Ys ) ) ) )
     => ( X = Y ) ) ).

% prefixes_Cons(1)
thf(fact_849_prefixes__Cons_I1_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ X @ Xs ) @ ( set_list_nat2 @ ( prefixes_nat @ ( cons_nat @ Y @ Ys ) ) ) )
     => ( X = Y ) ) ).

% prefixes_Cons(1)
thf(fact_850_prefixes__Cons_I1_J,axiom,
    ! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ ( set_list_list_nat2 @ ( prefixes_list_nat @ ( cons_list_nat @ Y @ Ys ) ) ) )
     => ( X = Y ) ) ).

% prefixes_Cons(1)
thf(fact_851_prefixes__Cons_I1_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a] :
      ( ( member5342144027231129785list_a @ ( cons_list_list_a @ X @ Xs ) @ ( set_list_list_list_a2 @ ( prefixes_list_list_a @ ( cons_list_list_a @ Y @ Ys ) ) ) )
     => ( X = Y ) ) ).

% prefixes_Cons(1)
thf(fact_852_prefixes__set__Nil,axiom,
    ! [Xs: list_nat] : ( member_list_nat @ nil_nat @ ( set_list_nat2 @ ( prefixes_nat @ Xs ) ) ) ).

% prefixes_set_Nil
thf(fact_853_prefixes__set__Nil,axiom,
    ! [Xs: list_a] : ( member_list_a @ nil_a @ ( set_list_a2 @ ( prefixes_a @ Xs ) ) ) ).

% prefixes_set_Nil
thf(fact_854_prefixes__prepend,axiom,
    ! [Xs6: list_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member_list_nat @ Xs6 @ ( set_list_nat2 @ ( prefixes_nat @ Xs ) ) )
     => ( member_list_nat @ ( append_nat @ Ys @ Xs6 ) @ ( set_list_nat2 @ ( prefixes_nat @ ( append_nat @ Ys @ Xs ) ) ) ) ) ).

% prefixes_prepend
thf(fact_855_prefixes__prepend,axiom,
    ! [Xs6: list_a,Xs: list_a,Ys: list_a] :
      ( ( member_list_a @ Xs6 @ ( set_list_a2 @ ( prefixes_a @ Xs ) ) )
     => ( member_list_a @ ( append_a @ Ys @ Xs6 ) @ ( set_list_a2 @ ( prefixes_a @ ( append_a @ Ys @ Xs ) ) ) ) ) ).

% prefixes_prepend
thf(fact_856_prefixes__set__ob,axiom,
    ! [Xs: list_nat,Xss2: list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( prefixes_nat @ Xss2 ) ) )
     => ~ ! [Xs5: list_nat] :
            ( Xss2
           != ( append_nat @ Xs @ Xs5 ) ) ) ).

% prefixes_set_ob
thf(fact_857_prefixes__set__ob,axiom,
    ! [Xs: list_a,Xss2: list_a] :
      ( ( member_list_a @ Xs @ ( set_list_a2 @ ( prefixes_a @ Xss2 ) ) )
     => ~ ! [Xs5: list_a] :
            ( Xss2
           != ( append_a @ Xs @ Xs5 ) ) ) ).

% prefixes_set_ob
thf(fact_858_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( ( Xs = nil_list_a )
       => ( ( butlast_list_a @ ( cons_list_a @ X @ Xs ) )
          = nil_list_a ) )
      & ( ( Xs != nil_list_a )
       => ( ( butlast_list_a @ ( cons_list_a @ X @ Xs ) )
          = ( cons_list_a @ X @ ( butlast_list_a @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_859_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_a,X: a] :
      ( ( ( Xs = nil_a )
       => ( ( butlast_a @ ( cons_a @ X @ Xs ) )
          = nil_a ) )
      & ( ( Xs != nil_a )
       => ( ( butlast_a @ ( cons_a @ X @ Xs ) )
          = ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_860_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
          = nil_nat ) )
      & ( ( Xs != nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_861_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( ( Xs = nil_list_nat )
       => ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs ) )
          = nil_list_nat ) )
      & ( ( Xs != nil_list_nat )
       => ( ( butlast_list_nat @ ( cons_list_nat @ X @ Xs ) )
          = ( cons_list_nat @ X @ ( butlast_list_nat @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_862_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( ( Xs = nil_list_list_a )
       => ( ( butlast_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
          = nil_list_list_a ) )
      & ( ( Xs != nil_list_list_a )
       => ( ( butlast_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
          = ( cons_list_list_a @ X @ ( butlast_list_list_a @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_863_butlast__append,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( ( Ys = nil_a )
       => ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
          = ( butlast_a @ Xs ) ) )
      & ( ( Ys != nil_a )
       => ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
          = ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_864_butlast__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
          = ( butlast_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
          = ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_865_Cons__in__subseqsD,axiom,
    ! [Y: list_a,Ys: list_list_a,Xs: list_list_a] :
      ( ( member_list_list_a @ ( cons_list_a @ Y @ Ys ) @ ( set_list_list_a2 @ ( subseqs_list_a @ Xs ) ) )
     => ( member_list_list_a @ Ys @ ( set_list_list_a2 @ ( subseqs_list_a @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_866_Cons__in__subseqsD,axiom,
    ! [Y: a,Ys: list_a,Xs: list_a] :
      ( ( member_list_a @ ( cons_a @ Y @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
     => ( member_list_a @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_867_Cons__in__subseqsD,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_868_Cons__in__subseqsD,axiom,
    ! [Y: list_nat,Ys: list_list_nat,Xs: list_list_nat] :
      ( ( member_list_list_nat @ ( cons_list_nat @ Y @ Ys ) @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs ) ) )
     => ( member_list_list_nat @ Ys @ ( set_list_list_nat2 @ ( subseqs_list_nat @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_869_Cons__in__subseqsD,axiom,
    ! [Y: list_list_a,Ys: list_list_list_a,Xs: list_list_list_a] :
      ( ( member5342144027231129785list_a @ ( cons_list_list_a @ Y @ Ys ) @ ( set_list_list_list_a2 @ ( subseqs_list_list_a @ Xs ) ) )
     => ( member5342144027231129785list_a @ Ys @ ( set_list_list_list_a2 @ ( subseqs_list_list_a @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_870_List_Oinsert__def,axiom,
    ( insert_list_a
    = ( ^ [X4: list_a,Xs3: list_list_a] : ( if_list_list_a @ ( member_list_a @ X4 @ ( set_list_a2 @ Xs3 ) ) @ Xs3 @ ( cons_list_a @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_871_List_Oinsert__def,axiom,
    ( insert_a
    = ( ^ [X4: a,Xs3: list_a] : ( if_list_a @ ( member_a @ X4 @ ( set_a2 @ Xs3 ) ) @ Xs3 @ ( cons_a @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_872_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X4: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X4 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_873_List_Oinsert__def,axiom,
    ( insert_list_nat
    = ( ^ [X4: list_nat,Xs3: list_list_nat] : ( if_list_list_nat @ ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_list_nat @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_874_List_Oinsert__def,axiom,
    ( insert_list_list_a
    = ( ^ [X4: list_list_a,Xs3: list_list_list_a] : ( if_list_list_list_a @ ( member_list_list_a @ X4 @ ( set_list_list_a2 @ Xs3 ) ) @ Xs3 @ ( cons_list_list_a @ X4 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_875_Nil__notin__lex,axiom,
    ! [Ys: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R ) ) ).

% Nil_notin_lex
thf(fact_876_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys ) @ ( lex_nat @ R ) ) ).

% Nil_notin_lex
thf(fact_877_Nil2__notin__lex,axiom,
    ! [Xs: list_a,R: set_Product_prod_a_a] :
      ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R ) ) ).

% Nil2_notin_lex
thf(fact_878_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( lex_nat @ R ) ) ).

% Nil2_notin_lex
thf(fact_879_lex__append__leftI,axiom,
    ! [Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_880_prefixes__prefix__suffix__ob,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( member_list_nat @ A @ ( set_list_nat2 @ ( prefixes_nat @ ( append_nat @ B @ C ) ) ) )
     => ( ~ ( member_list_nat @ A @ ( set_list_nat2 @ ( prefixes_nat @ B ) ) )
       => ~ ! [C2: list_nat] :
              ( ? [C3: list_nat] :
                  ( C
                  = ( append_nat @ C2 @ C3 ) )
             => ( ( A
                  = ( append_nat @ B @ C2 ) )
               => ( C2 = nil_nat ) ) ) ) ) ).

% prefixes_prefix_suffix_ob
thf(fact_881_prefixes__prefix__suffix__ob,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( member_list_a @ A @ ( set_list_a2 @ ( prefixes_a @ ( append_a @ B @ C ) ) ) )
     => ( ~ ( member_list_a @ A @ ( set_list_a2 @ ( prefixes_a @ B ) ) )
       => ~ ! [C2: list_a] :
              ( ? [C3: list_a] :
                  ( C
                  = ( append_a @ C2 @ C3 ) )
             => ( ( A
                  = ( append_a @ B @ C2 ) )
               => ( C2 = nil_a ) ) ) ) ) ).

% prefixes_prefix_suffix_ob
thf(fact_882_lexord__partial__trans,axiom,
    ! [Xs: list_list_a,R: set_Pr4048851178543822343list_a,Ys: list_list_a,Zs: list_list_a] :
      ( ! [X3: list_a,Y3: list_a,Z5: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
         => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y3 ) @ R )
           => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y3 @ Z5 ) @ R )
             => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Z5 ) @ R ) ) ) )
     => ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lexord_list_a @ R ) )
       => ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lexord_list_a @ R ) )
         => ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs ) @ ( lexord_list_a @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_883_prefixes_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( prefixes_list_a @ ( cons_list_a @ X @ Xs ) )
      = ( cons_list_list_a @ nil_list_a @ ( map_li8713736314956022724list_a @ ( cons_list_a @ X ) @ ( prefixes_list_a @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_884_prefixes_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( prefixes_a @ ( cons_a @ X @ Xs ) )
      = ( cons_list_a @ nil_a @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_885_prefixes_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
      = ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_886_prefixes_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( prefixes_list_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( cons_list_list_nat @ nil_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( prefixes_list_nat @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_887_prefixes_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( prefixes_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( cons_l8577867181958743494list_a @ nil_list_list_a @ ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X ) @ ( prefixes_list_list_a @ Xs ) ) ) ) ).

% prefixes.simps(2)
thf(fact_888_can__select__set__list__ex1,axiom,
    ! [P: list_a > $o,A5: list_list_a] :
      ( ( can_select_list_a @ P @ ( set_list_a2 @ A5 ) )
      = ( list_ex1_list_a @ P @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_889_append__butlast__last__id,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
     => ( ( append_list_a @ ( butlast_list_a @ Xs ) @ ( cons_list_a @ ( last_list_a @ Xs ) @ nil_list_a ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_890_append__butlast__last__id,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_891_append__butlast__last__id,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_892_append__butlast__last__id,axiom,
    ! [Xs: list_list_nat] :
      ( ( Xs != nil_list_nat )
     => ( ( append_list_nat @ ( butlast_list_nat @ Xs ) @ ( cons_list_nat @ ( last_list_nat @ Xs ) @ nil_list_nat ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_893_append__butlast__last__id,axiom,
    ! [Xs: list_list_list_a] :
      ( ( Xs != nil_list_list_a )
     => ( ( append_list_list_a @ ( butlast_list_list_a @ Xs ) @ ( cons_list_list_a @ ( last_list_list_a @ Xs ) @ nil_list_list_a ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_894_Cons__in__lex,axiom,
    ! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( lex_list_a @ R ) )
      = ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
          & ( ( size_s349497388124573686list_a @ Xs )
            = ( size_s349497388124573686list_a @ Ys ) ) )
        | ( ( X = Y )
          & ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_895_Cons__in__lex,axiom,
    ! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
      = ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
          & ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) ) )
        | ( ( X = Y )
          & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_896_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_897_Cons__in__lex,axiom,
    ! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys: list_list_nat,R: set_Pr3451248702717554689st_nat] :
      ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys ) ) @ ( lex_list_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ R )
          & ( ( size_s3023201423986296836st_nat @ Xs )
            = ( size_s3023201423986296836st_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Xs @ Ys ) @ ( lex_list_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_898_Cons__in__lex,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Y: list_list_a,Ys: list_list_list_a,R: set_Pr5382606609415531783list_a] :
      ( ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ ( cons_list_list_a @ X @ Xs ) @ ( cons_list_list_a @ Y @ Ys ) ) @ ( lex_list_list_a @ R ) )
      = ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X @ Y ) @ R )
          & ( ( size_s2403821588304063868list_a @ Xs )
            = ( size_s2403821588304063868list_a @ Ys ) ) )
        | ( ( X = Y )
          & ( member8667732009277439056list_a @ ( produc1091363791885468951list_a @ Xs @ Ys ) @ ( lex_list_list_a @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_899_Cons_OIH,axiom,
    ! [Res: produc2922778490715282969list_a] :
      ( ( ( produc3726311466776224373list_a @ ( simple2673724192819300657lper_a @ x @ yssa @ Res ) )
        = ( ( produc3726311466776224373list_a @ Res )
          | ( member_list_a @ nil_a @ ( set_list_a2 @ yssa ) ) ) )
      & ( ( set_list_a2 @ ( produc4969414473911996599list_a @ ( simple2673724192819300657lper_a @ x @ yssa @ Res ) ) )
        = ( sup_sup_set_list_a @ ( set_list_a2 @ ( produc4969414473911996599list_a @ Res ) )
          @ ( collect_list_a
            @ ^ [Ys3: list_a] : ( member_list_a @ ( cons_a @ x @ Ys3 ) @ ( set_list_a2 @ yssa ) ) ) ) ) ) ).

% Cons.IH
thf(fact_900_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_901_the__elem__set,axiom,
    ! [X: list_a] :
      ( ( the_elem_list_a @ ( set_list_a2 @ ( cons_list_a @ X @ nil_list_a ) ) )
      = X ) ).

% the_elem_set
thf(fact_902_the__elem__set,axiom,
    ! [X: a] :
      ( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
      = X ) ).

% the_elem_set
thf(fact_903_the__elem__set,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% the_elem_set
thf(fact_904_the__elem__set,axiom,
    ! [X: list_nat] :
      ( ( the_elem_list_nat @ ( set_list_nat2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = X ) ).

% the_elem_set
thf(fact_905_the__elem__set,axiom,
    ! [X: list_list_a] :
      ( ( the_elem_list_list_a @ ( set_list_list_a2 @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = X ) ).

% the_elem_set
thf(fact_906_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_list_a,X: list_a,Ys: list_list_a] :
      ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) )
        = Ys )
      = ( ( Ys != nil_list_a )
        & ( ( butlast_list_a @ Ys )
          = Xs )
        & ( ( last_list_a @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_907_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_a,X: a,Ys: list_a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
        = Ys )
      = ( ( Ys != nil_a )
        & ( ( butlast_a @ Ys )
          = Xs )
        & ( ( last_a @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_908_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = Ys )
      = ( ( Ys != nil_nat )
        & ( ( butlast_nat @ Ys )
          = Xs )
        & ( ( last_nat @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_909_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_list_nat,X: list_nat,Ys: list_list_nat] :
      ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) )
        = Ys )
      = ( ( Ys != nil_list_nat )
        & ( ( butlast_list_nat @ Ys )
          = Xs )
        & ( ( last_list_nat @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_910_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_list_list_a,X: list_list_a,Ys: list_list_list_a] :
      ( ( ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) )
        = Ys )
      = ( ( Ys != nil_list_list_a )
        & ( ( butlast_list_list_a @ Ys )
          = Xs )
        & ( ( last_list_list_a @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_911_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X4: nat] : X4 )
    = ( ^ [Xs3: list_nat] : Xs3 ) ) ).

% map_ident
thf(fact_912_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_913_length__map,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_914_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_a > list_list_a,Al: list_P369347992569427103list_a] :
      ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ ( map_ra3570251311891529249list_a @ F2 @ Al ) )
      = ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) ) ).

% map_fst_map_ran
thf(fact_915_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_nat > list_list_a,Al: list_P3026376796534914631st_nat] :
      ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ ( map_ra8120288265412832055list_a @ F2 @ Al ) )
      = ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_916_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_list_a > list_list_a,Al: list_P1915399164375025317list_a] :
      ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ ( map_ra6112735818830951707list_a @ F2 @ Al ) )
      = ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) ) ).

% map_fst_map_ran
thf(fact_917_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_a > list_list_nat,Al: list_P369347992569427103list_a] :
      ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ ( map_ra2045802175598081305st_nat @ F2 @ Al ) )
      = ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) ) ).

% map_fst_map_ran
thf(fact_918_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_nat > list_list_nat,Al: list_P3026376796534914631st_nat] :
      ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ ( map_ra5394953335731719811st_nat @ F2 @ Al ) )
      = ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_919_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_list_a > list_list_nat,Al: list_P1915399164375025317list_a] :
      ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ ( map_ra3239984559097762719st_nat @ F2 @ Al ) )
      = ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) ) ).

% map_fst_map_ran
thf(fact_920_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_a > list_list_list_a,Al: list_P369347992569427103list_a] :
      ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ ( map_ra8396789647597898279list_a @ F2 @ Al ) )
      = ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) ) ).

% map_fst_map_ran
thf(fact_921_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_nat > list_list_list_a,Al: list_P3026376796534914631st_nat] :
      ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ ( map_ra9120075116080084029list_a @ F2 @ Al ) )
      = ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) ) ).

% map_fst_map_ran
thf(fact_922_map__fst__map__ran,axiom,
    ! [F2: $o > list_list_list_a > list_list_list_a,Al: list_P1915399164375025317list_a] :
      ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ ( map_ra4487667581546282017list_a @ F2 @ Al ) )
      = ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) ) ).

% map_fst_map_ran
thf(fact_923_set__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_append
thf(fact_924_set__append,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( set_list_a2 @ ( append_list_a @ Xs @ Ys ) )
      = ( sup_sup_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) ) ) ).

% set_append
thf(fact_925_last__appendR,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys != nil_a )
     => ( ( last_a @ ( append_a @ Xs @ Ys ) )
        = ( last_a @ Ys ) ) ) ).

% last_appendR
thf(fact_926_last__appendR,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Ys ) ) ) ).

% last_appendR
thf(fact_927_last__appendL,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys = nil_a )
     => ( ( last_a @ ( append_a @ Xs @ Ys ) )
        = ( last_a @ Xs ) ) ) ).

% last_appendL
thf(fact_928_last__appendL,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys = nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Xs ) ) ) ).

% last_appendL
thf(fact_929_last__snoc,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( last_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
      = X ) ).

% last_snoc
thf(fact_930_last__snoc,axiom,
    ! [Xs: list_a,X: a] :
      ( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = X ) ).

% last_snoc
thf(fact_931_last__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% last_snoc
thf(fact_932_last__snoc,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( last_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = X ) ).

% last_snoc
thf(fact_933_last__snoc,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( last_list_list_a @ ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = X ) ).

% last_snoc
thf(fact_934_concat__map__singleton,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( concat_nat
        @ ( map_nat_list_nat
          @ ^ [X4: nat] : ( cons_nat @ ( F2 @ X4 ) @ nil_nat )
          @ Xs ) )
      = ( map_nat_nat @ F2 @ Xs ) ) ).

% concat_map_singleton
thf(fact_935_suffixes__snoc,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( suffixes_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
      = ( cons_list_list_a @ nil_list_a
        @ ( map_li8713736314956022724list_a
          @ ^ [Ys3: list_list_a] : ( append_list_a @ Ys3 @ ( cons_list_a @ X @ nil_list_a ) )
          @ ( suffixes_list_a @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_936_suffixes__snoc,axiom,
    ! [Xs: list_a,X: a] :
      ( ( suffixes_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = ( cons_list_a @ nil_a
        @ ( map_list_a_list_a
          @ ^ [Ys3: list_a] : ( append_a @ Ys3 @ ( cons_a @ X @ nil_a ) )
          @ ( suffixes_a @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_937_suffixes__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( suffixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = ( cons_list_nat @ nil_nat
        @ ( map_li7225945977422193158st_nat
          @ ^ [Ys3: list_nat] : ( append_nat @ Ys3 @ ( cons_nat @ X @ nil_nat ) )
          @ ( suffixes_nat @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_938_suffixes__snoc,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( suffixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
      = ( cons_list_list_nat @ nil_list_nat
        @ ( map_li2855073862107769254st_nat
          @ ^ [Ys3: list_list_nat] : ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ nil_list_nat ) )
          @ ( suffixes_list_nat @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_939_suffixes__snoc,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( suffixes_list_list_a @ ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ nil_list_list_a ) ) )
      = ( cons_l8577867181958743494list_a @ nil_list_list_a
        @ ( map_li6096488569700017860list_a
          @ ^ [Ys3: list_list_list_a] : ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ nil_list_list_a ) )
          @ ( suffixes_list_list_a @ Xs ) ) ) ) ).

% suffixes_snoc
thf(fact_940_concat__pair__set,axiom,
    ! [Ys: list_list_a,Xs: list_list_a] :
      ( ( set_Pr2906193453920572092list_a
        @ ( concat1786428182313682536list_a
          @ ( map_li8762436591914366061list_a
            @ ^ [X4: list_a] : ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec943055143889122450list_a
        @ ^ [Xy: produc9164743771328383783list_a] :
            ( ( member_list_a @ ( produc3698117735987127555list_a @ Xy ) @ ( set_list_a2 @ Xs ) )
            & ( member_list_a @ ( produc8617614985401127493list_a @ Xy ) @ ( set_list_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_941_concat__pair__set,axiom,
    ! [Ys: list_list_list_a,Xs: list_o] :
      ( ( set_Pr6916019737621569582list_a
        @ ( concat3647740114120321498list_a
          @ ( map_o_6649159865426111947list_a
            @ ^ [X4: $o] : ( map_li1526341635108603219list_a @ ( produc7347373609238049417list_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec7533289355003596036list_a
        @ ^ [Xy: produc2922778490715282969list_a] :
            ( ( member_o @ ( produc3726311466776224373list_a @ Xy ) @ ( set_o2 @ Xs ) )
            & ( member_list_list_a @ ( produc4969414473911996599list_a @ Xy ) @ ( set_list_list_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_942_concat__pair__set,axiom,
    ! [Ys: list_list_list_nat,Xs: list_o] :
      ( ( set_Pr7556831416313966818st_nat
        @ ( concat6350014399496557622st_nat
          @ ( map_o_8982006751275110939st_nat
            @ ^ [X4: $o] : ( map_li2622555147623879021st_nat @ ( produc8892817347036706481st_nat @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec477485074338355852st_nat
        @ ^ [Xy: produc7974263126369132087st_nat] :
            ( ( member_o @ ( produc3357746125391881157st_nat @ Xy ) @ ( set_o2 @ Xs ) )
            & ( member_list_list_nat @ ( produc4024048262274189571st_nat @ Xy ) @ ( set_list_list_nat2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_943_concat__pair__set,axiom,
    ! [Ys: list_l7815035709764258326list_a,Xs: list_o] :
      ( ( set_Pr1118946995495890996list_a
        @ ( concat2355107953516655584list_a
          @ ( map_o_6639142526742297041list_a
            @ ^ [X4: $o] : ( map_li7157115275402008147list_a @ ( produc8548274197880810127list_a @ X4 ) @ Ys )
            @ Xs ) ) )
      = ( collec1912989214694242826list_a
        @ ^ [Xy: produc2086622085225315487list_a] :
            ( ( member_o @ ( produc2785537127425143931list_a @ Xy ) @ ( set_o2 @ Xs ) )
            & ( member5342144027231129785list_a @ ( produc1153002459082627517list_a @ Xy ) @ ( set_list_list_list_a2 @ Ys ) ) ) ) ) ).

% concat_pair_set
thf(fact_944_map__pair__fst,axiom,
    ! [F2: $o > list_list_a,Xs: list_o] :
      ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a
        @ ( map_o_4843186313287130437list_a
          @ ^ [X4: $o] : ( produc7347373609238049417list_a @ X4 @ ( F2 @ X4 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_945_map__pair__fst,axiom,
    ! [F2: $o > list_list_nat,Xs: list_o] :
      ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat
        @ ( map_o_567819337388026891st_nat
          @ ^ [X4: $o] : ( produc8892817347036706481st_nat @ X4 @ ( F2 @ X4 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_946_map__pair__fst,axiom,
    ! [F2: $o > list_list_list_a,Xs: list_o] :
      ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a
        @ ( map_o_8366433958082000331list_a
          @ ^ [X4: $o] : ( produc8548274197880810127list_a @ X4 @ ( F2 @ X4 ) )
          @ Xs ) )
      = Xs ) ).

% map_pair_fst
thf(fact_947_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X4: nat] : X4
        @ T )
      = T ) ).

% list.map_ident
thf(fact_948_in__set__product__lists__length,axiom,
    ! [Xs: list_a,Xss2: list_list_a] :
      ( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss2 ) ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_949_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_950_can__select__def,axiom,
    ( can_select_list_a
    = ( ^ [P6: list_a > $o,A6: set_list_a] :
        ? [X4: list_a] :
          ( ( member_list_a @ X4 @ A6 )
          & ( P6 @ X4 )
          & ! [Y5: list_a] :
              ( ( ( member_list_a @ Y5 @ A6 )
                & ( P6 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_951_Shift__def,axiom,
    ( bNF_Gr7042794125918077091list_a
    = ( ^ [Kl3: set_list_list_a,K5: list_a] :
          ( collect_list_list_a
          @ ^ [Kl4: list_list_a] : ( member_list_list_a @ ( cons_list_a @ K5 @ Kl4 ) @ Kl3 ) ) ) ) ).

% Shift_def
thf(fact_952_Shift__def,axiom,
    ( bNF_Greatest_Shift_a
    = ( ^ [Kl3: set_list_a,K5: a] :
          ( collect_list_a
          @ ^ [Kl4: list_a] : ( member_list_a @ ( cons_a @ K5 @ Kl4 ) @ Kl3 ) ) ) ) ).

% Shift_def
thf(fact_953_Shift__def,axiom,
    ( bNF_Gr1872714664788909425ft_nat
    = ( ^ [Kl3: set_list_nat,K5: nat] :
          ( collect_list_nat
          @ ^ [Kl4: list_nat] : ( member_list_nat @ ( cons_nat @ K5 @ Kl4 ) @ Kl3 ) ) ) ) ).

% Shift_def
thf(fact_954_Shift__def,axiom,
    ( bNF_Gr9051742241863529473st_nat
    = ( ^ [Kl3: set_list_list_nat,K5: list_nat] :
          ( collec5989764272469232197st_nat
          @ ^ [Kl4: list_list_nat] : ( member_list_list_nat @ ( cons_list_nat @ K5 @ Kl4 ) @ Kl3 ) ) ) ) ).

% Shift_def
thf(fact_955_Shift__def,axiom,
    ( bNF_Gr1120979937343206697list_a
    = ( ^ [Kl3: set_list_list_list_a,K5: list_list_a] :
          ( collec1292721268053437947list_a
          @ ^ [Kl4: list_list_list_a] : ( member5342144027231129785list_a @ ( cons_list_list_a @ K5 @ Kl4 ) @ Kl3 ) ) ) ) ).

% Shift_def
thf(fact_956_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_list_nat,Xss2: list_list_list_nat] :
      ( ( produc6783906451316923569st_nat @ ( cons_list_list_nat @ Xs @ Xss2 ) )
      = ( concat_list_list_nat
        @ ( map_li4109527381804435110st_nat
          @ ^ [X4: list_nat] : ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X4 ) @ ( produc6783906451316923569st_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_957_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_list_list_a,Xss2: list_l7815035709764258326list_a] :
      ( ( produc1008196158183698553list_a @ ( cons_l8577867181958743494list_a @ Xs @ Xss2 ) )
      = ( concat7147740552513348305list_a
        @ ( map_li6887844498753631312list_a
          @ ^ [X4: list_list_a] : ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X4 ) @ ( produc1008196158183698553list_a @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_958_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_a,Xss2: list_list_a] :
      ( ( product_lists_a @ ( cons_list_a @ Xs @ Xss2 ) )
      = ( concat_list_a
        @ ( map_a_list_list_a
          @ ^ [X4: a] : ( map_list_a_list_a @ ( cons_a @ X4 ) @ ( product_lists_a @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_959_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_nat,Xss2: list_list_nat] :
      ( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss2 ) )
      = ( concat_list_nat
        @ ( map_na6205611841492582150st_nat
          @ ^ [X4: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X4 ) @ ( product_lists_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_960_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_list_a,Xss2: list_list_list_a] :
      ( ( product_lists_list_a @ ( cons_list_list_a @ Xs @ Xss2 ) )
      = ( concat_list_list_a
        @ ( map_li3674916623447280208list_a
          @ ^ [X4: list_a] : ( map_li8713736314956022724list_a @ ( cons_list_a @ X4 ) @ ( product_lists_list_a @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_961_pair__list__eqI,axiom,
    ! [Xs: list_P369347992569427103list_a,Ys: list_P369347992569427103list_a] :
      ( ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Xs )
        = ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Ys ) )
     => ( ( ( map_Pr5214606335154893685list_a @ produc4969414473911996599list_a @ Xs )
          = ( map_Pr5214606335154893685list_a @ produc4969414473911996599list_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_962_pair__list__eqI,axiom,
    ! [Xs: list_P3026376796534914631st_nat,Ys: list_P3026376796534914631st_nat] :
      ( ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Xs )
        = ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Ys ) )
     => ( ( ( map_Pr3525702398142076653st_nat @ produc4024048262274189571st_nat @ Xs )
          = ( map_Pr3525702398142076653st_nat @ produc4024048262274189571st_nat @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_963_pair__list__eqI,axiom,
    ! [Xs: list_P1915399164375025317list_a,Ys: list_P1915399164375025317list_a] :
      ( ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Xs )
        = ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Ys ) )
     => ( ( ( map_Pr7479673503296696053list_a @ produc1153002459082627517list_a @ Xs )
          = ( map_Pr7479673503296696053list_a @ produc1153002459082627517list_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_964_Succ__def,axiom,
    ( bNF_Gr4634511371912843295list_a
    = ( ^ [Kl3: set_list_list_a,Kl4: list_list_a] :
          ( collect_list_a
          @ ^ [K5: list_a] : ( member_list_list_a @ ( append_list_a @ Kl4 @ ( cons_list_a @ K5 @ nil_list_a ) ) @ Kl3 ) ) ) ) ).

% Succ_def
thf(fact_965_Succ__def,axiom,
    ( bNF_Greatest_Succ_a
    = ( ^ [Kl3: set_list_a,Kl4: list_a] :
          ( collect_a
          @ ^ [K5: a] : ( member_list_a @ ( append_a @ Kl4 @ ( cons_a @ K5 @ nil_a ) ) @ Kl3 ) ) ) ) ).

% Succ_def
thf(fact_966_Succ__def,axiom,
    ( bNF_Gr6352880689984616693cc_nat
    = ( ^ [Kl3: set_list_nat,Kl4: list_nat] :
          ( collect_nat
          @ ^ [K5: nat] : ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K5 @ nil_nat ) ) @ Kl3 ) ) ) ) ).

% Succ_def
thf(fact_967_Succ__def,axiom,
    ( bNF_Gr3053708287304744325st_nat
    = ( ^ [Kl3: set_list_list_nat,Kl4: list_list_nat] :
          ( collect_list_nat
          @ ^ [K5: list_nat] : ( member_list_list_nat @ ( append_list_nat @ Kl4 @ ( cons_list_nat @ K5 @ nil_list_nat ) ) @ Kl3 ) ) ) ) ).

% Succ_def
thf(fact_968_Succ__def,axiom,
    ( bNF_Gr6039074496463509157list_a
    = ( ^ [Kl3: set_list_list_list_a,Kl4: list_list_list_a] :
          ( collect_list_list_a
          @ ^ [K5: list_list_a] : ( member5342144027231129785list_a @ ( append_list_list_a @ Kl4 @ ( cons_list_list_a @ K5 @ nil_list_list_a ) ) @ Kl3 ) ) ) ) ).

% Succ_def
thf(fact_969_subseqs_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( subseqs_list_a @ ( cons_list_a @ X @ Xs ) )
      = ( append_list_list_a @ ( map_li8713736314956022724list_a @ ( cons_list_a @ X ) @ ( subseqs_list_a @ Xs ) ) @ ( subseqs_list_a @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_970_subseqs_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( subseqs_a @ ( cons_a @ X @ Xs ) )
      = ( append_list_a @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( subseqs_a @ Xs ) ) @ ( subseqs_a @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_971_subseqs_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( subseqs_nat @ ( cons_nat @ X @ Xs ) )
      = ( append_list_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( subseqs_nat @ Xs ) ) @ ( subseqs_nat @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_972_subseqs_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( subseqs_list_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append_list_list_nat @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( subseqs_list_nat @ Xs ) ) @ ( subseqs_list_nat @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_973_subseqs_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( subseqs_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append6004555433696164139list_a @ ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X ) @ ( subseqs_list_list_a @ Xs ) ) @ ( subseqs_list_list_a @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_974_list__induct2,axiom,
    ! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_a @ nil_a )
       => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_975_list__induct2,axiom,
    ! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_a @ nil_nat )
       => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_976_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_nat @ nil_a )
       => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_977_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_978_list__induct2,axiom,
    ! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_list_a @ nil_a )
       => ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a] :
              ( ( ( size_s349497388124573686list_a @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_979_list__induct2,axiom,
    ! [Xs: list_list_a,Ys: list_nat,P: list_list_a > list_nat > $o] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_list_a @ nil_nat )
       => ( ! [X3: list_a,Xs2: list_list_a,Y3: nat,Ys2: list_nat] :
              ( ( ( size_s349497388124573686list_a @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_980_list__induct2,axiom,
    ! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ( P @ nil_a @ nil_list_a )
       => ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_s349497388124573686list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_981_list__induct2,axiom,
    ! [Xs: list_a,Ys: list_list_nat,P: list_a > list_list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( P @ nil_a @ nil_list_nat )
       => ( ! [X3: a,Xs2: list_a,Y3: list_nat,Ys2: list_list_nat] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_s3023201423986296836st_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_982_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_list_a,P: list_nat > list_list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ( P @ nil_nat @ nil_list_a )
       => ( ! [X3: nat,Xs2: list_nat,Y3: list_a,Ys2: list_list_a] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s349497388124573686list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_983_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,P: list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_list_nat )
       => ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s3023201423986296836st_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_984_list__induct3,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( P @ nil_a @ nil_a @ nil_a )
         => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a] :
                ( ( ( size_size_list_a @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_a @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_985_list__induct3,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_a @ nil_a @ nil_nat )
         => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z5: nat,Zs3: list_nat] :
                ( ( ( size_size_list_a @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_986_list__induct3,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( P @ nil_a @ nil_nat @ nil_a )
         => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z5: a,Zs3: list_a] :
                ( ( ( size_size_list_a @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_a @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_987_list__induct3,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_a @ nil_nat @ nil_nat )
         => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z5: nat,Zs3: list_nat] :
                ( ( ( size_size_list_a @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_988_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( P @ nil_nat @ nil_a @ nil_a )
         => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_a @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_989_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_a @ nil_nat )
         => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a,Z5: nat,Zs3: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_990_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_a )
         => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z5: a,Zs3: list_a] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_a @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_991_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,Z5: nat,Zs3: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys2 ) )
               => ( ( ( size_size_list_nat @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_992_list__induct3,axiom,
    ! [Xs: list_list_a,Ys: list_a,Zs: list_a,P: list_list_a > list_a > list_a > $o] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( P @ nil_list_a @ nil_a @ nil_a )
         => ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a] :
                ( ( ( size_s349497388124573686list_a @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_a @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_993_list__induct3,axiom,
    ! [Xs: list_list_a,Ys: list_a,Zs: list_nat,P: list_list_a > list_a > list_nat > $o] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_list_a @ nil_a @ nil_nat )
         => ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z5: nat,Zs3: list_nat] :
                ( ( ( size_s349497388124573686list_a @ Xs2 )
                  = ( size_size_list_a @ Ys2 ) )
               => ( ( ( size_size_list_a @ Ys2 )
                    = ( size_size_list_nat @ Zs3 ) )
                 => ( ( P @ Xs2 @ Ys2 @ Zs3 )
                   => ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_994_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
           => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_a @ Zs3 ) )
                   => ( ( ( size_size_list_a @ Zs3 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_995_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_a > list_a > list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_a @ nil_a @ nil_a @ nil_nat )
           => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_a @ Zs3 ) )
                   => ( ( ( size_size_list_a @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_996_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_a,P: list_a > list_a > list_nat > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_a @ nil_a @ nil_nat @ nil_a )
           => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z5: nat,Zs3: list_nat,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_997_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_nat,P: list_a > list_a > list_nat > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_a @ nil_a @ nil_nat @ nil_nat )
           => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z5: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_998_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_a,P: list_a > list_nat > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_a @ nil_nat @ nil_a @ nil_a )
           => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_a @ Zs3 ) )
                   => ( ( ( size_size_list_a @ Zs3 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_999_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_nat,P: list_a > list_nat > list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_a @ nil_nat @ nil_a @ nil_nat )
           => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z5: a,Zs3: list_a,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_a @ Zs3 ) )
                   => ( ( ( size_size_list_a @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1000_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
           => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z5: nat,Zs3: list_nat,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1001_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat,Z5: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_a @ Xs2 )
                    = ( size_size_list_nat @ Ys2 ) )
                 => ( ( ( size_size_list_nat @ Ys2 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z5 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1002_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
           => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_a @ Zs3 ) )
                   => ( ( ( size_size_list_a @ Zs3 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1003_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs ) )
       => ( ( ( size_size_list_a @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
           => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a,Z5: a,Zs3: list_a,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_a @ Ys2 ) )
                 => ( ( ( size_size_list_a @ Ys2 )
                      = ( size_size_list_a @ Zs3 ) )
                   => ( ( ( size_size_list_a @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z5 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1004_pow__list_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( pow_list_list_a @ ( cons_list_a @ X @ Xs ) )
      = ( append_list_list_a @ ( pow_list_list_a @ Xs ) @ ( map_li8713736314956022724list_a @ ( cons_list_a @ X ) @ ( pow_list_list_a @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_1005_pow__list_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( pow_list_a @ ( cons_a @ X @ Xs ) )
      = ( append_list_a @ ( pow_list_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( pow_list_a @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_1006_pow__list_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( pow_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( append_list_nat @ ( pow_list_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( pow_list_nat @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_1007_pow__list_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( pow_list_list_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append_list_list_nat @ ( pow_list_list_nat @ Xs ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X ) @ ( pow_list_list_nat @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_1008_pow__list_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( pow_list_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append6004555433696164139list_a @ ( pow_list_list_list_a @ Xs ) @ ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X ) @ ( pow_list_list_list_a @ Xs ) ) ) ) ).

% pow_list.simps(2)
thf(fact_1009_last_Osimps,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( ( Xs = nil_list_a )
       => ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_list_a )
       => ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
          = ( last_list_a @ Xs ) ) ) ) ).

% last.simps
thf(fact_1010_last_Osimps,axiom,
    ! [Xs: list_a,X: a] :
      ( ( ( Xs = nil_a )
       => ( ( last_a @ ( cons_a @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_a )
       => ( ( last_a @ ( cons_a @ X @ Xs ) )
          = ( last_a @ Xs ) ) ) ) ).

% last.simps
thf(fact_1011_last_Osimps,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
          = ( last_nat @ Xs ) ) ) ) ).

% last.simps
thf(fact_1012_last_Osimps,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( ( Xs = nil_list_nat )
       => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_list_nat )
       => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
          = ( last_list_nat @ Xs ) ) ) ) ).

% last.simps
thf(fact_1013_last_Osimps,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( ( Xs = nil_list_list_a )
       => ( ( last_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_list_list_a )
       => ( ( last_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
          = ( last_list_list_a @ Xs ) ) ) ) ).

% last.simps
thf(fact_1014_last__ConsL,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( Xs = nil_list_a )
     => ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1015_last__ConsL,axiom,
    ! [Xs: list_a,X: a] :
      ( ( Xs = nil_a )
     => ( ( last_a @ ( cons_a @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1016_last__ConsL,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs = nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1017_last__ConsL,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( Xs = nil_list_nat )
     => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1018_last__ConsL,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( Xs = nil_list_list_a )
     => ( ( last_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1019_last__ConsR,axiom,
    ! [Xs: list_list_a,X: list_a] :
      ( ( Xs != nil_list_a )
     => ( ( last_list_a @ ( cons_list_a @ X @ Xs ) )
        = ( last_list_a @ Xs ) ) ) ).

% last_ConsR
thf(fact_1020_last__ConsR,axiom,
    ! [Xs: list_a,X: a] :
      ( ( Xs != nil_a )
     => ( ( last_a @ ( cons_a @ X @ Xs ) )
        = ( last_a @ Xs ) ) ) ).

% last_ConsR
thf(fact_1021_last__ConsR,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
        = ( last_nat @ Xs ) ) ) ).

% last_ConsR
thf(fact_1022_last__ConsR,axiom,
    ! [Xs: list_list_nat,X: list_nat] :
      ( ( Xs != nil_list_nat )
     => ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
        = ( last_list_nat @ Xs ) ) ) ).

% last_ConsR
thf(fact_1023_last__ConsR,axiom,
    ! [Xs: list_list_list_a,X: list_list_a] :
      ( ( Xs != nil_list_list_a )
     => ( ( last_list_list_a @ ( cons_list_list_a @ X @ Xs ) )
        = ( last_list_list_a @ Xs ) ) ) ).

% last_ConsR
thf(fact_1024_last__in__set,axiom,
    ! [As: list_a] :
      ( ( As != nil_a )
     => ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).

% last_in_set
thf(fact_1025_last__in__set,axiom,
    ! [As: list_nat] :
      ( ( As != nil_nat )
     => ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).

% last_in_set
thf(fact_1026_last__in__set,axiom,
    ! [As: list_list_a] :
      ( ( As != nil_list_a )
     => ( member_list_a @ ( last_list_a @ As ) @ ( set_list_a2 @ As ) ) ) ).

% last_in_set
thf(fact_1027_last__append,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( ( Ys = nil_a )
       => ( ( last_a @ ( append_a @ Xs @ Ys ) )
          = ( last_a @ Xs ) ) )
      & ( ( Ys != nil_a )
       => ( ( last_a @ ( append_a @ Xs @ Ys ) )
          = ( last_a @ Ys ) ) ) ) ).

% last_append
thf(fact_1028_last__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Ys ) ) ) ) ).

% last_append
thf(fact_1029_longest__common__suffix,axiom,
    ! [Xs: list_a,Ys: list_a] :
    ? [Ss: list_a,Xs5: list_a,Ys6: list_a] :
      ( ( Xs
        = ( append_a @ Xs5 @ Ss ) )
      & ( Ys
        = ( append_a @ Ys6 @ Ss ) )
      & ( ( Xs5 = nil_a )
        | ( Ys6 = nil_a )
        | ( ( last_a @ Xs5 )
         != ( last_a @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1030_longest__common__suffix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ss: list_nat,Xs5: list_nat,Ys6: list_nat] :
      ( ( Xs
        = ( append_nat @ Xs5 @ Ss ) )
      & ( Ys
        = ( append_nat @ Ys6 @ Ss ) )
      & ( ( Xs5 = nil_nat )
        | ( Ys6 = nil_nat )
        | ( ( last_nat @ Xs5 )
         != ( last_nat @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1031_last__map,axiom,
    ! [Xs: list_nat,F2: nat > nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( map_nat_nat @ F2 @ Xs ) )
        = ( F2 @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_1032_rev__induct2,axiom,
    ! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_a @ nil_a )
       => ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1033_rev__induct2,axiom,
    ! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_a @ nil_nat )
       => ( ! [X3: a,Xs2: list_a,Y3: nat,Ys2: list_nat] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1034_rev__induct2,axiom,
    ! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_nat @ nil_a )
       => ( ! [X3: nat,Xs2: list_nat,Y3: a,Ys2: list_a] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1035_rev__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1036_rev__induct2,axiom,
    ! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P @ nil_list_a @ nil_a )
       => ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a] :
              ( ( ( size_s349497388124573686list_a @ Xs2 )
                = ( size_size_list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X3 @ nil_list_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1037_rev__induct2,axiom,
    ! [Xs: list_list_a,Ys: list_nat,P: list_list_a > list_nat > $o] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_list_a @ nil_nat )
       => ( ! [X3: list_a,Xs2: list_list_a,Y3: nat,Ys2: list_nat] :
              ( ( ( size_s349497388124573686list_a @ Xs2 )
                = ( size_size_list_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_list_a @ Xs2 @ ( cons_list_a @ X3 @ nil_list_a ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1038_rev__induct2,axiom,
    ! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ( P @ nil_a @ nil_list_a )
       => ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_s349497388124573686list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_list_a @ Ys2 @ ( cons_list_a @ Y3 @ nil_list_a ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1039_rev__induct2,axiom,
    ! [Xs: list_a,Ys: list_list_nat,P: list_a > list_list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( P @ nil_a @ nil_list_nat )
       => ( ! [X3: a,Xs2: list_a,Y3: list_nat,Ys2: list_list_nat] :
              ( ( ( size_size_list_a @ Xs2 )
                = ( size_s3023201423986296836st_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1040_rev__induct2,axiom,
    ! [Xs: list_nat,Ys: list_list_a,P: list_nat > list_list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ( P @ nil_nat @ nil_list_a )
       => ( ! [X3: nat,Xs2: list_nat,Y3: list_a,Ys2: list_list_a] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s349497388124573686list_a @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_list_a @ Ys2 @ ( cons_list_a @ Y3 @ nil_list_a ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1041_rev__induct2,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,P: list_nat > list_list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_list_nat )
       => ( ! [X3: nat,Xs2: list_nat,Y3: list_nat,Ys2: list_list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s3023201423986296836st_nat @ Ys2 ) )
             => ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% rev_induct2
thf(fact_1042_same__length__different,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( Xs != Ys )
     => ( ( ( size_s349497388124573686list_a @ Xs )
          = ( size_s349497388124573686list_a @ Ys ) )
       => ? [Pre: list_list_a,X3: list_a,Xs5: list_list_a,Y3: list_a,Ys6: list_list_a] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X3 @ nil_list_a ) @ Xs5 ) ) )
            & ( Ys
              = ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y3 @ nil_list_a ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1043_same__length__different,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
       => ? [Pre: list_a,X3: a,Xs5: list_a,Y3: a,Ys6: list_a] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs5 ) ) )
            & ( Ys
              = ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1044_same__length__different,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X3: nat,Xs5: list_nat,Y3: nat,Ys6: list_nat] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs5 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1045_same__length__different,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ? [Pre: list_list_nat,X3: list_nat,Xs5: list_list_nat,Y3: list_nat,Ys6: list_list_nat] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ X3 @ nil_list_nat ) @ Xs5 ) ) )
            & ( Ys
              = ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ Y3 @ nil_list_nat ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1046_same__length__different,axiom,
    ! [Xs: list_list_list_a,Ys: list_list_list_a] :
      ( ( Xs != Ys )
     => ( ( ( size_s2403821588304063868list_a @ Xs )
          = ( size_s2403821588304063868list_a @ Ys ) )
       => ? [Pre: list_list_list_a,X3: list_list_a,Xs5: list_list_list_a,Y3: list_list_a,Ys6: list_list_list_a] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append_list_list_a @ Pre @ ( append_list_list_a @ ( cons_list_list_a @ X3 @ nil_list_list_a ) @ Xs5 ) ) )
            & ( Ys
              = ( append_list_list_a @ Pre @ ( append_list_list_a @ ( cons_list_list_a @ Y3 @ nil_list_list_a ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1047_lexord__sufE,axiom,
    ! [Xs: list_nat,Zs: list_nat,Ys: list_nat,Qs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Zs ) @ ( append_nat @ Ys @ Qs ) ) @ ( lexord_nat @ R ) )
     => ( ( Xs != Ys )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ( ( size_size_list_nat @ Zs )
              = ( size_size_list_nat @ Qs ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_1048_lex__append__rightI,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Vs: list_nat,Us: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) )
     => ( ( ( size_size_list_nat @ Vs )
          = ( size_size_list_nat @ Us ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_1049_pow__list_Oelims,axiom,
    ! [X: list_list_nat,Y: list_list_list_nat] :
      ( ( ( pow_list_list_nat @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y
           != ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_list_nat @ ( pow_list_list_nat @ Xs2 ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X3 ) @ ( pow_list_list_nat @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_1050_pow__list_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_l7815035709764258326list_a] :
      ( ( ( pow_list_list_list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y
           != ( cons_l8577867181958743494list_a @ nil_list_list_a @ nil_list_list_list_a ) ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append6004555433696164139list_a @ ( pow_list_list_list_a @ Xs2 ) @ ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X3 ) @ ( pow_list_list_list_a @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_1051_pow__list_Oelims,axiom,
    ! [X: list_a,Y: list_list_a] :
      ( ( ( pow_list_a @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y
           != ( cons_list_a @ nil_a @ nil_list_a ) ) )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( X
                = ( cons_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_a @ ( pow_list_a @ Xs2 ) @ ( map_list_a_list_a @ ( cons_a @ X3 ) @ ( pow_list_a @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_1052_pow__list_Oelims,axiom,
    ! [X: list_nat,Y: list_list_nat] :
      ( ( ( pow_list_nat @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y
           != ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_nat @ ( pow_list_nat @ Xs2 ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X3 ) @ ( pow_list_nat @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_1053_pow__list_Oelims,axiom,
    ! [X: list_list_a,Y: list_list_list_a] :
      ( ( ( pow_list_list_a @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y
           != ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) )
       => ~ ! [X3: list_a,Xs2: list_list_a] :
              ( ( X
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append_list_list_a @ ( pow_list_list_a @ Xs2 ) @ ( map_li8713736314956022724list_a @ ( cons_list_a @ X3 ) @ ( pow_list_list_a @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.elims
thf(fact_1054_pow__list_Opelims,axiom,
    ! [X: list_list_nat,Y: list_list_list_nat] :
      ( ( ( pow_list_list_nat @ X )
        = Y )
     => ( ( accp_list_list_nat @ pow_li3926171397405599314st_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y
                = ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) )
             => ~ ( accp_list_list_nat @ pow_li3926171397405599314st_nat @ nil_list_nat ) ) )
         => ~ ! [X3: list_nat,Xs2: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_list_nat @ ( pow_list_list_nat @ Xs2 ) @ ( map_li2855073862107769254st_nat @ ( cons_list_nat @ X3 ) @ ( pow_list_list_nat @ Xs2 ) ) ) )
                 => ~ ( accp_list_list_nat @ pow_li3926171397405599314st_nat @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_1055_pow__list_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_l7815035709764258326list_a] :
      ( ( ( pow_list_list_list_a @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ pow_li1502696855258329496list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y
                = ( cons_l8577867181958743494list_a @ nil_list_list_a @ nil_list_list_list_a ) )
             => ~ ( accp_l3666201008454774169list_a @ pow_li1502696855258329496list_a @ nil_list_list_a ) ) )
         => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append6004555433696164139list_a @ ( pow_list_list_list_a @ Xs2 ) @ ( map_li6096488569700017860list_a @ ( cons_list_list_a @ X3 ) @ ( pow_list_list_list_a @ Xs2 ) ) ) )
                 => ~ ( accp_l3666201008454774169list_a @ pow_li1502696855258329496list_a @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_1056_pow__list_Opelims,axiom,
    ! [X: list_a,Y: list_list_a] :
      ( ( ( pow_list_a @ X )
        = Y )
     => ( ( accp_list_a @ pow_list_rel_a @ X )
       => ( ( ( X = nil_a )
           => ( ( Y
                = ( cons_list_a @ nil_a @ nil_list_a ) )
             => ~ ( accp_list_a @ pow_list_rel_a @ nil_a ) ) )
         => ~ ! [X3: a,Xs2: list_a] :
                ( ( X
                  = ( cons_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_a @ ( pow_list_a @ Xs2 ) @ ( map_list_a_list_a @ ( cons_a @ X3 ) @ ( pow_list_a @ Xs2 ) ) ) )
                 => ~ ( accp_list_a @ pow_list_rel_a @ ( cons_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_1057_pow__list_Opelims,axiom,
    ! [X: list_nat,Y: list_list_nat] :
      ( ( ( pow_list_nat @ X )
        = Y )
     => ( ( accp_list_nat @ pow_list_rel_nat @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y
                = ( cons_list_nat @ nil_nat @ nil_list_nat ) )
             => ~ ( accp_list_nat @ pow_list_rel_nat @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_nat @ ( pow_list_nat @ Xs2 ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X3 ) @ ( pow_list_nat @ Xs2 ) ) ) )
                 => ~ ( accp_list_nat @ pow_list_rel_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_1058_pow__list_Opelims,axiom,
    ! [X: list_list_a,Y: list_list_list_a] :
      ( ( ( pow_list_list_a @ X )
        = Y )
     => ( ( accp_list_list_a @ pow_list_rel_list_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y
                = ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) )
             => ~ ( accp_list_list_a @ pow_list_rel_list_a @ nil_list_a ) ) )
         => ~ ! [X3: list_a,Xs2: list_list_a] :
                ( ( X
                  = ( cons_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append_list_list_a @ ( pow_list_list_a @ Xs2 ) @ ( map_li8713736314956022724list_a @ ( cons_list_a @ X3 ) @ ( pow_list_list_a @ Xs2 ) ) ) )
                 => ~ ( accp_list_list_a @ pow_list_rel_list_a @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% pow_list.pelims
thf(fact_1059_set__union,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( set_list_a2 @ ( union_list_a @ Xs @ Ys ) )
      = ( sup_sup_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) ) ) ).

% set_union
thf(fact_1060_prefix__pairs_Oelims,axiom,
    ! [X: list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( prefix_pairs_list_a @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y != nil_Pr6382449733212294445list_a ) )
       => ~ ! [V: list_a,Va: list_list_a] :
              ( ( X
                = ( cons_list_a @ V @ Va ) )
             => ( Y
               != ( append8923838579933563074list_a @ ( prefix_pairs_list_a @ ( butlast_list_a @ ( cons_list_a @ V @ Va ) ) )
                  @ ( map_li2617854710493732705list_a
                    @ ^ [Ys3: list_list_a] : ( produc8696003437204565271list_a @ Ys3 @ ( cons_list_a @ V @ Va ) )
                    @ ( butlast_list_list_a @ ( prefixes_list_a @ ( cons_list_a @ V @ Va ) ) ) ) ) ) ) ) ) ).

% prefix_pairs.elims
thf(fact_1061_prefix__pairs_Oelims,axiom,
    ! [X: list_a,Y: list_P321204300973800749list_a] :
      ( ( ( prefix_pairs_a @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y != nil_Pr3188421586756112173list_a ) )
       => ~ ! [V: a,Va: list_a] :
              ( ( X
                = ( cons_a @ V @ Va ) )
             => ( Y
               != ( append622457529216362434list_a @ ( prefix_pairs_a @ ( butlast_a @ ( cons_a @ V @ Va ) ) )
                  @ ( map_li1163028862020805223list_a
                    @ ^ [Ys3: list_a] : ( produc6837034575241423639list_a @ Ys3 @ ( cons_a @ V @ Va ) )
                    @ ( butlast_list_a @ ( prefixes_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).

% prefix_pairs.elims
thf(fact_1062_prefix__pairs_Oelims,axiom,
    ! [X: list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( prefix_pairs_nat @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != nil_Pr8413428694792600231st_nat ) )
       => ~ ! [V: nat,Va: list_nat] :
              ( ( X
                = ( cons_nat @ V @ Va ) )
             => ( Y
               != ( append2623875052807961020st_nat @ ( prefix_pairs_nat @ ( butlast_nat @ ( cons_nat @ V @ Va ) ) )
                  @ ( map_li6665866383299509243st_nat
                    @ ^ [Ys3: list_nat] : ( produc2694037385005941721st_nat @ Ys3 @ ( cons_nat @ V @ Va ) )
                    @ ( butlast_list_nat @ ( prefixes_nat @ ( cons_nat @ V @ Va ) ) ) ) ) ) ) ) ) ).

% prefix_pairs.elims
thf(fact_1063_prefix__pairs_Oelims,axiom,
    ! [X: list_list_nat,Y: list_P4223930160327851207st_nat] :
      ( ( ( prefix2089531372049794650st_nat @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y != nil_Pr7692610383271629383st_nat ) )
       => ~ ! [V: list_nat,Va: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ V @ Va ) )
             => ( Y
               != ( append2641727691416532060st_nat @ ( prefix2089531372049794650st_nat @ ( butlast_list_nat @ ( cons_list_nat @ V @ Va ) ) )
                  @ ( map_li8160113138763195787st_nat
                    @ ^ [Ys3: list_list_nat] : ( produc7129799990162260089st_nat @ Ys3 @ ( cons_list_nat @ V @ Va ) )
                    @ ( butlas6429778205849610142st_nat @ ( prefixes_list_nat @ ( cons_list_nat @ V @ Va ) ) ) ) ) ) ) ) ) ).

% prefix_pairs.elims
thf(fact_1064_prefix__pairs_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_P2731179207972443437list_a] :
      ( ( ( prefix3627290527299430288list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y != nil_Pr8028441414664233261list_a ) )
       => ~ ! [V: list_list_a,Va: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ V @ Va ) )
             => ( Y
               != ( append1777735555505536962list_a @ ( prefix3627290527299430288list_a @ ( butlast_list_list_a @ ( cons_list_list_a @ V @ Va ) ) )
                  @ ( map_li4810980779913120475list_a
                    @ ^ [Ys3: list_list_list_a] : ( produc1091363791885468951list_a @ Ys3 @ ( cons_list_list_a @ V @ Va ) )
                    @ ( butlas1038664195803223394list_a @ ( prefixes_list_list_a @ ( cons_list_list_a @ V @ Va ) ) ) ) ) ) ) ) ) ).

% prefix_pairs.elims
thf(fact_1065_linear__order__from__list__position_H_Oelims,axiom,
    ! [X: list_list_a,Y: list_P321204300973800749list_a] :
      ( ( ( linear3336838573582304814list_a @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y != nil_Pr3188421586756112173list_a ) )
       => ~ ! [X3: list_a,Xs2: list_list_a] :
              ( ( X
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ X3 @ X3 ) @ ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X3 ) @ Xs2 ) @ ( linear3336838573582304814list_a @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_1066_linear__order__from__list__position_H_Oelims,axiom,
    ! [X: list_a,Y: list_P1396940483166286381od_a_a] :
      ( ( ( linear6964191089970808488tion_a @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y != nil_Product_prod_a_a ) )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( X
                = ( cons_a @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs2 ) @ ( linear6964191089970808488tion_a @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_1067_linear__order__from__list__position_H_Oelims,axiom,
    ! [X: list_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( linear8051522407387036070on_nat @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Xs2 ) @ ( linear8051522407387036070on_nat @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_1068_linear__order__from__list__position_H_Oelims,axiom,
    ! [X: list_list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( linear8696861518193291702st_nat @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y != nil_Pr8413428694792600231st_nat ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_P5007559046487125591st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X3 ) @ Xs2 ) @ ( linear8696861518193291702st_nat @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_1069_linear__order__from__list__position_H_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( linear3551333125237012788list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y != nil_Pr6382449733212294445list_a ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_P4475838960125360349list_a @ ( produc8696003437204565271list_a @ X3 @ X3 ) @ ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X3 ) @ Xs2 ) @ ( linear3551333125237012788list_a @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.elims
thf(fact_1070_prefix__pairs_Osimps_I2_J,axiom,
    ! [V2: list_a,Va2: list_list_a] :
      ( ( prefix_pairs_list_a @ ( cons_list_a @ V2 @ Va2 ) )
      = ( append8923838579933563074list_a @ ( prefix_pairs_list_a @ ( butlast_list_a @ ( cons_list_a @ V2 @ Va2 ) ) )
        @ ( map_li2617854710493732705list_a
          @ ^ [Ys3: list_list_a] : ( produc8696003437204565271list_a @ Ys3 @ ( cons_list_a @ V2 @ Va2 ) )
          @ ( butlast_list_list_a @ ( prefixes_list_a @ ( cons_list_a @ V2 @ Va2 ) ) ) ) ) ) ).

% prefix_pairs.simps(2)
thf(fact_1071_prefix__pairs_Osimps_I2_J,axiom,
    ! [V2: a,Va2: list_a] :
      ( ( prefix_pairs_a @ ( cons_a @ V2 @ Va2 ) )
      = ( append622457529216362434list_a @ ( prefix_pairs_a @ ( butlast_a @ ( cons_a @ V2 @ Va2 ) ) )
        @ ( map_li1163028862020805223list_a
          @ ^ [Ys3: list_a] : ( produc6837034575241423639list_a @ Ys3 @ ( cons_a @ V2 @ Va2 ) )
          @ ( butlast_list_a @ ( prefixes_a @ ( cons_a @ V2 @ Va2 ) ) ) ) ) ) ).

% prefix_pairs.simps(2)
thf(fact_1072_prefix__pairs_Osimps_I2_J,axiom,
    ! [V2: nat,Va2: list_nat] :
      ( ( prefix_pairs_nat @ ( cons_nat @ V2 @ Va2 ) )
      = ( append2623875052807961020st_nat @ ( prefix_pairs_nat @ ( butlast_nat @ ( cons_nat @ V2 @ Va2 ) ) )
        @ ( map_li6665866383299509243st_nat
          @ ^ [Ys3: list_nat] : ( produc2694037385005941721st_nat @ Ys3 @ ( cons_nat @ V2 @ Va2 ) )
          @ ( butlast_list_nat @ ( prefixes_nat @ ( cons_nat @ V2 @ Va2 ) ) ) ) ) ) ).

% prefix_pairs.simps(2)
thf(fact_1073_prefix__pairs_Osimps_I2_J,axiom,
    ! [V2: list_nat,Va2: list_list_nat] :
      ( ( prefix2089531372049794650st_nat @ ( cons_list_nat @ V2 @ Va2 ) )
      = ( append2641727691416532060st_nat @ ( prefix2089531372049794650st_nat @ ( butlast_list_nat @ ( cons_list_nat @ V2 @ Va2 ) ) )
        @ ( map_li8160113138763195787st_nat
          @ ^ [Ys3: list_list_nat] : ( produc7129799990162260089st_nat @ Ys3 @ ( cons_list_nat @ V2 @ Va2 ) )
          @ ( butlas6429778205849610142st_nat @ ( prefixes_list_nat @ ( cons_list_nat @ V2 @ Va2 ) ) ) ) ) ) ).

% prefix_pairs.simps(2)
thf(fact_1074_prefix__pairs_Osimps_I2_J,axiom,
    ! [V2: list_list_a,Va2: list_list_list_a] :
      ( ( prefix3627290527299430288list_a @ ( cons_list_list_a @ V2 @ Va2 ) )
      = ( append1777735555505536962list_a @ ( prefix3627290527299430288list_a @ ( butlast_list_list_a @ ( cons_list_list_a @ V2 @ Va2 ) ) )
        @ ( map_li4810980779913120475list_a
          @ ^ [Ys3: list_list_list_a] : ( produc1091363791885468951list_a @ Ys3 @ ( cons_list_list_a @ V2 @ Va2 ) )
          @ ( butlas1038664195803223394list_a @ ( prefixes_list_list_a @ ( cons_list_list_a @ V2 @ Va2 ) ) ) ) ) ) ).

% prefix_pairs.simps(2)
thf(fact_1075_prefix__pairs_Osimps_I1_J,axiom,
    ( ( prefix_pairs_a @ nil_a )
    = nil_Pr3188421586756112173list_a ) ).

% prefix_pairs.simps(1)
thf(fact_1076_prefix__pairs_Osimps_I1_J,axiom,
    ( ( prefix_pairs_nat @ nil_nat )
    = nil_Pr8413428694792600231st_nat ) ).

% prefix_pairs.simps(1)
thf(fact_1077_linear__order__from__list__position_H_Osimps_I1_J,axiom,
    ( ( linear6964191089970808488tion_a @ nil_a )
    = nil_Product_prod_a_a ) ).

% linear_order_from_list_position'.simps(1)
thf(fact_1078_linear__order__from__list__position_H_Osimps_I1_J,axiom,
    ( ( linear8051522407387036070on_nat @ nil_nat )
    = nil_Pr5478986624290739719at_nat ) ).

% linear_order_from_list_position'.simps(1)
thf(fact_1079_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( linear3336838573582304814list_a @ ( cons_list_a @ X @ Xs ) )
      = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ X @ X ) @ ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X ) @ Xs ) @ ( linear3336838573582304814list_a @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_1080_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( linear6964191089970808488tion_a @ ( cons_a @ X @ Xs ) )
      = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X @ X ) @ ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X ) @ Xs ) @ ( linear6964191089970808488tion_a @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_1081_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( linear8051522407387036070on_nat @ ( cons_nat @ X @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Xs ) @ ( linear8051522407387036070on_nat @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_1082_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( linear8696861518193291702st_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( cons_P5007559046487125591st_nat @ ( produc2694037385005941721st_nat @ X @ X ) @ ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X ) @ Xs ) @ ( linear8696861518193291702st_nat @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_1083_linear__order__from__list__position_H_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( linear3551333125237012788list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( cons_P4475838960125360349list_a @ ( produc8696003437204565271list_a @ X @ X ) @ ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X ) @ Xs ) @ ( linear3551333125237012788list_a @ Xs ) ) ) ) ).

% linear_order_from_list_position'.simps(2)
thf(fact_1084_prefix__pairs_Opelims,axiom,
    ! [X: list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( prefix_pairs_list_a @ X )
        = Y )
     => ( ( accp_list_list_a @ prefix3835138820318204227list_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y = nil_Pr6382449733212294445list_a )
             => ~ ( accp_list_list_a @ prefix3835138820318204227list_a @ nil_list_a ) ) )
         => ~ ! [V: list_a,Va: list_list_a] :
                ( ( X
                  = ( cons_list_a @ V @ Va ) )
               => ( ( Y
                    = ( append8923838579933563074list_a @ ( prefix_pairs_list_a @ ( butlast_list_a @ ( cons_list_a @ V @ Va ) ) )
                      @ ( map_li2617854710493732705list_a
                        @ ^ [Ys3: list_list_a] : ( produc8696003437204565271list_a @ Ys3 @ ( cons_list_a @ V @ Va ) )
                        @ ( butlast_list_list_a @ ( prefixes_list_a @ ( cons_list_a @ V @ Va ) ) ) ) ) )
                 => ~ ( accp_list_list_a @ prefix3835138820318204227list_a @ ( cons_list_a @ V @ Va ) ) ) ) ) ) ) ).

% prefix_pairs.pelims
thf(fact_1085_prefix__pairs_Opelims,axiom,
    ! [X: list_a,Y: list_P321204300973800749list_a] :
      ( ( ( prefix_pairs_a @ X )
        = Y )
     => ( ( accp_list_a @ prefix_pairs_rel_a @ X )
       => ( ( ( X = nil_a )
           => ( ( Y = nil_Pr3188421586756112173list_a )
             => ~ ( accp_list_a @ prefix_pairs_rel_a @ nil_a ) ) )
         => ~ ! [V: a,Va: list_a] :
                ( ( X
                  = ( cons_a @ V @ Va ) )
               => ( ( Y
                    = ( append622457529216362434list_a @ ( prefix_pairs_a @ ( butlast_a @ ( cons_a @ V @ Va ) ) )
                      @ ( map_li1163028862020805223list_a
                        @ ^ [Ys3: list_a] : ( produc6837034575241423639list_a @ Ys3 @ ( cons_a @ V @ Va ) )
                        @ ( butlast_list_a @ ( prefixes_a @ ( cons_a @ V @ Va ) ) ) ) ) )
                 => ~ ( accp_list_a @ prefix_pairs_rel_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ).

% prefix_pairs.pelims
thf(fact_1086_prefix__pairs_Opelims,axiom,
    ! [X: list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( prefix_pairs_nat @ X )
        = Y )
     => ( ( accp_list_nat @ prefix_pairs_rel_nat @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = nil_Pr8413428694792600231st_nat )
             => ~ ( accp_list_nat @ prefix_pairs_rel_nat @ nil_nat ) ) )
         => ~ ! [V: nat,Va: list_nat] :
                ( ( X
                  = ( cons_nat @ V @ Va ) )
               => ( ( Y
                    = ( append2623875052807961020st_nat @ ( prefix_pairs_nat @ ( butlast_nat @ ( cons_nat @ V @ Va ) ) )
                      @ ( map_li6665866383299509243st_nat
                        @ ^ [Ys3: list_nat] : ( produc2694037385005941721st_nat @ Ys3 @ ( cons_nat @ V @ Va ) )
                        @ ( butlast_list_nat @ ( prefixes_nat @ ( cons_nat @ V @ Va ) ) ) ) ) )
                 => ~ ( accp_list_nat @ prefix_pairs_rel_nat @ ( cons_nat @ V @ Va ) ) ) ) ) ) ) ).

% prefix_pairs.pelims
thf(fact_1087_prefix__pairs_Opelims,axiom,
    ! [X: list_list_nat,Y: list_P4223930160327851207st_nat] :
      ( ( ( prefix2089531372049794650st_nat @ X )
        = Y )
     => ( ( accp_list_list_nat @ prefix7414232386363521889st_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y = nil_Pr7692610383271629383st_nat )
             => ~ ( accp_list_list_nat @ prefix7414232386363521889st_nat @ nil_list_nat ) ) )
         => ~ ! [V: list_nat,Va: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ V @ Va ) )
               => ( ( Y
                    = ( append2641727691416532060st_nat @ ( prefix2089531372049794650st_nat @ ( butlast_list_nat @ ( cons_list_nat @ V @ Va ) ) )
                      @ ( map_li8160113138763195787st_nat
                        @ ^ [Ys3: list_list_nat] : ( produc7129799990162260089st_nat @ Ys3 @ ( cons_list_nat @ V @ Va ) )
                        @ ( butlas6429778205849610142st_nat @ ( prefixes_list_nat @ ( cons_list_nat @ V @ Va ) ) ) ) ) )
                 => ~ ( accp_list_list_nat @ prefix7414232386363521889st_nat @ ( cons_list_nat @ V @ Va ) ) ) ) ) ) ) ).

% prefix_pairs.pelims
thf(fact_1088_prefix__pairs_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_P2731179207972443437list_a] :
      ( ( ( prefix3627290527299430288list_a @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ prefix1672832661974955465list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y = nil_Pr8028441414664233261list_a )
             => ~ ( accp_l3666201008454774169list_a @ prefix1672832661974955465list_a @ nil_list_list_a ) ) )
         => ~ ! [V: list_list_a,Va: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ V @ Va ) )
               => ( ( Y
                    = ( append1777735555505536962list_a @ ( prefix3627290527299430288list_a @ ( butlast_list_list_a @ ( cons_list_list_a @ V @ Va ) ) )
                      @ ( map_li4810980779913120475list_a
                        @ ^ [Ys3: list_list_list_a] : ( produc1091363791885468951list_a @ Ys3 @ ( cons_list_list_a @ V @ Va ) )
                        @ ( butlas1038664195803223394list_a @ ( prefixes_list_list_a @ ( cons_list_list_a @ V @ Va ) ) ) ) ) )
                 => ~ ( accp_l3666201008454774169list_a @ prefix1672832661974955465list_a @ ( cons_list_list_a @ V @ Va ) ) ) ) ) ) ) ).

% prefix_pairs.pelims
thf(fact_1089_linear__order__from__list__position_H_Opelims,axiom,
    ! [X: list_list_a,Y: list_P321204300973800749list_a] :
      ( ( ( linear3336838573582304814list_a @ X )
        = Y )
     => ( ( accp_list_list_a @ linear1361599701548034719list_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y = nil_Pr3188421586756112173list_a )
             => ~ ( accp_list_list_a @ linear1361599701548034719list_a @ nil_list_a ) ) )
         => ~ ! [X3: list_a,Xs2: list_list_a] :
                ( ( X
                  = ( cons_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ X3 @ X3 ) @ ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X3 ) @ Xs2 ) @ ( linear3336838573582304814list_a @ Xs2 ) ) ) )
                 => ~ ( accp_list_list_a @ linear1361599701548034719list_a @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_1090_linear__order__from__list__position_H_Opelims,axiom,
    ! [X: list_a,Y: list_P1396940483166286381od_a_a] :
      ( ( ( linear6964191089970808488tion_a @ X )
        = Y )
     => ( ( accp_list_a @ linear4418539559401215897_rel_a @ X )
       => ( ( ( X = nil_a )
           => ( ( Y = nil_Product_prod_a_a )
             => ~ ( accp_list_a @ linear4418539559401215897_rel_a @ nil_a ) ) )
         => ~ ! [X3: a,Xs2: list_a] :
                ( ( X
                  = ( cons_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X3 @ X3 ) @ ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs2 ) @ ( linear6964191089970808488tion_a @ Xs2 ) ) ) )
                 => ~ ( accp_list_a @ linear4418539559401215897_rel_a @ ( cons_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_1091_linear__order__from__list__position_H_Opelims,axiom,
    ! [X: list_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( linear8051522407387036070on_nat @ X )
        = Y )
     => ( ( accp_list_nat @ linear1904149156627414645el_nat @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_list_nat @ linear1904149156627414645el_nat @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Xs2 ) @ ( linear8051522407387036070on_nat @ Xs2 ) ) ) )
                 => ~ ( accp_list_nat @ linear1904149156627414645el_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_1092_linear__order__from__list__position_H_Opelims,axiom,
    ! [X: list_list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( linear8696861518193291702st_nat @ X )
        = Y )
     => ( ( accp_list_list_nat @ linear4429276285293871877st_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y = nil_Pr8413428694792600231st_nat )
             => ~ ( accp_list_list_nat @ linear4429276285293871877st_nat @ nil_list_nat ) ) )
         => ~ ! [X3: list_nat,Xs2: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( cons_P5007559046487125591st_nat @ ( produc2694037385005941721st_nat @ X3 @ X3 ) @ ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X3 ) @ Xs2 ) @ ( linear8696861518193291702st_nat @ Xs2 ) ) ) )
                 => ~ ( accp_list_list_nat @ linear4429276285293871877st_nat @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_1093_linear__order__from__list__position_H_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( linear3551333125237012788list_a @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ linear2270011334602109221list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y = nil_Pr6382449733212294445list_a )
             => ~ ( accp_l3666201008454774169list_a @ linear2270011334602109221list_a @ nil_list_list_a ) ) )
         => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( cons_P4475838960125360349list_a @ ( produc8696003437204565271list_a @ X3 @ X3 ) @ ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X3 ) @ Xs2 ) @ ( linear3551333125237012788list_a @ Xs2 ) ) ) )
                 => ~ ( accp_l3666201008454774169list_a @ linear2270011334602109221list_a @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% linear_order_from_list_position'.pelims
thf(fact_1094_product__code,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( produc9042601602475682002list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
      = ( set_Pr2906193453920572092list_a
        @ ( concat1786428182313682536list_a
          @ ( map_li8762436591914366061list_a
            @ ^ [X4: list_a] : ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_1095_product__code,axiom,
    ! [Xs: list_o,Ys: list_list_list_a] :
      ( ( produc5023185770809875908list_a @ ( set_o2 @ Xs ) @ ( set_list_list_a2 @ Ys ) )
      = ( set_Pr6916019737621569582list_a
        @ ( concat3647740114120321498list_a
          @ ( map_o_6649159865426111947list_a
            @ ^ [X4: $o] : ( map_li1526341635108603219list_a @ ( produc7347373609238049417list_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_1096_product__code,axiom,
    ! [Xs: list_o,Ys: list_list_list_nat] :
      ( ( produc2897000267543773878st_nat @ ( set_o2 @ Xs ) @ ( set_list_list_nat2 @ Ys ) )
      = ( set_Pr7556831416313966818st_nat
        @ ( concat6350014399496557622st_nat
          @ ( map_o_8982006751275110939st_nat
            @ ^ [X4: $o] : ( map_li2622555147623879021st_nat @ ( produc8892817347036706481st_nat @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_1097_product__code,axiom,
    ! [Xs: list_o,Ys: list_l7815035709764258326list_a] :
      ( ( produc238215798039182410list_a @ ( set_o2 @ Xs ) @ ( set_list_list_list_a2 @ Ys ) )
      = ( set_Pr1118946995495890996list_a
        @ ( concat2355107953516655584list_a
          @ ( map_o_6639142526742297041list_a
            @ ^ [X4: $o] : ( map_li7157115275402008147list_a @ ( produc8548274197880810127list_a @ X4 ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_1098_list__ordered__pairs_Oelims,axiom,
    ! [X: list_list_a,Y: list_P321204300973800749list_a] :
      ( ( ( list_o4514482613960654667list_a @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y != nil_Pr3188421586756112173list_a ) )
       => ~ ! [X3: list_a,Xs2: list_list_a] :
              ( ( X
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X3 ) @ Xs2 ) @ ( list_o4514482613960654667list_a @ Xs2 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_1099_list__ordered__pairs_Oelims,axiom,
    ! [X: list_a,Y: list_P1396940483166286381od_a_a] :
      ( ( ( list_ordered_pairs_a @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y != nil_Product_prod_a_a ) )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( X
                = ( cons_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs2 ) @ ( list_ordered_pairs_a @ Xs2 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_1100_list__ordered__pairs_Oelims,axiom,
    ! [X: list_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( list_o6794894459886620873rs_nat @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Xs2 ) @ ( list_o6794894459886620873rs_nat @ Xs2 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_1101_list__ordered__pairs_Oelims,axiom,
    ! [X: list_list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( list_o1383297816974482265st_nat @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y != nil_Pr8413428694792600231st_nat ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X3 ) @ Xs2 ) @ ( list_o1383297816974482265st_nat @ Xs2 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_1102_list__ordered__pairs_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( list_o7931928007360004049list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y != nil_Pr6382449733212294445list_a ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X3 ) @ Xs2 ) @ ( list_o7931928007360004049list_a @ Xs2 ) ) ) ) ) ) ).

% list_ordered_pairs.elims
thf(fact_1103_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X: list_list_a,Y: list_P321204300973800749list_a] :
      ( ( ( non_sy4546686295413189007list_a @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y != nil_Pr3188421586756112173list_a ) )
       => ~ ! [X3: list_a,Xs2: list_list_a] :
              ( ( X
                = ( cons_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X3 ) @ Xs2 ) @ ( non_sy4546686295413189007list_a @ Xs2 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_1104_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X: list_a,Y: list_P1396940483166286381od_a_a] :
      ( ( ( non_sym_dist_pairs_a @ X )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y != nil_Product_prod_a_a ) )
       => ~ ! [X3: a,Xs2: list_a] :
              ( ( X
                = ( cons_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs2 ) @ ( non_sym_dist_pairs_a @ Xs2 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_1105_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X: list_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( non_sy1784707333927398277rs_nat @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Xs2 ) @ ( non_sy1784707333927398277rs_nat @ Xs2 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_1106_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X: list_list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( non_sy8159205245201868821st_nat @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y != nil_Pr8413428694792600231st_nat ) )
       => ~ ! [X3: list_nat,Xs2: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X3 ) @ Xs2 ) @ ( non_sy8159205245201868821st_nat @ Xs2 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_1107_non__sym__dist__pairs_H_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( non_sy2820802293568272405list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y != nil_Pr6382449733212294445list_a ) )
       => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ X3 @ Xs2 ) )
             => ( Y
               != ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X3 ) @ Xs2 ) @ ( non_sy2820802293568272405list_a @ Xs2 ) ) ) ) ) ) ).

% non_sym_dist_pairs'.elims
thf(fact_1108_non__sym__dist__pairs_H_Osimps_I1_J,axiom,
    ( ( non_sym_dist_pairs_a @ nil_a )
    = nil_Product_prod_a_a ) ).

% non_sym_dist_pairs'.simps(1)
thf(fact_1109_non__sym__dist__pairs_H_Osimps_I1_J,axiom,
    ( ( non_sy1784707333927398277rs_nat @ nil_nat )
    = nil_Pr5478986624290739719at_nat ) ).

% non_sym_dist_pairs'.simps(1)
thf(fact_1110_list__ordered__pairs_Osimps_I1_J,axiom,
    ( ( list_ordered_pairs_a @ nil_a )
    = nil_Product_prod_a_a ) ).

% list_ordered_pairs.simps(1)
thf(fact_1111_list__ordered__pairs_Osimps_I1_J,axiom,
    ( ( list_o6794894459886620873rs_nat @ nil_nat )
    = nil_Pr5478986624290739719at_nat ) ).

% list_ordered_pairs.simps(1)
thf(fact_1112_list__ordered__pairs__set__containment,axiom,
    ! [X: list_a,Xs: list_list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ( member_list_a @ Y @ ( set_list_a2 @ Xs ) )
       => ( ( X != Y )
         => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( set_Pr2906193453920572092list_a @ ( list_o4514482613960654667list_a @ Xs ) ) )
            | ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ X ) @ ( set_Pr2906193453920572092list_a @ ( list_o4514482613960654667list_a @ Xs ) ) ) ) ) ) ) ).

% list_ordered_pairs_set_containment
thf(fact_1113_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( non_sy4546686295413189007list_a @ ( cons_list_a @ X @ Xs ) )
      = ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X ) @ Xs ) @ ( non_sy4546686295413189007list_a @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_1114_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( non_sym_dist_pairs_a @ ( cons_a @ X @ Xs ) )
      = ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X ) @ Xs ) @ ( non_sym_dist_pairs_a @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_1115_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( non_sy1784707333927398277rs_nat @ ( cons_nat @ X @ Xs ) )
      = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Xs ) @ ( non_sy1784707333927398277rs_nat @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_1116_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( non_sy8159205245201868821st_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X ) @ Xs ) @ ( non_sy8159205245201868821st_nat @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_1117_non__sym__dist__pairs_H_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( non_sy2820802293568272405list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X ) @ Xs ) @ ( non_sy2820802293568272405list_a @ Xs ) ) ) ).

% non_sym_dist_pairs'.simps(2)
thf(fact_1118_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( list_o4514482613960654667list_a @ ( cons_list_a @ X @ Xs ) )
      = ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X ) @ Xs ) @ ( list_o4514482613960654667list_a @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_1119_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( list_ordered_pairs_a @ ( cons_a @ X @ Xs ) )
      = ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X ) @ Xs ) @ ( list_ordered_pairs_a @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_1120_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( list_o6794894459886620873rs_nat @ ( cons_nat @ X @ Xs ) )
      = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Xs ) @ ( list_o6794894459886620873rs_nat @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_1121_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( list_o1383297816974482265st_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X ) @ Xs ) @ ( list_o1383297816974482265st_nat @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_1122_list__ordered__pairs_Osimps_I2_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( list_o7931928007360004049list_a @ ( cons_list_list_a @ X @ Xs ) )
      = ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X ) @ Xs ) @ ( list_o7931928007360004049list_a @ Xs ) ) ) ).

% list_ordered_pairs.simps(2)
thf(fact_1123_list__ordered__pairs_Opelims,axiom,
    ! [X: list_list_a,Y: list_P321204300973800749list_a] :
      ( ( ( list_o4514482613960654667list_a @ X )
        = Y )
     => ( ( accp_list_list_a @ list_o8690503879171716546list_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y = nil_Pr3188421586756112173list_a )
             => ~ ( accp_list_list_a @ list_o8690503879171716546list_a @ nil_list_a ) ) )
         => ~ ! [X3: list_a,Xs2: list_list_a] :
                ( ( X
                  = ( cons_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X3 ) @ Xs2 ) @ ( list_o4514482613960654667list_a @ Xs2 ) ) )
                 => ~ ( accp_list_list_a @ list_o8690503879171716546list_a @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_1124_list__ordered__pairs_Opelims,axiom,
    ! [X: list_a,Y: list_P1396940483166286381od_a_a] :
      ( ( ( list_ordered_pairs_a @ X )
        = Y )
     => ( ( accp_list_a @ list_o4545026682469208124_rel_a @ X )
       => ( ( ( X = nil_a )
           => ( ( Y = nil_Product_prod_a_a )
             => ~ ( accp_list_a @ list_o4545026682469208124_rel_a @ nil_a ) ) )
         => ~ ! [X3: a,Xs2: list_a] :
                ( ( X
                  = ( cons_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs2 ) @ ( list_ordered_pairs_a @ Xs2 ) ) )
                 => ~ ( accp_list_a @ list_o4545026682469208124_rel_a @ ( cons_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_1125_list__ordered__pairs_Opelims,axiom,
    ! [X: list_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( list_o6794894459886620873rs_nat @ X )
        = Y )
     => ( ( accp_list_nat @ list_o5556262271194772242el_nat @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_list_nat @ list_o5556262271194772242el_nat @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Xs2 ) @ ( list_o6794894459886620873rs_nat @ Xs2 ) ) )
                 => ~ ( accp_list_nat @ list_o5556262271194772242el_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_1126_list__ordered__pairs_Opelims,axiom,
    ! [X: list_list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( list_o1383297816974482265st_nat @ X )
        = Y )
     => ( ( accp_list_list_nat @ list_o7370866980707665186st_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y = nil_Pr8413428694792600231st_nat )
             => ~ ( accp_list_list_nat @ list_o7370866980707665186st_nat @ nil_list_nat ) ) )
         => ~ ! [X3: list_nat,Xs2: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X3 ) @ Xs2 ) @ ( list_o1383297816974482265st_nat @ Xs2 ) ) )
                 => ~ ( accp_list_list_nat @ list_o7370866980707665186st_nat @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_1127_list__ordered__pairs_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( list_o7931928007360004049list_a @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ list_o1222659641216536264list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y = nil_Pr6382449733212294445list_a )
             => ~ ( accp_l3666201008454774169list_a @ list_o1222659641216536264list_a @ nil_list_list_a ) ) )
         => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X3 ) @ Xs2 ) @ ( list_o7931928007360004049list_a @ Xs2 ) ) )
                 => ~ ( accp_l3666201008454774169list_a @ list_o1222659641216536264list_a @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% list_ordered_pairs.pelims
thf(fact_1128_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X: list_list_a,Y: list_P321204300973800749list_a] :
      ( ( ( non_sy4546686295413189007list_a @ X )
        = Y )
     => ( ( accp_list_list_a @ non_sy5541662938789404798list_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y = nil_Pr3188421586756112173list_a )
             => ~ ( accp_list_list_a @ non_sy5541662938789404798list_a @ nil_list_a ) ) )
         => ~ ! [X3: list_a,Xs2: list_list_a] :
                ( ( X
                  = ( cons_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append622457529216362434list_a @ ( map_li1163028862020805223list_a @ ( produc6837034575241423639list_a @ X3 ) @ Xs2 ) @ ( non_sy4546686295413189007list_a @ Xs2 ) ) )
                 => ~ ( accp_list_list_a @ non_sy5541662938789404798list_a @ ( cons_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_1129_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X: list_a,Y: list_P1396940483166286381od_a_a] :
      ( ( ( non_sym_dist_pairs_a @ X )
        = Y )
     => ( ( accp_list_a @ non_sy8358513589392214264_rel_a @ X )
       => ( ( ( X = nil_a )
           => ( ( Y = nil_Product_prod_a_a )
             => ~ ( accp_list_a @ non_sy8358513589392214264_rel_a @ nil_a ) ) )
         => ~ ! [X3: a,Xs2: list_a] :
                ( ( X
                  = ( cons_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append5335208819046833346od_a_a @ ( map_a_7860052162900579309od_a_a @ ( product_Pair_a_a @ X3 ) @ Xs2 ) @ ( non_sym_dist_pairs_a @ Xs2 ) ) )
                 => ~ ( accp_list_a @ non_sy8358513589392214264_rel_a @ ( cons_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_1130_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X: list_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( non_sy1784707333927398277rs_nat @ X )
        = Y )
     => ( ( accp_list_nat @ non_sy6628168677365187414el_nat @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_list_nat @ non_sy6628168677365187414el_nat @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X3 ) @ Xs2 ) @ ( non_sy1784707333927398277rs_nat @ Xs2 ) ) )
                 => ~ ( accp_list_nat @ non_sy6628168677365187414el_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_1131_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X: list_list_nat,Y: list_P7940050157051400743st_nat] :
      ( ( ( non_sy8159205245201868821st_nat @ X )
        = Y )
     => ( ( accp_list_list_nat @ non_sy2220477380124107622st_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y = nil_Pr8413428694792600231st_nat )
             => ~ ( accp_list_list_nat @ non_sy2220477380124107622st_nat @ nil_list_nat ) ) )
         => ~ ! [X3: list_nat,Xs2: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append2623875052807961020st_nat @ ( map_li6665866383299509243st_nat @ ( produc2694037385005941721st_nat @ X3 ) @ Xs2 ) @ ( non_sy8159205245201868821st_nat @ Xs2 ) ) )
                 => ~ ( accp_list_list_nat @ non_sy2220477380124107622st_nat @ ( cons_list_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_1132_non__sym__dist__pairs_H_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_P3352141576573427245list_a] :
      ( ( ( non_sy2820802293568272405list_a @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ non_sy3806674336369519492list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y = nil_Pr6382449733212294445list_a )
             => ~ ( accp_l3666201008454774169list_a @ non_sy3806674336369519492list_a @ nil_list_list_a ) ) )
         => ~ ! [X3: list_list_a,Xs2: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ X3 @ Xs2 ) )
               => ( ( Y
                    = ( append8923838579933563074list_a @ ( map_li2617854710493732705list_a @ ( produc8696003437204565271list_a @ X3 ) @ Xs2 ) @ ( non_sy2820802293568272405list_a @ Xs2 ) ) )
                 => ~ ( accp_l3666201008454774169list_a @ non_sy3806674336369519492list_a @ ( cons_list_list_a @ X3 @ Xs2 ) ) ) ) ) ) ) ).

% non_sym_dist_pairs'.pelims
thf(fact_1133_product__concat__map,axiom,
    ( produc2437005677598146871list_a
    = ( ^ [Xs3: list_o,Ys3: list_list_list_a] :
          ( concat3647740114120321498list_a
          @ ( map_o_6649159865426111947list_a
            @ ^ [X4: $o] : ( map_li1526341635108603219list_a @ ( produc7347373609238049417list_a @ X4 ) @ Ys3 )
            @ Xs3 ) ) ) ) ).

% product_concat_map
thf(fact_1134_product__concat__map,axiom,
    ( produc591783268217190531st_nat
    = ( ^ [Xs3: list_o,Ys3: list_list_list_nat] :
          ( concat6350014399496557622st_nat
          @ ( map_o_8982006751275110939st_nat
            @ ^ [X4: $o] : ( map_li2622555147623879021st_nat @ ( produc8892817347036706481st_nat @ X4 ) @ Ys3 )
            @ Xs3 ) ) ) ) ).

% product_concat_map
thf(fact_1135_product__concat__map,axiom,
    ( produc3636039701811874365list_a
    = ( ^ [Xs3: list_o,Ys3: list_l7815035709764258326list_a] :
          ( concat2355107953516655584list_a
          @ ( map_o_6639142526742297041list_a
            @ ^ [X4: $o] : ( map_li7157115275402008147list_a @ ( produc8548274197880810127list_a @ X4 ) @ Ys3 )
            @ Xs3 ) ) ) ) ).

% product_concat_map
thf(fact_1136_set__relcomp,axiom,
    ! [Xys: list_P4002435161011370285od_o_o,Yzs: list_P369347992569427103list_a] :
      ( ( relcom4444278369425376969list_a @ ( set_Product_prod_o_o2 @ Xys ) @ ( set_Pr6916019737621569582list_a @ Yzs ) )
      = ( set_Pr6916019737621569582list_a
        @ ( concat3647740114120321498list_a
          @ ( map_Pr6666266010173600124list_a
            @ ^ [Xy: product_prod_o_o] :
                ( concat3647740114120321498list_a
                @ ( map_Pr2543668955598506378list_a
                  @ ^ [Yz: produc2922778490715282969list_a] :
                      ( if_lis8333758933708535141list_a
                      @ ( ( product_snd_o_o @ Xy )
                        = ( produc3726311466776224373list_a @ Yz ) )
                      @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( product_fst_o_o @ Xy ) @ ( produc4969414473911996599list_a @ Yz ) ) @ nil_Pr1488683542974317855list_a )
                      @ nil_Pr1488683542974317855list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1137_set__relcomp,axiom,
    ! [Xys: list_P4002435161011370285od_o_o,Yzs: list_P3026376796534914631st_nat] :
      ( ( relcom5279109012999557745st_nat @ ( set_Product_prod_o_o2 @ Xys ) @ ( set_Pr7556831416313966818st_nat @ Yzs ) )
      = ( set_Pr7556831416313966818st_nat
        @ ( concat6350014399496557622st_nat
          @ ( map_Pr7742271846857602090st_nat
            @ ^ [Xy: product_prod_o_o] :
                ( concat6350014399496557622st_nat
                @ ( map_Pr1495674614592622788st_nat
                  @ ^ [Yz: produc7974263126369132087st_nat] :
                      ( if_lis773845013453699073st_nat
                      @ ( ( product_snd_o_o @ Xy )
                        = ( produc3357746125391881157st_nat @ Yz ) )
                      @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ ( product_fst_o_o @ Xy ) @ ( produc4024048262274189571st_nat @ Yz ) ) @ nil_Pr1938078369830740401st_nat )
                      @ nil_Pr1938078369830740401st_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1138_set__relcomp,axiom,
    ! [Xys: list_P4002435161011370285od_o_o,Yzs: list_P1915399164375025317list_a] :
      ( ( relcom5509885912654421711list_a @ ( set_Product_prod_o_o2 @ Xys ) @ ( set_Pr1118946995495890996list_a @ Yzs ) )
      = ( set_Pr1118946995495890996list_a
        @ ( concat2355107953516655584list_a
          @ ( map_Pr3177652381921472770list_a
            @ ^ [Xy: product_prod_o_o] :
                ( concat2355107953516655584list_a
                @ ( map_Pr947491794279195018list_a
                  @ ^ [Yz: produc2086622085225315487list_a] :
                      ( if_lis924885042395737195list_a
                      @ ( ( product_snd_o_o @ Xy )
                        = ( produc2785537127425143931list_a @ Yz ) )
                      @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ ( product_fst_o_o @ Xy ) @ ( produc1153002459082627517list_a @ Yz ) ) @ nil_Pr7514062926313711781list_a )
                      @ nil_Pr7514062926313711781list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1139_set__relcomp,axiom,
    ! [Xys: list_P369347992569427103list_a,Yzs: list_P3352141576573427245list_a] :
      ( ( relcom4954185536521216343list_a @ ( set_Pr6916019737621569582list_a @ Xys ) @ ( set_Pr1480943573297899964list_a @ Yzs ) )
      = ( set_Pr6916019737621569582list_a
        @ ( concat3647740114120321498list_a
          @ ( map_Pr2543668955598506378list_a
            @ ^ [Xy: produc2922778490715282969list_a] :
                ( concat3647740114120321498list_a
                @ ( map_Pr7011287121014099068list_a
                  @ ^ [Yz: produc7709606177366032167list_a] :
                      ( if_lis8333758933708535141list_a
                      @ ( ( produc4969414473911996599list_a @ Xy )
                        = ( produc7159145514487520003list_a @ Yz ) )
                      @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ Xy ) @ ( produc1797960232697640517list_a @ Yz ) ) @ nil_Pr1488683542974317855list_a )
                      @ nil_Pr1488683542974317855list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1140_set__relcomp,axiom,
    ! [Xys: list_P369347992569427103list_a,Yzs: list_P7928193943908101433st_nat] :
      ( ( relcom6284725622913142883st_nat @ ( set_Pr6916019737621569582list_a @ Xys ) @ ( set_Pr4936324642459402196st_nat @ Yzs ) )
      = ( set_Pr7556831416313966818st_nat
        @ ( concat6350014399496557622st_nat
          @ ( map_Pr8709084425266877596st_nat
            @ ^ [Xy: produc2922778490715282969list_a] :
                ( concat6350014399496557622st_nat
                @ ( map_Pr1483391071705932982st_nat
                  @ ^ [Yz: produc7772328520602695081st_nat] :
                      ( if_lis773845013453699073st_nat
                      @ ( ( produc4969414473911996599list_a @ Xy )
                        = ( produc8505036810882991031st_nat @ Yz ) )
                      @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ ( produc3726311466776224373list_a @ Xy ) @ ( produc7937458302913330165st_nat @ Yz ) ) @ nil_Pr1938078369830740401st_nat )
                      @ nil_Pr1938078369830740401st_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1141_set__relcomp,axiom,
    ! [Xys: list_P3026376796534914631st_nat,Yzs: list_P1351122815364470303list_a] :
      ( ( relcom3135839675873117825list_a @ ( set_Pr7556831416313966818st_nat @ Xys ) @ ( set_Pr2617593194225220538list_a @ Yzs ) )
      = ( set_Pr6916019737621569582list_a
        @ ( concat3647740114120321498list_a
          @ ( map_Pr677606885770823522list_a
            @ ^ [Xy: produc7974263126369132087st_nat] :
                ( concat3647740114120321498list_a
                @ ( map_Pr666448489100695946list_a
                  @ ^ [Yz: produc5453597072368513423list_a] :
                      ( if_lis8333758933708535141list_a
                      @ ( ( produc4024048262274189571st_nat @ Xy )
                        = ( produc5356150863842965973list_a @ Yz ) )
                      @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( produc3357746125391881157st_nat @ Xy ) @ ( produc4788572355873305107list_a @ Yz ) ) @ nil_Pr1488683542974317855list_a )
                      @ nil_Pr1488683542974317855list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1142_set__relcomp,axiom,
    ! [Xys: list_P3026376796534914631st_nat,Yzs: list_P4223930160327851207st_nat] :
      ( ( relcom2982748354405717177st_nat @ ( set_Pr7556831416313966818st_nat @ Xys ) @ ( set_Pr5052533350369348182st_nat @ Yzs ) )
      = ( set_Pr7556831416313966818st_nat
        @ ( concat6350014399496557622st_nat
          @ ( map_Pr1495674614592622788st_nat
            @ ^ [Xy: produc7974263126369132087st_nat] :
                ( concat6350014399496557622st_nat
                @ ( map_Pr1282673640243126468st_nat
                  @ ^ [Yz: produc4326814125627636033st_nat] :
                      ( if_lis773845013453699073st_nat
                      @ ( ( produc4024048262274189571st_nat @ Xy )
                        = ( produc5419024975087449189st_nat @ Yz ) )
                      @ ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ ( produc3357746125391881157st_nat @ Xy ) @ ( produc6000943452103592103st_nat @ Yz ) ) @ nil_Pr1938078369830740401st_nat )
                      @ nil_Pr1938078369830740401st_nat )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1143_set__relcomp,axiom,
    ! [Xys: list_P369347992569427103list_a,Yzs: list_P6040676878955417907list_a] :
      ( ( relcom2559849793647598173list_a @ ( set_Pr6916019737621569582list_a @ Xys ) @ ( set_Pr8548807919238929090list_a @ Yzs ) )
      = ( set_Pr1118946995495890996list_a
        @ ( concat2355107953516655584list_a
          @ ( map_Pr8967674728321051152list_a
            @ ^ [Xy: produc2922778490715282969list_a] :
                ( concat2355107953516655584list_a
                @ ( map_Pr3121340218037232252list_a
                  @ ^ [Yz: produc6212782398408408749list_a] :
                      ( if_lis924885042395737195list_a
                      @ ( ( produc4969414473911996599list_a @ Xy )
                        = ( produc4606020466041982473list_a @ Yz ) )
                      @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ ( produc3726311466776224373list_a @ Xy ) @ ( produc3299966549393252427list_a @ Yz ) ) @ nil_Pr7514062926313711781list_a )
                      @ nil_Pr7514062926313711781list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1144_set__relcomp,axiom,
    ! [Xys: list_P3026376796534914631st_nat,Yzs: list_P6103169657716828197list_a] :
      ( ( relcom2223375429488861831list_a @ ( set_Pr7556831416313966818st_nat @ Xys ) @ ( set_Pr4901817304773088704list_a @ Yzs ) )
      = ( set_Pr1118946995495890996list_a
        @ ( concat2355107953516655584list_a
          @ ( map_Pr8739944256166517736list_a
            @ ^ [Xy: produc7974263126369132087st_nat] :
                ( concat2355107953516655584list_a
                @ ( map_Pr2852096833643737482list_a
                  @ ^ [Yz: produc3007176892860194069list_a] :
                      ( if_lis924885042395737195list_a
                      @ ( ( produc4024048262274189571st_nat @ Xy )
                        = ( produc1472907236648737755list_a @ Yz ) )
                      @ ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ ( produc3357746125391881157st_nat @ Xy ) @ ( produc339964094158675225list_a @ Yz ) ) @ nil_Pr7514062926313711781list_a )
                      @ nil_Pr7514062926313711781list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1145_set__relcomp,axiom,
    ! [Xys: list_P1915399164375025317list_a,Yzs: list_P4615593650166146855list_a] :
      ( ( relcom275795964880651601list_a @ ( set_Pr1118946995495890996list_a @ Xys ) @ ( set_Pr1808266184328424118list_a @ Yzs ) )
      = ( set_Pr6916019737621569582list_a
        @ ( concat3647740114120321498list_a
          @ ( map_Pr8142448161384941060list_a
            @ ^ [Xy: produc2086622085225315487list_a] :
                ( concat3647740114120321498list_a
                @ ( map_Pr4145540806009976066list_a
                  @ ^ [Yz: produc8695612700352679585list_a] :
                      ( if_lis8333758933708535141list_a
                      @ ( ( produc1153002459082627517list_a @ Xy )
                        = ( produc2321966637275035901list_a @ Yz ) )
                      @ ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ ( produc2785537127425143931list_a @ Xy ) @ ( produc1015912720626305855list_a @ Yz ) ) @ nil_Pr1488683542974317855list_a )
                      @ nil_Pr1488683542974317855list_a )
                  @ Yzs ) )
            @ Xys ) ) ) ) ).

% set_relcomp
thf(fact_1146_List_Oproduct_Osimps_I2_J,axiom,
    ! [X: $o,Xs: list_o,Ys: list_list_list_a] :
      ( ( produc2437005677598146871list_a @ ( cons_o @ X @ Xs ) @ Ys )
      = ( append950407882155939892list_a @ ( map_li1526341635108603219list_a @ ( produc7347373609238049417list_a @ X ) @ Ys ) @ ( produc2437005677598146871list_a @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_1147_List_Oproduct_Osimps_I2_J,axiom,
    ! [X: $o,Xs: list_o,Ys: list_list_list_nat] :
      ( ( produc591783268217190531st_nat @ ( cons_o @ X @ Xs ) @ Ys )
      = ( append686362243666240348st_nat @ ( map_li2622555147623879021st_nat @ ( produc8892817347036706481st_nat @ X ) @ Ys ) @ ( produc591783268217190531st_nat @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_1148_List_Oproduct_Osimps_I2_J,axiom,
    ! [X: $o,Xs: list_o,Ys: list_l7815035709764258326list_a] :
      ( ( produc3636039701811874365list_a @ ( cons_o @ X @ Xs ) @ Ys )
      = ( append6767612481087204666list_a @ ( map_li7157115275402008147list_a @ ( produc8548274197880810127list_a @ X ) @ Ys ) @ ( produc3636039701811874365list_a @ Xs @ Ys ) ) ) ).

% List.product.simps(2)
thf(fact_1149_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Product_prod_o_o,S2: set_Pr9210667514870872697list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom4444278369425376969list_a @ R @ S2 ) )
     => ~ ! [B3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ B3 ) @ R )
           => ~ ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1150_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Product_prod_o_o,S2: set_Pr9088102690956498413st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom5279109012999557745st_nat @ R @ S2 ) )
     => ~ ! [B3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ B3 ) @ R )
           => ~ ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1151_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_list_a,R: set_Product_prod_o_o,S2: set_Pr4230474820173600639list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A1 @ A22 ) @ ( relcom5509885912654421711list_a @ R @ S2 ) )
     => ~ ! [B3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A1 @ B3 ) @ R )
           => ~ ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1152_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr5382606609415531783list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom4954185536521216343list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_a] :
            ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ B3 ) @ R )
           => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1153_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Pr9088102690956498413st_nat,S2: set_Pr7190871385473524933list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom3135839675873117825list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_nat] :
            ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ B3 ) @ R )
           => ~ ( member2537990086590096678list_a @ ( produc1647677298281351873list_a @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1154_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Pr9210667514870872697list_a,S2: set_Pr4544570477162380255st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom6284725622913142883st_nat @ R @ S2 ) )
     => ~ ! [B3: list_list_a] :
            ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ B3 ) @ R )
           => ~ ( member4856721534824278336st_nat @ ( produc4796563245321376931st_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1155_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Pr9088102690956498413st_nat,S2: set_Pr1190453367779242145st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom2982748354405717177st_nat @ R @ S2 ) )
     => ~ ! [B3: list_list_nat] :
            ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ B3 ) @ R )
           => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1156_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Pr4230474820173600639list_a,S2: set_Pr2948258354300535041list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom275795964880651601list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_list_a] :
            ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A1 @ B3 ) @ R )
           => ~ ( member555686232370031178list_a @ ( produc3691048671751139601list_a @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1157_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Pr4230474820173600639list_a,S2: set_Pr4341903704603966821st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom5566656909361316329st_nat @ R @ S2 ) )
     => ~ ! [B3: list_list_list_a] :
            ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A1 @ B3 ) @ R )
           => ~ ( member7739137107329898438st_nat @ ( produc8334297056750347817st_nat @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1158_relcomp_Ocases,axiom,
    ! [A1: $o,A22: list_list_list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr4373341583089806093list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A1 @ A22 ) @ ( relcom2559849793647598173list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_a] :
            ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ B3 ) @ R )
           => ~ ( member7296227967280536150list_a @ ( produc5975102500518086173list_a @ B3 @ A22 ) @ S2 ) ) ) ).

% relcomp.cases
thf(fact_1159_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Product_prod_o_o,S2: set_Pr9210667514870872697list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom4444278369425376969list_a @ R @ S2 ) )
      = ( ? [A4: $o,B4: $o,C4: list_list_a] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B4 ) @ R )
            & ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1160_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Product_prod_o_o,S2: set_Pr9088102690956498413st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom5279109012999557745st_nat @ R @ S2 ) )
      = ( ? [A4: $o,B4: $o,C4: list_list_nat] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B4 ) @ R )
            & ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1161_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_list_a,R: set_Product_prod_o_o,S2: set_Pr4230474820173600639list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A1 @ A22 ) @ ( relcom5509885912654421711list_a @ R @ S2 ) )
      = ( ? [A4: $o,B4: $o,C4: list_list_list_a] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A4 @ B4 ) @ R )
            & ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1162_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr5382606609415531783list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom4954185536521216343list_a @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_a,C4: list_list_a] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A4 @ B4 ) @ R )
            & ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1163_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Pr9088102690956498413st_nat,S2: set_Pr7190871385473524933list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom3135839675873117825list_a @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_nat,C4: list_list_a] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A4 @ B4 ) @ R )
            & ( member2537990086590096678list_a @ ( produc1647677298281351873list_a @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1164_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Pr9210667514870872697list_a,S2: set_Pr4544570477162380255st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom6284725622913142883st_nat @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_a,C4: list_list_nat] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A4 @ B4 ) @ R )
            & ( member4856721534824278336st_nat @ ( produc4796563245321376931st_nat @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1165_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Pr9088102690956498413st_nat,S2: set_Pr1190453367779242145st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom2982748354405717177st_nat @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_nat,C4: list_list_nat] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A4 @ B4 ) @ R )
            & ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1166_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_a,R: set_Pr4230474820173600639list_a,S2: set_Pr2948258354300535041list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A1 @ A22 ) @ ( relcom275795964880651601list_a @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_list_a,C4: list_list_a] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A4 @ B4 ) @ R )
            & ( member555686232370031178list_a @ ( produc3691048671751139601list_a @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1167_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_nat,R: set_Pr4230474820173600639list_a,S2: set_Pr4341903704603966821st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A1 @ A22 ) @ ( relcom5566656909361316329st_nat @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_list_a,C4: list_list_nat] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A4 @ B4 ) @ R )
            & ( member7739137107329898438st_nat @ ( produc8334297056750347817st_nat @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1168_relcomp_Osimps,axiom,
    ! [A1: $o,A22: list_list_list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr4373341583089806093list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A1 @ A22 ) @ ( relcom2559849793647598173list_a @ R @ S2 ) )
      = ( ? [A4: $o,B4: list_list_a,C4: list_list_list_a] :
            ( ( A1 = A4 )
            & ( A22 = C4 )
            & ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A4 @ B4 ) @ R )
            & ( member7296227967280536150list_a @ ( produc5975102500518086173list_a @ B4 @ C4 ) @ S2 ) ) ) ) ).

% relcomp.simps
thf(fact_1169_relcomp_OrelcompI,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o,C: list_list_a,S2: set_Pr9210667514870872697list_a] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
     => ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ B @ C ) @ S2 )
       => ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom4444278369425376969list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1170_relcomp_OrelcompI,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o,C: list_list_nat,S2: set_Pr9088102690956498413st_nat] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
     => ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ B @ C ) @ S2 )
       => ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom5279109012999557745st_nat @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1171_relcomp_OrelcompI,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o,C: list_list_list_a,S2: set_Pr4230474820173600639list_a] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
     => ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ B @ C ) @ S2 )
       => ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ C ) @ ( relcom5509885912654421711list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1172_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_a,R: set_Pr9210667514870872697list_a,C: list_list_a,S2: set_Pr5382606609415531783list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ B ) @ R )
     => ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ B @ C ) @ S2 )
       => ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom4954185536521216343list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1173_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_a,R: set_Pr9210667514870872697list_a,C: list_list_nat,S2: set_Pr4544570477162380255st_nat] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ B ) @ R )
     => ( ( member4856721534824278336st_nat @ ( produc4796563245321376931st_nat @ B @ C ) @ S2 )
       => ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom6284725622913142883st_nat @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1174_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_nat,R: set_Pr9088102690956498413st_nat,C: list_list_a,S2: set_Pr7190871385473524933list_a] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ B ) @ R )
     => ( ( member2537990086590096678list_a @ ( produc1647677298281351873list_a @ B @ C ) @ S2 )
       => ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom3135839675873117825list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1175_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_nat,R: set_Pr9088102690956498413st_nat,C: list_list_nat,S2: set_Pr1190453367779242145st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ B ) @ R )
     => ( ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ B @ C ) @ S2 )
       => ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom2982748354405717177st_nat @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1176_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_a,R: set_Pr9210667514870872697list_a,C: list_list_list_a,S2: set_Pr4373341583089806093list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ B ) @ R )
     => ( ( member7296227967280536150list_a @ ( produc5975102500518086173list_a @ B @ C ) @ S2 )
       => ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ C ) @ ( relcom2559849793647598173list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1177_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_nat,R: set_Pr9088102690956498413st_nat,C: list_list_list_a,S2: set_Pr4220907699740535755list_a] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ B ) @ R )
     => ( ( member459125995552069932list_a @ ( produc4991015576877893319list_a @ B @ C ) @ S2 )
       => ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ C ) @ ( relcom2223375429488861831list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1178_relcomp_OrelcompI,axiom,
    ! [A: $o,B: list_list_list_a,R: set_Pr4230474820173600639list_a,C: list_list_a,S2: set_Pr2948258354300535041list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ B ) @ R )
     => ( ( member555686232370031178list_a @ ( produc3691048671751139601list_a @ B @ C ) @ S2 )
       => ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom275795964880651601list_a @ R @ S2 ) ) ) ) ).

% relcomp.relcompI
thf(fact_1179_relcompE,axiom,
    ! [Xz: produc2922778490715282969list_a,R: set_Product_prod_o_o,S2: set_Pr9210667514870872697list_a] :
      ( ( member3472001477584205762list_a @ Xz @ ( relcom4444278369425376969list_a @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: $o,Z5: list_list_a] :
            ( ( Xz
              = ( produc7347373609238049417list_a @ X3 @ Z5 ) )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R )
             => ~ ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1180_relcompE,axiom,
    ! [Xz: produc7974263126369132087st_nat,R: set_Product_prod_o_o,S2: set_Pr9088102690956498413st_nat] :
      ( ( member5290084443887197262st_nat @ Xz @ ( relcom5279109012999557745st_nat @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: $o,Z5: list_list_nat] :
            ( ( Xz
              = ( produc8892817347036706481st_nat @ X3 @ Z5 ) )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R )
             => ~ ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1181_relcompE,axiom,
    ! [Xz: produc2086622085225315487list_a,R: set_Product_prod_o_o,S2: set_Pr4230474820173600639list_a] :
      ( ( member767485897379589576list_a @ Xz @ ( relcom5509885912654421711list_a @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: $o,Z5: list_list_list_a] :
            ( ( Xz
              = ( produc8548274197880810127list_a @ X3 @ Z5 ) )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R )
             => ~ ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1182_relcompE,axiom,
    ! [Xz: produc2922778490715282969list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr5382606609415531783list_a] :
      ( ( member3472001477584205762list_a @ Xz @ ( relcom4954185536521216343list_a @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_a,Z5: list_list_a] :
            ( ( Xz
              = ( produc7347373609238049417list_a @ X3 @ Z5 ) )
           => ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ X3 @ Y3 ) @ R )
             => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1183_relcompE,axiom,
    ! [Xz: produc2922778490715282969list_a,R: set_Pr9088102690956498413st_nat,S2: set_Pr7190871385473524933list_a] :
      ( ( member3472001477584205762list_a @ Xz @ ( relcom3135839675873117825list_a @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_nat,Z5: list_list_a] :
            ( ( Xz
              = ( produc7347373609238049417list_a @ X3 @ Z5 ) )
           => ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ X3 @ Y3 ) @ R )
             => ~ ( member2537990086590096678list_a @ ( produc1647677298281351873list_a @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1184_relcompE,axiom,
    ! [Xz: produc7974263126369132087st_nat,R: set_Pr9210667514870872697list_a,S2: set_Pr4544570477162380255st_nat] :
      ( ( member5290084443887197262st_nat @ Xz @ ( relcom6284725622913142883st_nat @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_a,Z5: list_list_nat] :
            ( ( Xz
              = ( produc8892817347036706481st_nat @ X3 @ Z5 ) )
           => ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ X3 @ Y3 ) @ R )
             => ~ ( member4856721534824278336st_nat @ ( produc4796563245321376931st_nat @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1185_relcompE,axiom,
    ! [Xz: produc7974263126369132087st_nat,R: set_Pr9088102690956498413st_nat,S2: set_Pr1190453367779242145st_nat] :
      ( ( member5290084443887197262st_nat @ Xz @ ( relcom2982748354405717177st_nat @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_nat,Z5: list_list_nat] :
            ( ( Xz
              = ( produc8892817347036706481st_nat @ X3 @ Z5 ) )
           => ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1186_relcompE,axiom,
    ! [Xz: produc2922778490715282969list_a,R: set_Pr4230474820173600639list_a,S2: set_Pr2948258354300535041list_a] :
      ( ( member3472001477584205762list_a @ Xz @ ( relcom275795964880651601list_a @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_list_a,Z5: list_list_a] :
            ( ( Xz
              = ( produc7347373609238049417list_a @ X3 @ Z5 ) )
           => ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ X3 @ Y3 ) @ R )
             => ~ ( member555686232370031178list_a @ ( produc3691048671751139601list_a @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1187_relcompE,axiom,
    ! [Xz: produc7974263126369132087st_nat,R: set_Pr4230474820173600639list_a,S2: set_Pr4341903704603966821st_nat] :
      ( ( member5290084443887197262st_nat @ Xz @ ( relcom5566656909361316329st_nat @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_list_a,Z5: list_list_nat] :
            ( ( Xz
              = ( produc8892817347036706481st_nat @ X3 @ Z5 ) )
           => ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ X3 @ Y3 ) @ R )
             => ~ ( member7739137107329898438st_nat @ ( produc8334297056750347817st_nat @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1188_relcompE,axiom,
    ! [Xz: produc2086622085225315487list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr4373341583089806093list_a] :
      ( ( member767485897379589576list_a @ Xz @ ( relcom2559849793647598173list_a @ R @ S2 ) )
     => ~ ! [X3: $o,Y3: list_list_a,Z5: list_list_list_a] :
            ( ( Xz
              = ( produc8548274197880810127list_a @ X3 @ Z5 ) )
           => ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ X3 @ Y3 ) @ R )
             => ~ ( member7296227967280536150list_a @ ( produc5975102500518086173list_a @ Y3 @ Z5 ) @ S2 ) ) ) ) ).

% relcompE
thf(fact_1189_pred__equals__eq2,axiom,
    ! [R2: set_Pr9210667514870872697list_a,S3: set_Pr9210667514870872697list_a] :
      ( ( ( ^ [X4: $o,Y5: list_list_a] : ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: $o,Y5: list_list_a] : ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_1190_pred__equals__eq2,axiom,
    ! [R2: set_Pr9088102690956498413st_nat,S3: set_Pr9088102690956498413st_nat] :
      ( ( ( ^ [X4: $o,Y5: list_list_nat] : ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: $o,Y5: list_list_nat] : ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_1191_pred__equals__eq2,axiom,
    ! [R2: set_Pr4230474820173600639list_a,S3: set_Pr4230474820173600639list_a] :
      ( ( ( ^ [X4: $o,Y5: list_list_list_a] : ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: $o,Y5: list_list_list_a] : ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_1192_relcompEpair,axiom,
    ! [A: $o,C: list_list_a,R: set_Product_prod_o_o,S2: set_Pr9210667514870872697list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom4444278369425376969list_a @ R @ S2 ) )
     => ~ ! [B3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B3 ) @ R )
           => ~ ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1193_relcompEpair,axiom,
    ! [A: $o,C: list_list_nat,R: set_Product_prod_o_o,S2: set_Pr9088102690956498413st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom5279109012999557745st_nat @ R @ S2 ) )
     => ~ ! [B3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B3 ) @ R )
           => ~ ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1194_relcompEpair,axiom,
    ! [A: $o,C: list_list_list_a,R: set_Product_prod_o_o,S2: set_Pr4230474820173600639list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ C ) @ ( relcom5509885912654421711list_a @ R @ S2 ) )
     => ~ ! [B3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B3 ) @ R )
           => ~ ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1195_relcompEpair,axiom,
    ! [A: $o,C: list_list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr5382606609415531783list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom4954185536521216343list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_a] :
            ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ B3 ) @ R )
           => ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1196_relcompEpair,axiom,
    ! [A: $o,C: list_list_a,R: set_Pr9088102690956498413st_nat,S2: set_Pr7190871385473524933list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom3135839675873117825list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_nat] :
            ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ B3 ) @ R )
           => ~ ( member2537990086590096678list_a @ ( produc1647677298281351873list_a @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1197_relcompEpair,axiom,
    ! [A: $o,C: list_list_nat,R: set_Pr9210667514870872697list_a,S2: set_Pr4544570477162380255st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom6284725622913142883st_nat @ R @ S2 ) )
     => ~ ! [B3: list_list_a] :
            ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ B3 ) @ R )
           => ~ ( member4856721534824278336st_nat @ ( produc4796563245321376931st_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1198_relcompEpair,axiom,
    ! [A: $o,C: list_list_nat,R: set_Pr9088102690956498413st_nat,S2: set_Pr1190453367779242145st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom2982748354405717177st_nat @ R @ S2 ) )
     => ~ ! [B3: list_list_nat] :
            ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ B3 ) @ R )
           => ~ ( member8680655010358287850st_nat @ ( produc7129799990162260089st_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1199_relcompEpair,axiom,
    ! [A: $o,C: list_list_a,R: set_Pr4230474820173600639list_a,S2: set_Pr2948258354300535041list_a] :
      ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ C ) @ ( relcom275795964880651601list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_list_a] :
            ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ B3 ) @ R )
           => ~ ( member555686232370031178list_a @ ( produc3691048671751139601list_a @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1200_relcompEpair,axiom,
    ! [A: $o,C: list_list_nat,R: set_Pr4230474820173600639list_a,S2: set_Pr4341903704603966821st_nat] :
      ( ( member5290084443887197262st_nat @ ( produc8892817347036706481st_nat @ A @ C ) @ ( relcom5566656909361316329st_nat @ R @ S2 ) )
     => ~ ! [B3: list_list_list_a] :
            ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ B3 ) @ R )
           => ~ ( member7739137107329898438st_nat @ ( produc8334297056750347817st_nat @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1201_relcompEpair,axiom,
    ! [A: $o,C: list_list_list_a,R: set_Pr9210667514870872697list_a,S2: set_Pr4373341583089806093list_a] :
      ( ( member767485897379589576list_a @ ( produc8548274197880810127list_a @ A @ C ) @ ( relcom2559849793647598173list_a @ R @ S2 ) )
     => ~ ! [B3: list_list_a] :
            ( ( member3472001477584205762list_a @ ( produc7347373609238049417list_a @ A @ B3 ) @ R )
           => ~ ( member7296227967280536150list_a @ ( produc5975102500518086173list_a @ B3 @ C ) @ S2 ) ) ) ).

% relcompEpair
thf(fact_1202_update__keys,axiom,
    ! [K3: $o,Al: list_P369347992569427103list_a,V2: list_list_a] :
      ( ( ( member_o @ K3 @ ( set_o2 @ ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) ) )
       => ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ ( update_o_list_list_a @ K3 @ V2 @ Al ) )
          = ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) ) )
      & ( ~ ( member_o @ K3 @ ( set_o2 @ ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) ) )
       => ( ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ ( update_o_list_list_a @ K3 @ V2 @ Al ) )
          = ( append_o @ ( map_Pr5241763764686268419st_a_o @ produc3726311466776224373list_a @ Al ) @ ( cons_o @ K3 @ nil_o ) ) ) ) ) ).

% update_keys
thf(fact_1203_update__keys,axiom,
    ! [K3: $o,Al: list_P3026376796534914631st_nat,V2: list_list_nat] :
      ( ( ( member_o @ K3 @ ( set_o2 @ ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) ) )
       => ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ ( update2175254896392664972st_nat @ K3 @ V2 @ Al ) )
          = ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) ) )
      & ( ~ ( member_o @ K3 @ ( set_o2 @ ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) ) )
       => ( ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ ( update2175254896392664972st_nat @ K3 @ V2 @ Al ) )
          = ( append_o @ ( map_Pr6074390301017694683_nat_o @ produc3357746125391881157st_nat @ Al ) @ ( cons_o @ K3 @ nil_o ) ) ) ) ) ).

% update_keys
thf(fact_1204_update__keys,axiom,
    ! [K3: $o,Al: list_P1915399164375025317list_a,V2: list_list_list_a] :
      ( ( ( member_o @ K3 @ ( set_o2 @ ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) ) )
       => ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ ( update3637859629751153652list_a @ K3 @ V2 @ Al ) )
          = ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) ) )
      & ( ~ ( member_o @ K3 @ ( set_o2 @ ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) ) )
       => ( ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ ( update3637859629751153652list_a @ K3 @ V2 @ Al ) )
          = ( append_o @ ( map_Pr3417367988113718653st_a_o @ produc2785537127425143931list_a @ Al ) @ ( cons_o @ K3 @ nil_o ) ) ) ) ) ).

% update_keys
thf(fact_1205_Id__on__set,axiom,
    ! [Xs: list_list_a] :
      ( ( id_on_list_a @ ( set_list_a2 @ Xs ) )
      = ( set_Pr2906193453920572092list_a
        @ ( map_li1163028862020805223list_a
          @ ^ [X4: list_a] : ( produc6837034575241423639list_a @ X4 @ X4 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1206_Id__onI,axiom,
    ! [A: list_a,A5: set_list_a] :
      ( ( member_list_a @ A @ A5 )
     => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ A ) @ ( id_on_list_a @ A5 ) ) ) ).

% Id_onI
thf(fact_1207_Id__on__iff,axiom,
    ! [X: list_a,Y: list_a,A5: set_list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( id_on_list_a @ A5 ) )
      = ( ( X = Y )
        & ( member_list_a @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1208_Id__on__eqI,axiom,
    ! [A: list_a,B: list_a,A5: set_list_a] :
      ( ( A = B )
     => ( ( member_list_a @ A @ A5 )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B ) @ ( id_on_list_a @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1209_Id__onE,axiom,
    ! [C: produc9164743771328383783list_a,A5: set_list_a] :
      ( ( member8191768239178080336list_a @ C @ ( id_on_list_a @ A5 ) )
     => ~ ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A5 )
           => ( C
             != ( produc6837034575241423639list_a @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1210_update_Osimps_I2_J,axiom,
    ! [P2: produc2922778490715282969list_a,K3: $o,V2: list_list_a,Ps2: list_P369347992569427103list_a] :
      ( ( ( ( produc3726311466776224373list_a @ P2 )
          = K3 )
       => ( ( update_o_list_list_a @ K3 @ V2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V2 ) @ Ps2 ) ) )
      & ( ( ( produc3726311466776224373list_a @ P2 )
          = ~ K3 )
       => ( ( update_o_list_list_a @ K3 @ V2 @ ( cons_P3051609457517836495list_a @ P2 @ Ps2 ) )
          = ( cons_P3051609457517836495list_a @ P2 @ ( update_o_list_list_a @ K3 @ V2 @ Ps2 ) ) ) ) ) ).

% update.simps(2)
thf(fact_1211_update_Osimps_I2_J,axiom,
    ! [P2: produc7974263126369132087st_nat,K3: $o,V2: list_list_nat,Ps2: list_P3026376796534914631st_nat] :
      ( ( ( ( produc3357746125391881157st_nat @ P2 )
          = K3 )
       => ( ( update2175254896392664972st_nat @ K3 @ V2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V2 ) @ Ps2 ) ) )
      & ( ( ( produc3357746125391881157st_nat @ P2 )
          = ~ K3 )
       => ( ( update2175254896392664972st_nat @ K3 @ V2 @ ( cons_P5527778705151500801st_nat @ P2 @ Ps2 ) )
          = ( cons_P5527778705151500801st_nat @ P2 @ ( update2175254896392664972st_nat @ K3 @ V2 @ Ps2 ) ) ) ) ) ).

% update.simps(2)
thf(fact_1212_update_Osimps_I2_J,axiom,
    ! [P2: produc2086622085225315487list_a,K3: $o,V2: list_list_list_a,Ps2: list_P1915399164375025317list_a] :
      ( ( ( ( produc2785537127425143931list_a @ P2 )
          = K3 )
       => ( ( update3637859629751153652list_a @ K3 @ V2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V2 ) @ Ps2 ) ) )
      & ( ( ( produc2785537127425143931list_a @ P2 )
          = ~ K3 )
       => ( ( update3637859629751153652list_a @ K3 @ V2 @ ( cons_P2654393883304477781list_a @ P2 @ Ps2 ) )
          = ( cons_P2654393883304477781list_a @ P2 @ ( update3637859629751153652list_a @ K3 @ V2 @ Ps2 ) ) ) ) ) ).

% update.simps(2)
thf(fact_1213_update_Osimps_I1_J,axiom,
    ! [K3: $o,V2: list_list_a] :
      ( ( update_o_list_list_a @ K3 @ V2 @ nil_Pr1488683542974317855list_a )
      = ( cons_P3051609457517836495list_a @ ( produc7347373609238049417list_a @ K3 @ V2 ) @ nil_Pr1488683542974317855list_a ) ) ).

% update.simps(1)
thf(fact_1214_update_Osimps_I1_J,axiom,
    ! [K3: $o,V2: list_list_nat] :
      ( ( update2175254896392664972st_nat @ K3 @ V2 @ nil_Pr1938078369830740401st_nat )
      = ( cons_P5527778705151500801st_nat @ ( produc8892817347036706481st_nat @ K3 @ V2 ) @ nil_Pr1938078369830740401st_nat ) ) ).

% update.simps(1)
thf(fact_1215_update_Osimps_I1_J,axiom,
    ! [K3: $o,V2: list_list_list_a] :
      ( ( update3637859629751153652list_a @ K3 @ V2 @ nil_Pr7514062926313711781list_a )
      = ( cons_P2654393883304477781list_a @ ( produc8548274197880810127list_a @ K3 @ V2 ) @ nil_Pr7514062926313711781list_a ) ) ).

% update.simps(1)
thf(fact_1216_transpose_Opinduct,axiom,
    ! [A0: list_list_list_nat,P: list_list_list_nat > $o] :
      ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ A0 )
     => ( ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ nil_list_list_nat )
         => ( P @ nil_list_list_nat ) )
       => ( ! [Xss: list_list_list_nat] :
              ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_list_list_nat @ nil_list_nat @ Xss ) ) ) )
         => ( ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
                ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_list_list_nat @ Xs2
                      @ ( concat_list_list_nat
                        @ ( map_li5769348595424326838st_nat
                          @ ( case_l9045269829974911560st_nat @ nil_list_list_nat
                            @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_list_nat @ T2 @ nil_list_list_nat ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_1217_transpose_Opinduct,axiom,
    ! [A0: list_l7815035709764258326list_a,P: list_l7815035709764258326list_a > $o] :
      ( ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ A0 )
     => ( ( ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ nil_list_list_list_a )
         => ( P @ nil_list_list_list_a ) )
       => ( ! [Xss: list_l7815035709764258326list_a] :
              ( ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ ( cons_l8577867181958743494list_a @ nil_list_list_a @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_l8577867181958743494list_a @ nil_list_list_a @ Xss ) ) ) )
         => ( ! [X3: list_list_a,Xs2: list_list_list_a,Xss: list_l7815035709764258326list_a] :
                ( ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_l8577867181958743494list_a @ Xs2
                      @ ( concat7147740552513348305list_a
                        @ ( map_li996746621073191882list_a
                          @ ( case_l4623913378254156506list_a @ nil_list_list_list_a
                            @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_l8577867181958743494list_a @ T2 @ nil_list_list_list_a ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_1218_transpose_Opinduct,axiom,
    ! [A0: list_list_a,P: list_list_a > $o] :
      ( ( accp_list_list_a @ transpose_rel_a @ A0 )
     => ( ( ( accp_list_list_a @ transpose_rel_a @ nil_list_a )
         => ( P @ nil_list_a ) )
       => ( ! [Xss: list_list_a] :
              ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_list_a @ nil_a @ Xss ) ) ) )
         => ( ! [X3: a,Xs2: list_a,Xss: list_list_a] :
                ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_list_a @ Xs2
                      @ ( concat_list_a
                        @ ( map_li5729356230488778442list_a
                          @ ( case_l8408404631611421914st_a_a @ nil_list_a
                            @ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_1219_transpose_Opinduct,axiom,
    ! [A0: list_list_nat,P: list_list_nat > $o] :
      ( ( accp_list_list_nat @ transpose_rel_nat @ A0 )
     => ( ( ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat )
         => ( P @ nil_list_nat ) )
       => ( ! [Xss: list_list_nat] :
              ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
                ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_list_nat @ Xs2
                      @ ( concat_list_nat
                        @ ( map_li960784813134754710st_nat
                          @ ( case_l3331202209248957608at_nat @ nil_list_nat
                            @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_1220_transpose_Opinduct,axiom,
    ! [A0: list_list_list_a,P: list_list_list_a > $o] :
      ( ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ A0 )
     => ( ( ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ nil_list_list_a )
         => ( P @ nil_list_list_a ) )
       => ( ! [Xss: list_list_list_a] :
              ( ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ ( cons_list_list_a @ nil_list_a @ Xss ) )
             => ( ( P @ Xss )
               => ( P @ ( cons_list_list_a @ nil_list_a @ Xss ) ) ) )
         => ( ! [X3: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
                ( ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) )
               => ( ( P
                    @ ( cons_list_list_a @ Xs2
                      @ ( concat_list_list_a
                        @ ( map_li5227692475714150986list_a
                          @ ( case_l8090992918877587674list_a @ nil_list_list_a
                            @ ^ [H: list_a,T2: list_list_a] : ( cons_list_list_a @ T2 @ nil_list_list_a ) )
                          @ Xss ) ) ) )
                 => ( P @ ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_1221_transpose_Oelims,axiom,
    ! [X: list_list_list_nat,Y: list_list_list_nat] :
      ( ( ( transpose_list_nat @ X )
        = Y )
     => ( ( ( X = nil_list_list_nat )
         => ( Y != nil_list_list_nat ) )
       => ( ! [Xss: list_list_list_nat] :
              ( ( X
                = ( cons_list_list_nat @ nil_list_nat @ Xss ) )
             => ( Y
               != ( transpose_list_nat @ Xss ) ) )
         => ~ ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
                ( ( X
                  = ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) )
               => ( Y
                 != ( cons_list_list_nat
                    @ ( cons_list_nat @ X3
                      @ ( concat_list_nat
                        @ ( map_li2855073862107769254st_nat
                          @ ( case_l2985193972497081272st_nat @ nil_list_nat
                            @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
                          @ Xss ) ) )
                    @ ( transpose_list_nat
                      @ ( cons_list_list_nat @ Xs2
                        @ ( concat_list_list_nat
                          @ ( map_li5769348595424326838st_nat
                            @ ( case_l9045269829974911560st_nat @ nil_list_list_nat
                              @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_list_nat @ T2 @ nil_list_list_nat ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_1222_transpose_Oelims,axiom,
    ! [X: list_l7815035709764258326list_a,Y: list_l7815035709764258326list_a] :
      ( ( ( transp8616149884816826138list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_list_a )
         => ( Y != nil_list_list_list_a ) )
       => ( ! [Xss: list_l7815035709764258326list_a] :
              ( ( X
                = ( cons_l8577867181958743494list_a @ nil_list_list_a @ Xss ) )
             => ( Y
               != ( transp8616149884816826138list_a @ Xss ) ) )
         => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Xss: list_l7815035709764258326list_a] :
                ( ( X
                  = ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Xss ) )
               => ( Y
                 != ( cons_l8577867181958743494list_a
                    @ ( cons_list_list_a @ X3
                      @ ( concat_list_list_a
                        @ ( map_li6096488569700017860list_a
                          @ ( case_l8039372830345901536list_a @ nil_list_list_a
                            @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_list_list_a @ H @ nil_list_list_a ) )
                          @ Xss ) ) )
                    @ ( transp8616149884816826138list_a
                      @ ( cons_l8577867181958743494list_a @ Xs2
                        @ ( concat7147740552513348305list_a
                          @ ( map_li996746621073191882list_a
                            @ ( case_l4623913378254156506list_a @ nil_list_list_list_a
                              @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_l8577867181958743494list_a @ T2 @ nil_list_list_list_a ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_1223_transpose_Oelims,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( ( transpose_a @ X )
        = Y )
     => ( ( ( X = nil_list_a )
         => ( Y != nil_list_a ) )
       => ( ! [Xss: list_list_a] :
              ( ( X
                = ( cons_list_a @ nil_a @ Xss ) )
             => ( Y
               != ( transpose_a @ Xss ) ) )
         => ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
                ( ( X
                  = ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
               => ( Y
                 != ( cons_list_a
                    @ ( cons_a @ X3
                      @ ( concat_a
                        @ ( map_list_a_list_a
                          @ ( case_list_list_a_a @ nil_a
                            @ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
                          @ Xss ) ) )
                    @ ( transpose_a
                      @ ( cons_list_a @ Xs2
                        @ ( concat_list_a
                          @ ( map_li5729356230488778442list_a
                            @ ( case_l8408404631611421914st_a_a @ nil_list_a
                              @ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_1224_transpose_Oelims,axiom,
    ! [X: list_list_nat,Y: list_list_nat] :
      ( ( ( transpose_nat @ X )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( Y != nil_list_nat ) )
       => ( ! [Xss: list_list_nat] :
              ( ( X
                = ( cons_list_nat @ nil_nat @ Xss ) )
             => ( Y
               != ( transpose_nat @ Xss ) ) )
         => ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
               => ( Y
                 != ( cons_list_nat
                    @ ( cons_nat @ X3
                      @ ( concat_nat
                        @ ( map_li7225945977422193158st_nat
                          @ ( case_l2340614614379431832at_nat @ nil_nat
                            @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
                          @ Xss ) ) )
                    @ ( transpose_nat
                      @ ( cons_list_nat @ Xs2
                        @ ( concat_list_nat
                          @ ( map_li960784813134754710st_nat
                            @ ( case_l3331202209248957608at_nat @ nil_list_nat
                              @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_1225_transpose_Oelims,axiom,
    ! [X: list_list_list_a,Y: list_list_list_a] :
      ( ( ( transpose_list_a @ X )
        = Y )
     => ( ( ( X = nil_list_list_a )
         => ( Y != nil_list_list_a ) )
       => ( ! [Xss: list_list_list_a] :
              ( ( X
                = ( cons_list_list_a @ nil_list_a @ Xss ) )
             => ( Y
               != ( transpose_list_a @ Xss ) ) )
         => ~ ! [X3: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) )
               => ( Y
                 != ( cons_list_list_a
                    @ ( cons_list_a @ X3
                      @ ( concat_list_a
                        @ ( map_li8713736314956022724list_a
                          @ ( case_l6560536634501641056list_a @ nil_list_a
                            @ ^ [H: list_a,T2: list_list_a] : ( cons_list_a @ H @ nil_list_a ) )
                          @ Xss ) ) )
                    @ ( transpose_list_a
                      @ ( cons_list_list_a @ Xs2
                        @ ( concat_list_list_a
                          @ ( map_li5227692475714150986list_a
                            @ ( case_l8090992918877587674list_a @ nil_list_list_a
                              @ ^ [H: list_a,T2: list_list_a] : ( cons_list_list_a @ T2 @ nil_list_list_a ) )
                            @ Xss ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_1226_transpose_Osimps_I3_J,axiom,
    ! [X: list_nat,Xs: list_list_nat,Xss2: list_list_list_nat] :
      ( ( transpose_list_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ Xss2 ) )
      = ( cons_list_list_nat
        @ ( cons_list_nat @ X
          @ ( concat_list_nat
            @ ( map_li2855073862107769254st_nat
              @ ( case_l2985193972497081272st_nat @ nil_list_nat
                @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
              @ Xss2 ) ) )
        @ ( transpose_list_nat
          @ ( cons_list_list_nat @ Xs
            @ ( concat_list_list_nat
              @ ( map_li5769348595424326838st_nat
                @ ( case_l9045269829974911560st_nat @ nil_list_list_nat
                  @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_list_nat @ T2 @ nil_list_list_nat ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_1227_transpose_Osimps_I3_J,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Xss2: list_l7815035709764258326list_a] :
      ( ( transp8616149884816826138list_a @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X @ Xs ) @ Xss2 ) )
      = ( cons_l8577867181958743494list_a
        @ ( cons_list_list_a @ X
          @ ( concat_list_list_a
            @ ( map_li6096488569700017860list_a
              @ ( case_l8039372830345901536list_a @ nil_list_list_a
                @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_list_list_a @ H @ nil_list_list_a ) )
              @ Xss2 ) ) )
        @ ( transp8616149884816826138list_a
          @ ( cons_l8577867181958743494list_a @ Xs
            @ ( concat7147740552513348305list_a
              @ ( map_li996746621073191882list_a
                @ ( case_l4623913378254156506list_a @ nil_list_list_list_a
                  @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_l8577867181958743494list_a @ T2 @ nil_list_list_list_a ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_1228_transpose_Osimps_I3_J,axiom,
    ! [X: a,Xs: list_a,Xss2: list_list_a] :
      ( ( transpose_a @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ Xss2 ) )
      = ( cons_list_a
        @ ( cons_a @ X
          @ ( concat_a
            @ ( map_list_a_list_a
              @ ( case_list_list_a_a @ nil_a
                @ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
              @ Xss2 ) ) )
        @ ( transpose_a
          @ ( cons_list_a @ Xs
            @ ( concat_list_a
              @ ( map_li5729356230488778442list_a
                @ ( case_l8408404631611421914st_a_a @ nil_list_a
                  @ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_1229_transpose_Osimps_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Xss2: list_list_nat] :
      ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ Xss2 ) )
      = ( cons_list_nat
        @ ( cons_nat @ X
          @ ( concat_nat
            @ ( map_li7225945977422193158st_nat
              @ ( case_l2340614614379431832at_nat @ nil_nat
                @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
              @ Xss2 ) ) )
        @ ( transpose_nat
          @ ( cons_list_nat @ Xs
            @ ( concat_list_nat
              @ ( map_li960784813134754710st_nat
                @ ( case_l3331202209248957608at_nat @ nil_list_nat
                  @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_1230_transpose_Osimps_I3_J,axiom,
    ! [X: list_a,Xs: list_list_a,Xss2: list_list_list_a] :
      ( ( transpose_list_a @ ( cons_list_list_a @ ( cons_list_a @ X @ Xs ) @ Xss2 ) )
      = ( cons_list_list_a
        @ ( cons_list_a @ X
          @ ( concat_list_a
            @ ( map_li8713736314956022724list_a
              @ ( case_l6560536634501641056list_a @ nil_list_a
                @ ^ [H: list_a,T2: list_list_a] : ( cons_list_a @ H @ nil_list_a ) )
              @ Xss2 ) ) )
        @ ( transpose_list_a
          @ ( cons_list_list_a @ Xs
            @ ( concat_list_list_a
              @ ( map_li5227692475714150986list_a
                @ ( case_l8090992918877587674list_a @ nil_list_list_a
                  @ ^ [H: list_a,T2: list_list_a] : ( cons_list_list_a @ T2 @ nil_list_list_a ) )
                @ Xss2 ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_1231_transpose_Opelims,axiom,
    ! [X: list_list_list_nat,Y: list_list_list_nat] :
      ( ( ( transpose_list_nat @ X )
        = Y )
     => ( ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ X )
       => ( ( ( X = nil_list_list_nat )
           => ( ( Y = nil_list_list_nat )
             => ~ ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ nil_list_list_nat ) ) )
         => ( ! [Xss: list_list_list_nat] :
                ( ( X
                  = ( cons_list_list_nat @ nil_list_nat @ Xss ) )
               => ( ( Y
                    = ( transpose_list_nat @ Xss ) )
                 => ~ ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ nil_list_nat @ Xss ) ) ) )
           => ~ ! [X3: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
                  ( ( X
                    = ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_list_list_nat
                        @ ( cons_list_nat @ X3
                          @ ( concat_list_nat
                            @ ( map_li2855073862107769254st_nat
                              @ ( case_l2985193972497081272st_nat @ nil_list_nat
                                @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_nat @ H @ nil_list_nat ) )
                              @ Xss ) ) )
                        @ ( transpose_list_nat
                          @ ( cons_list_list_nat @ Xs2
                            @ ( concat_list_list_nat
                              @ ( map_li5769348595424326838st_nat
                                @ ( case_l9045269829974911560st_nat @ nil_list_list_nat
                                  @ ^ [H: list_nat,T2: list_list_nat] : ( cons_list_list_nat @ T2 @ nil_list_list_nat ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_l8765801942216208055st_nat @ transp7462230406613344747st_nat @ ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_1232_transpose_Opelims,axiom,
    ! [X: list_l7815035709764258326list_a,Y: list_l7815035709764258326list_a] :
      ( ( ( transp8616149884816826138list_a @ X )
        = Y )
     => ( ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ X )
       => ( ( ( X = nil_list_list_list_a )
           => ( ( Y = nil_list_list_list_a )
             => ~ ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ nil_list_list_list_a ) ) )
         => ( ! [Xss: list_l7815035709764258326list_a] :
                ( ( X
                  = ( cons_l8577867181958743494list_a @ nil_list_list_a @ Xss ) )
               => ( ( Y
                    = ( transp8616149884816826138list_a @ Xss ) )
                 => ~ ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ ( cons_l8577867181958743494list_a @ nil_list_list_a @ Xss ) ) ) )
           => ~ ! [X3: list_list_a,Xs2: list_list_list_a,Xss: list_l7815035709764258326list_a] :
                  ( ( X
                    = ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_l8577867181958743494list_a
                        @ ( cons_list_list_a @ X3
                          @ ( concat_list_list_a
                            @ ( map_li6096488569700017860list_a
                              @ ( case_l8039372830345901536list_a @ nil_list_list_a
                                @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_list_list_a @ H @ nil_list_list_a ) )
                              @ Xss ) ) )
                        @ ( transp8616149884816826138list_a
                          @ ( cons_l8577867181958743494list_a @ Xs2
                            @ ( concat7147740552513348305list_a
                              @ ( map_li996746621073191882list_a
                                @ ( case_l4623913378254156506list_a @ nil_list_list_list_a
                                  @ ^ [H: list_list_a,T2: list_list_list_a] : ( cons_l8577867181958743494list_a @ T2 @ nil_list_list_list_a ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_l6027334576914615327list_a @ transp4409832848296258239list_a @ ( cons_l8577867181958743494list_a @ ( cons_list_list_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_1233_transpose_Opelims,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( ( transpose_a @ X )
        = Y )
     => ( ( accp_list_list_a @ transpose_rel_a @ X )
       => ( ( ( X = nil_list_a )
           => ( ( Y = nil_list_a )
             => ~ ( accp_list_list_a @ transpose_rel_a @ nil_list_a ) ) )
         => ( ! [Xss: list_list_a] :
                ( ( X
                  = ( cons_list_a @ nil_a @ Xss ) )
               => ( ( Y
                    = ( transpose_a @ Xss ) )
                 => ~ ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss ) ) ) )
           => ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
                  ( ( X
                    = ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_list_a
                        @ ( cons_a @ X3
                          @ ( concat_a
                            @ ( map_list_a_list_a
                              @ ( case_list_list_a_a @ nil_a
                                @ ^ [H: a,T2: list_a] : ( cons_a @ H @ nil_a ) )
                              @ Xss ) ) )
                        @ ( transpose_a
                          @ ( cons_list_a @ Xs2
                            @ ( concat_list_a
                              @ ( map_li5729356230488778442list_a
                                @ ( case_l8408404631611421914st_a_a @ nil_list_a
                                  @ ^ [H: a,T2: list_a] : ( cons_list_a @ T2 @ nil_list_a ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_1234_transpose_Opelims,axiom,
    ! [X: list_list_nat,Y: list_list_nat] :
      ( ( ( transpose_nat @ X )
        = Y )
     => ( ( accp_list_list_nat @ transpose_rel_nat @ X )
       => ( ( ( X = nil_list_nat )
           => ( ( Y = nil_list_nat )
             => ~ ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat ) ) )
         => ( ! [Xss: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ nil_nat @ Xss ) )
               => ( ( Y
                    = ( transpose_nat @ Xss ) )
                 => ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
           => ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
                  ( ( X
                    = ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_list_nat
                        @ ( cons_nat @ X3
                          @ ( concat_nat
                            @ ( map_li7225945977422193158st_nat
                              @ ( case_l2340614614379431832at_nat @ nil_nat
                                @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
                              @ Xss ) ) )
                        @ ( transpose_nat
                          @ ( cons_list_nat @ Xs2
                            @ ( concat_list_nat
                              @ ( map_li960784813134754710st_nat
                                @ ( case_l3331202209248957608at_nat @ nil_list_nat
                                  @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_1235_transpose_Opelims,axiom,
    ! [X: list_list_list_a,Y: list_list_list_a] :
      ( ( ( transpose_list_a @ X )
        = Y )
     => ( ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ X )
       => ( ( ( X = nil_list_list_a )
           => ( ( Y = nil_list_list_a )
             => ~ ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ nil_list_list_a ) ) )
         => ( ! [Xss: list_list_list_a] :
                ( ( X
                  = ( cons_list_list_a @ nil_list_a @ Xss ) )
               => ( ( Y
                    = ( transpose_list_a @ Xss ) )
                 => ~ ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ ( cons_list_list_a @ nil_list_a @ Xss ) ) ) )
           => ~ ! [X3: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
                  ( ( X
                    = ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) )
                 => ( ( Y
                      = ( cons_list_list_a
                        @ ( cons_list_a @ X3
                          @ ( concat_list_a
                            @ ( map_li8713736314956022724list_a
                              @ ( case_l6560536634501641056list_a @ nil_list_a
                                @ ^ [H: list_a,T2: list_list_a] : ( cons_list_a @ H @ nil_list_a ) )
                              @ Xss ) ) )
                        @ ( transpose_list_a
                          @ ( cons_list_list_a @ Xs2
                            @ ( concat_list_list_a
                              @ ( map_li5227692475714150986list_a
                                @ ( case_l8090992918877587674list_a @ nil_list_list_a
                                  @ ^ [H: list_a,T2: list_list_a] : ( cons_list_list_a @ T2 @ nil_list_list_a ) )
                                @ Xss ) ) ) ) ) )
                   => ~ ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_1236_transpose_Opsimps_I2_J,axiom,
    ! [Xss2: list_list_a] :
      ( ( accp_list_list_a @ transpose_rel_a @ ( cons_list_a @ nil_a @ Xss2 ) )
     => ( ( transpose_a @ ( cons_list_a @ nil_a @ Xss2 ) )
        = ( transpose_a @ Xss2 ) ) ) ).

% transpose.psimps(2)
thf(fact_1237_transpose_Opsimps_I2_J,axiom,
    ! [Xss2: list_list_nat] :
      ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
     => ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
        = ( transpose_nat @ Xss2 ) ) ) ).

% transpose.psimps(2)
thf(fact_1238_transpose_Opsimps_I2_J,axiom,
    ! [Xss2: list_list_list_a] :
      ( ( accp_l3666201008454774169list_a @ transpose_rel_list_a @ ( cons_list_list_a @ nil_list_a @ Xss2 ) )
     => ( ( transpose_list_a @ ( cons_list_list_a @ nil_list_a @ Xss2 ) )
        = ( transpose_list_a @ Xss2 ) ) ) ).

% transpose.psimps(2)
thf(fact_1239_list_Odisc__eq__case_I2_J,axiom,
    ! [List: list_a] :
      ( ( List != nil_a )
      = ( case_list_o_a @ $false
        @ ^ [Uu2: a,Uv2: list_a] : $true
        @ List ) ) ).

% list.disc_eq_case(2)
thf(fact_1240_list_Odisc__eq__case_I2_J,axiom,
    ! [List: list_nat] :
      ( ( List != nil_nat )
      = ( case_list_o_nat @ $false
        @ ^ [Uu2: nat,Uv2: list_nat] : $true
        @ List ) ) ).

% list.disc_eq_case(2)
thf(fact_1241_list_Odisc__eq__case_I1_J,axiom,
    ! [List: list_nat] :
      ( ( List = nil_nat )
      = ( case_list_o_nat @ $true
        @ ^ [Uu2: nat,Uv2: list_nat] : $false
        @ List ) ) ).

% list.disc_eq_case(1)
thf(fact_1242_minimal__fixpoint__helper_I2_J,axiom,
    ! [F2: nat > nat,P: nat > $o,K3: nat,X: nat,X6: nat] :
      ( ( F2
        = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F2 @ ( suc @ X4 ) ) ) ) )
     => ( ! [X3: nat] :
            ( ( ord_less_eq_nat @ K3 @ X3 )
           => ( P @ X3 ) )
       => ( ( ord_less_eq_nat @ X @ X6 )
         => ( ( ord_less_nat @ X6 @ ( F2 @ X ) )
           => ~ ( P @ X6 ) ) ) ) ) ).

% minimal_fixpoint_helper(2)
thf(fact_1243_recursion__renaming__helper,axiom,
    ! [F1: nat > nat,P: nat > $o,F22: nat > nat,K3: nat] :
      ( ( F1
        = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F1 @ ( suc @ X4 ) ) ) ) )
     => ( ( F22
          = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F22 @ ( suc @ X4 ) ) ) ) )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ K3 @ X3 )
             => ( P @ X3 ) )
         => ( F1 = F22 ) ) ) ) ).

% recursion_renaming_helper
thf(fact_1244_minimal__fixpoint__helper_I1_J,axiom,
    ! [F2: nat > nat,P: nat > $o,K3: nat,X: nat] :
      ( ( F2
        = ( ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ X4 @ ( F2 @ ( suc @ X4 ) ) ) ) )
     => ( ! [X3: nat] :
            ( ( ord_less_eq_nat @ K3 @ X3 )
           => ( P @ X3 ) )
       => ( P @ ( F2 @ X ) ) ) ) ).

% minimal_fixpoint_helper(1)
thf(fact_1245_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_1246_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_1247_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1248_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1249_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1250_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1251_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1252_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1253_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1254_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_1255_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec

% Helper facts (19)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $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_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( if_list_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( if_list_list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
    ! [X: list_list_nat,Y: list_list_nat] :
      ( ( if_list_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
    ! [X: list_list_nat,Y: list_list_nat] :
      ( ( if_list_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
    ! [X: list_list_list_a,Y: list_list_list_a] :
      ( ( if_list_list_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_T,axiom,
    ! [X: list_list_list_a,Y: list_list_list_a] :
      ( ( if_list_list_list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
    ! [X: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( if_lis8333758933708535141list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J_T,axiom,
    ! [X: list_P369347992569427103list_a,Y: list_P369347992569427103list_a] :
      ( ( if_lis8333758933708535141list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( if_lis773845013453699073st_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: list_P3026376796534914631st_nat,Y: list_P3026376796534914631st_nat] :
      ( ( if_lis773845013453699073st_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_T,axiom,
    ! [X: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( if_lis924885042395737195list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_T,axiom,
    ! [X: list_P1915399164375025317list_a,Y: list_P1915399164375025317list_a] :
      ( ( if_lis924885042395737195list_a @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( simple2673724192819300657lper_a @ y @ ( cons_list_a @ ( cons_a @ y @ ys ) @ yssa ) @ resa )
    = ( simple2673724192819300657lper_a @ y @ yssa @ ( produc7347373609238049417list_a @ ( produc3726311466776224373list_a @ resa ) @ ( cons_list_a @ ys @ ( produc4969414473911996599list_a @ resa ) ) ) ) ) ).

%------------------------------------------------------------------------------