%------------------------------------------------------------------------------
% 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    : FO_Theory_Rewriting/0069_Context_Extensions/prob_00116_005409__18589126_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1781 ( 492 unt; 550 typ;   0 def)
%            Number of atoms       : 3916 (1712 equ;   0 cnn)
%            Maximal formula atoms :   17 (   3 avg)
%            Number of connectives : 17819 ( 391   ~;  70   |; 462   &;14796   @)
%                                         (   0 <=>;2100  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   8 avg)
%            Number of types       :  118 ( 117 usr)
%            Number of type conns  : 1302 (1302   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  436 ( 433 usr;  16 con; 0-4 aty)
%            Number of variables   : 4462 ( 104   ^;4016   !; 342   ?;4462   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 12:01:44.241
%------------------------------------------------------------------------------
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thf(ty_n_t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Ground____MCtxt__Ogmctxt_Itf__a_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
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thf(ty_n_t__Ground____Terms__Ogterm_Itf__a_J,type,
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thf(ty_n_t__Ground____Ctxt__Ogctxt_Itf__a_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__String__Ochar,type,
    char: $tType ).

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

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

% Explicit typings (433)
thf(sy_c_Context__Extensions_Ogctxtex__onp_001tf__a,type,
    contex6947914739494133532_onp_a: ( ground_gctxt_a > $o ) > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).

thf(sy_c_Context__Extensions_Ogmctxtex__onp_001tf__a,type,
    contex3086340098647126761_onp_a: ( ground_gmctxt_a > $o ) > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).

thf(sy_c_Ground__Closure_Oall__ctxt__closed_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ground2327783556906289298at_nat: set_Pr9093778441882193744at_nat > set_Pr8144746507299050889at_nat > $o ).

thf(sy_c_Ground__Closure_Oall__ctxt__closed_001tf__a,type,
    ground3158306204687651307osed_a: set_Pr4934435412358123699_a_nat > set_Pr8282532413760598727term_a > $o ).

thf(sy_c_Ground__Closure_Ogmctxt__cl_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ground1030676277180445380at_nat: set_Pr9093778441882193744at_nat > set_Pr8144746507299050889at_nat > set_Pr8144746507299050889at_nat ).

thf(sy_c_Ground__Closure_Ogmctxt__cl_001tf__a,type,
    ground_gmctxt_cl_a: set_Pr4934435412358123699_a_nat > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).

thf(sy_c_Ground__Closure_Ogmctxt__clp_001tf__a,type,
    ground_gmctxt_clp_a: ( a > nat > $o ) > ( ground_gterm_a > ground_gterm_a > $o ) > ground_gterm_a > ground_gterm_a > $o ).

thf(sy_c_Ground__Closure_Otrancl__list_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    ground4295926921573292569ctxt_a: set_Pr8705490787052945405ctxt_a > set_Pr8687807964536448157ctxt_a ).

thf(sy_c_Ground__Closure_Otrancl__list_001t__Ground____Terms__Ogterm_Itf__a_J,type,
    ground4492235380174378228term_a: set_Pr8282532413760598727term_a > set_Pr2711430308268100935term_a ).

thf(sy_c_Ground__Closure_Otrancl__list_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
    ground1737800716954789882term_a: set_Pr2711430308268100935term_a > set_Pr8594894544342482375term_a ).

thf(sy_c_Ground__Closure_Otrancl__list_001t__Nat__Onat,type,
    ground4984107042736062923st_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(sy_c_Ground__Closure_Otrancl__list_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
    ground583886336375474348term_a: set_Pr5156918327056800967term_a > set_Pr1780059727814202951term_a ).

thf(sy_c_Ground__Closure_Otrancl__listp_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
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thf(sy_c_Ground__Closure_Otrancl__listp_001t__Ground____Terms__Ogterm_Itf__a_J,type,
    ground1804610477406788024term_a: ( ground_gterm_a > ground_gterm_a > $o ) > list_Ground_gterm_a > list_Ground_gterm_a > $o ).

thf(sy_c_Ground__Closure_Otrancl__listp_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
    ground8433361165389822654term_a: ( list_Ground_gterm_a > list_Ground_gterm_a > $o ) > list_l4678906206682822203term_a > list_l4678906206682822203term_a > $o ).

thf(sy_c_Ground__Closure_Otrancl__listp_001t__Nat__Onat,type,
    ground2818829557537609223tp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_Ground__Closure_Otrancl__listp_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
    ground1143888467740470128term_a: ( produc1525979690750518759term_a > produc1525979690750518759term_a > $o ) > list_P2433084232546440685term_a > list_P2433084232546440685term_a > $o ).

thf(sy_c_Ground__MCtxt_Ocomp__gmctxt_001tf__a,type,
    ground_comp_gmctxt_a: set_Pr8705490787052945405ctxt_a ).

thf(sy_c_Ground__MCtxt_Ocomp__gmctxtp_001tf__a,type,
    ground4927988881900525590txtp_a: ground_gmctxt_a > ground_gmctxt_a > $o ).

thf(sy_c_Ground__MCtxt_Oeq__gfill_001tf__a,type,
    ground_eq_gfill_a: ground_gterm_a > produc2459281416120113548term_a > $o ).

thf(sy_c_Ground__MCtxt_Ofill__gholes_001tf__a,type,
    ground_fill_gholes_a: ground_gmctxt_a > list_Ground_gterm_a > ground_gterm_a ).

thf(sy_c_Ground__MCtxt_Ofill__gholes__gmctxt_001tf__a,type,
    ground2530127064035342964ctxt_a: ground_gmctxt_a > list_Ground_gmctxt_a > ground_gmctxt_a ).

thf(sy_c_Ground__MCtxt_Ofunas__gmctxt_001tf__a,type,
    ground4323624445104842080ctxt_a: ground_gmctxt_a > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Ground__MCtxt_Ogmctxt_OGMFun_001tf__a,type,
    ground_GMFun_a: a > list_Ground_gmctxt_a > ground_gmctxt_a ).

thf(sy_c_Ground__MCtxt_Ogmctxt__of__gterm_001tf__a,type,
    ground582559344758063963term_a: ground_gterm_a > ground_gmctxt_a ).

thf(sy_c_Ground__MCtxt_Ogmctxt__subtgm__at__fill__args_001tf__a_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    ground6121243438039047541ctxt_a: list_nat > ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_Ground__MCtxt_Ogmctxt__subtgm__at__fill__args_001tf__a_001t__Ground____Terms__Ogterm_Itf__a_J,type,
    ground8448531477516352152term_a: list_nat > ground_gmctxt_a > list_Ground_gterm_a > list_Ground_gterm_a ).

thf(sy_c_Ground__MCtxt_Ogmctxt__subtgm__at__fill__args_001tf__a_001t__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
    ground8506523905942776734term_a: list_nat > ground_gmctxt_a > list_l4678906206682822203term_a > list_l4678906206682822203term_a ).

thf(sy_c_Ground__MCtxt_Ogmctxt__subtgm__at__fill__args_001tf__a_001t__Nat__Onat,type,
    ground6209547762876788007_a_nat: list_nat > ground_gmctxt_a > list_nat > list_nat ).

thf(sy_c_Ground__MCtxt_Ogmctxt__subtgm__at__fill__args_001tf__a_001t__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J,type,
    ground7952841227834989648term_a: list_nat > ground_gmctxt_a > list_P2433084232546440685term_a > list_P2433084232546440685term_a ).

thf(sy_c_Ground__MCtxt_Ogterm__of__gmctxt_001tf__a,type,
    ground2297065866753446215ctxt_a: ground_gmctxt_a > ground_gterm_a ).

thf(sy_c_Ground__MCtxt_Oless__eq__gmctxt_001tf__a,type,
    ground6477899500129135038ctxt_a: ground_gmctxt_a > ground_gmctxt_a > $o ).

thf(sy_c_Ground__MCtxt_Onum__gholes_001tf__a,type,
    ground_num_gholes_a: ground_gmctxt_a > nat ).

thf(sy_c_Ground__MCtxt_Osup__gmctxt__args_001tf__a,type,
    ground7778985915038999110args_a: ground_gmctxt_a > ground_gmctxt_a > list_Ground_gmctxt_a ).

thf(sy_c_Ground__MCtxt_Ounfill__gholes_001tf__a,type,
    ground8214737007138946830oles_a: ground_gmctxt_a > ground_gterm_a > list_Ground_gterm_a ).

thf(sy_c_Ground__Terms_O_092_060T_062_092_060_094sub_062G_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ground_T_G_nat_nat: set_Pr9093778441882193744at_nat > set_Gr7543109056609072026at_nat ).

thf(sy_c_Ground__Terms_O_092_060T_062_092_060_094sub_062G_001tf__a,type,
    ground_T_G_a: set_Pr4934435412358123699_a_nat > set_Ground_gterm_a ).

thf(sy_c_Ground__Terms_O_092_060T_062_092_060_094sub_062Gp_001tf__a,type,
    ground_T_Gp_a: ( a > nat > $o ) > ground_gterm_a > $o ).

thf(sy_c_Ground__Terms_Ofunas__gterm_001tf__a,type,
    ground_funas_gterm_a: ground_gterm_a > set_Pr4934435412358123699_a_nat ).

thf(sy_c_Ground__Terms_Ogroot_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ground_groot_nat_nat: ground_gterm_nat_nat > produc8199716216217303280at_nat ).

thf(sy_c_Ground__Terms_Ogroot_001tf__a,type,
    ground_groot_a: ground_gterm_a > product_prod_a_nat ).

thf(sy_c_Ground__Terms_Ogterm_OGFun_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ground_GFun_nat_nat: ( nat > nat ) > list_G4348582881148899572at_nat > ground_gterm_nat_nat ).

thf(sy_c_Ground__Terms_Ogterm_OGFun_001tf__a,type,
    ground_GFun_a: a > list_Ground_gterm_a > ground_gterm_a ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Ground____MCtxt__Ogmctxt_Itf__a_J_J,type,
    minus_4616606335744337325ctxt_a: set_Ground_gmctxt_a > set_Ground_gmctxt_a > set_Ground_gmctxt_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Ground____Terms__Ogterm_Itf__a_J_J,type,
    minus_80403276131473398term_a: set_Ground_gterm_a > set_Ground_gterm_a > set_Ground_gterm_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
    minus_3383382438752481404term_a: set_li8588095516285831957term_a > set_li8588095516285831957term_a > set_li8588095516285831957term_a ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Ground____Terms__Ogterm_Itf__a_J_Mt__Ground____Terms__Ogterm_Itf__a_J_J_J,type,
    minus_551215046067487790term_a: set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a > set_Pr8282532413760598727term_a ).

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

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

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

thf(sy_c_List_Oappend_001t__Ground____MCtxt__Ogmctxt_Itf__a_J,type,
    append5238750768297430211ctxt_a: list_Ground_gmctxt_a > list_Ground_gmctxt_a > list_Ground_gmctxt_a ).

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thf(sy_v_P,type,
    p: ground_gmctxt_a > $o ).

thf(sy_v_Q,type,
    q: ground_gmctxt_a > $o ).

thf(sy_v__092_060R_062,type,
    r: set_Pr8282532413760598727term_a ).

thf(sy_v_s,type,
    s: ground_gterm_a ).

thf(sy_v_t,type,
    t: ground_gterm_a ).

thf(sy_v_thesis,type,
    thesis: $o ).

% Relevant facts (1226)
thf(fact_0_assms,axiom,
    member4351891425000572176term_a @ ( produc7419039783125785943term_a @ s @ t ) @ ( contex3086340098647126761_onp_a @ p @ ( contex3086340098647126761_onp_a @ q @ r ) ) ).

% assms
thf(fact_1_that,axiom,
    ! [C: ground_gmctxt_a,Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a,Ds: list_Ground_gmctxt_a,Us: list_l4678906206682822203term_a,Vs: list_l4678906206682822203term_a] :
      ( ( ground_eq_gfill_a @ s @ ( produc4720362732934906942term_a @ C @ Ss ) )
     => ( ( ground_eq_gfill_a @ t @ ( produc4720362732934906942term_a @ C @ Ts ) )
       => ( ( p @ C )
         => ( ! [X: ground_gmctxt_a] :
                ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Ds ) )
               => ( q @ X ) )
           => ( ( ( ground_num_gholes_a @ C )
                = ( size_s962548663205356226ctxt_a @ Ds ) )
             => ( ( ( size_s962548663205356226ctxt_a @ Ds )
                  = ( size_s1893836384466666913term_a @ Ss ) )
               => ( ( ( size_s1893836384466666913term_a @ Ss )
                    = ( size_s1893836384466666913term_a @ Ts ) )
                 => ( ( ( size_s1893836384466666913term_a @ Ts )
                      = ( size_s5321119603959729319term_a @ Us ) )
                   => ( ( ( size_s5321119603959729319term_a @ Us )
                        = ( size_s5321119603959729319term_a @ Vs ) )
                     => ( ! [I: nat] :
                            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
                           => ( ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Us @ I ) ) )
                              & ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Vs @ I ) ) ) ) )
                       => ( ! [I: nat] :
                              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
                             => ! [J: nat] :
                                  ( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Vs @ I ) ) )
                                 => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Us @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Vs @ I ) @ J ) ) @ r ) ) )
                         => thesis ) ) ) ) ) ) ) ) ) ) ) ).

% that
thf(fact_2_eqgfE_I2_J,axiom,
    ! [T: ground_gterm_a,D: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
      ( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss ) )
     => ( ( ground_num_gholes_a @ D )
        = ( size_s1893836384466666913term_a @ Ss ) ) ) ).

% eqgfE(2)
thf(fact_3_gmctxtex__onpE2,axiom,
    ! [S: ground_gterm_a,T: ground_gterm_a,P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
     => ~ ! [C2: ground_gmctxt_a,Us2: list_Ground_gterm_a] :
            ( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C2 @ Us2 ) )
           => ! [Vs2: list_Ground_gterm_a] :
                ( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ C2 @ Vs2 ) )
               => ( ( P @ C2 )
                 => ~ ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Vs2 ) )
                       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Us2 @ I2 ) @ ( nth_Ground_gterm_a @ Vs2 @ I2 ) ) @ R ) ) ) ) ) ) ).

% gmctxtex_onpE2
thf(fact_4_nth__mem,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a] :
      ( ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
     => ( member4351891425000572176term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ N ) @ ( set_Pr5405221470436175740term_a @ Xs ) ) ) ).

% nth_mem
thf(fact_5_nth__mem,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ N ) @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_6_nth__mem,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a] :
      ( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Xs @ N ) @ ( set_Ground_gterm_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_7_nth__mem,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( member5011579089780872286term_a @ ( nth_li6172171122907299772term_a @ Xs @ N ) @ ( set_li7546925467519454922term_a @ Xs ) ) ) ).

% nth_mem
thf(fact_8_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_9_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a,P: produc1525979690750518759term_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ! [X: produc1525979690750518759term_a] :
            ( ( member4351891425000572176term_a @ X @ ( set_Pr5405221470436175740term_a @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_Pr5943965302187034734term_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_10_list__ball__nth,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ! [X: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_Ground_gmctxt_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_11_list__ball__nth,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a,P: ground_gterm_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ! [X: ground_gterm_a] :
            ( ( member5261315023172567896term_a @ X @ ( set_Ground_gterm_a2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_Ground_gterm_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_12_list__ball__nth,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a,P: list_Ground_gterm_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ! [X: list_Ground_gterm_a] :
            ( ( member5011579089780872286term_a @ X @ ( set_li7546925467519454922term_a @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_li6172171122907299772term_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_13_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_14_in__set__conv__nth,axiom,
    ! [X2: produc1525979690750518759term_a,Xs: list_P2433084232546440685term_a] :
      ( ( member4351891425000572176term_a @ X2 @ ( set_Pr5405221470436175740term_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs ) )
            & ( ( nth_Pr5943965302187034734term_a @ Xs @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_15_in__set__conv__nth,axiom,
    ! [X2: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( ( nth_Ground_gmctxt_a @ Xs @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_16_in__set__conv__nth,axiom,
    ! [X2: ground_gterm_a,Xs: list_Ground_gterm_a] :
      ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs ) )
            & ( ( nth_Ground_gterm_a @ Xs @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_17_in__set__conv__nth,axiom,
    ! [X2: list_Ground_gterm_a,Xs: list_l4678906206682822203term_a] :
      ( ( member5011579089780872286term_a @ X2 @ ( set_li7546925467519454922term_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Xs ) )
            & ( ( nth_li6172171122907299772term_a @ Xs @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_18_in__set__conv__nth,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_19_in__set__idx,axiom,
    ! [X2: produc1525979690750518759term_a,Xs: list_P2433084232546440685term_a] :
      ( ( member4351891425000572176term_a @ X2 @ ( set_Pr5405221470436175740term_a @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Xs ) )
          & ( ( nth_Pr5943965302187034734term_a @ Xs @ I )
            = X2 ) ) ) ).

% in_set_idx
thf(fact_20_in__set__idx,axiom,
    ! [X2: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
          & ( ( nth_Ground_gmctxt_a @ Xs @ I )
            = X2 ) ) ) ).

% in_set_idx
thf(fact_21_in__set__idx,axiom,
    ! [X2: ground_gterm_a,Xs: list_Ground_gterm_a] :
      ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Xs ) )
          & ( ( nth_Ground_gterm_a @ Xs @ I )
            = X2 ) ) ) ).

% in_set_idx
thf(fact_22_in__set__idx,axiom,
    ! [X2: list_Ground_gterm_a,Xs: list_l4678906206682822203term_a] :
      ( ( member5011579089780872286term_a @ X2 @ ( set_li7546925467519454922term_a @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
          & ( ( nth_li6172171122907299772term_a @ Xs @ I )
            = X2 ) ) ) ).

% in_set_idx
thf(fact_23_in__set__idx,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
          & ( ( nth_nat @ Xs @ I )
            = X2 ) ) ) ).

% in_set_idx
thf(fact_24_all__nth__imp__all__set,axiom,
    ! [Xs: list_P2433084232546440685term_a,P: produc1525979690750518759term_a > $o,X2: produc1525979690750518759term_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Xs ) )
         => ( P @ ( nth_Pr5943965302187034734term_a @ Xs @ I ) ) )
     => ( ( member4351891425000572176term_a @ X2 @ ( set_Pr5405221470436175740term_a @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_25_all__nth__imp__all__set,axiom,
    ! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o,X2: ground_gmctxt_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
         => ( P @ ( nth_Ground_gmctxt_a @ Xs @ I ) ) )
     => ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_26_all__nth__imp__all__set,axiom,
    ! [Xs: list_Ground_gterm_a,P: ground_gterm_a > $o,X2: ground_gterm_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Xs ) )
         => ( P @ ( nth_Ground_gterm_a @ Xs @ I ) ) )
     => ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_27_all__nth__imp__all__set,axiom,
    ! [Xs: list_l4678906206682822203term_a,P: list_Ground_gterm_a > $o,X2: list_Ground_gterm_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
         => ( P @ ( nth_li6172171122907299772term_a @ Xs @ I ) ) )
     => ( ( member5011579089780872286term_a @ X2 @ ( set_li7546925467519454922term_a @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_28_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X2: nat] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I ) ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_29_all__set__conv__all__nth,axiom,
    ! [Xs: list_P2433084232546440685term_a,P: produc1525979690750518759term_a > $o] :
      ( ( ! [X3: produc1525979690750518759term_a] :
            ( ( member4351891425000572176term_a @ X3 @ ( set_Pr5405221470436175740term_a @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs ) )
           => ( P @ ( nth_Pr5943965302187034734term_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_30_all__set__conv__all__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ! [X3: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( P @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_31_all__set__conv__all__nth,axiom,
    ! [Xs: list_Ground_gterm_a,P: ground_gterm_a > $o] :
      ( ( ! [X3: ground_gterm_a] :
            ( ( member5261315023172567896term_a @ X3 @ ( set_Ground_gterm_a2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs ) )
           => ( P @ ( nth_Ground_gterm_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_32_all__set__conv__all__nth,axiom,
    ! [Xs: list_l4678906206682822203term_a,P: list_Ground_gterm_a > $o] :
      ( ( ! [X3: list_Ground_gterm_a] :
            ( ( member5011579089780872286term_a @ X3 @ ( set_li7546925467519454922term_a @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( P @ ( nth_li6172171122907299772term_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_33_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_34_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( ? [X3: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( P @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_35_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_Ground_gterm_a,P: ground_gterm_a > $o] :
      ( ( ? [X3: ground_gterm_a] :
            ( ( member5261315023172567896term_a @ X3 @ ( set_Ground_gterm_a2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs ) )
            & ( P @ ( nth_Ground_gterm_a @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_36_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_l4678906206682822203term_a,P: list_Ground_gterm_a > $o] :
      ( ( ? [X3: list_Ground_gterm_a] :
            ( ( member5011579089780872286term_a @ X3 @ ( set_li7546925467519454922term_a @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Xs ) )
            & ( P @ ( nth_li6172171122907299772term_a @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_37_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_38_ex__set__conv__ex__nth,axiom,
    ! [Xs: list_P2433084232546440685term_a,P: produc1525979690750518759term_a > $o] :
      ( ( ? [X3: produc1525979690750518759term_a] :
            ( ( member4351891425000572176term_a @ X3 @ ( set_Pr5405221470436175740term_a @ Xs ) )
            & ( P @ X3 ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs ) )
            & ( P @ ( nth_Pr5943965302187034734term_a @ Xs @ I3 ) ) ) ) ) ).

% ex_set_conv_ex_nth
thf(fact_39_nth__equalityI,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( ( nth_Ground_gmctxt_a @ Xs @ I )
              = ( nth_Ground_gmctxt_a @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_40_nth__equalityI,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Xs ) )
           => ( ( nth_Ground_gterm_a @ Xs @ I )
              = ( nth_Ground_gterm_a @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_41_nth__equalityI,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( ( nth_li6172171122907299772term_a @ Xs @ I )
              = ( nth_li6172171122907299772term_a @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_42_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_43_nth__equalityI,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s281034491502894169term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Xs ) )
           => ( ( nth_Pr5943965302187034734term_a @ Xs @ I )
              = ( nth_Pr5943965302187034734term_a @ Ys @ I ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_44_nth__equalityE,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s962548663205356226ctxt_a @ Xs )
            = ( size_s962548663205356226ctxt_a @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
               => ( ( nth_Ground_gmctxt_a @ Xs @ I2 )
                  = ( nth_Ground_gmctxt_a @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_45_nth__equalityE,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s1893836384466666913term_a @ Xs )
            = ( size_s1893836384466666913term_a @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Xs ) )
               => ( ( nth_Ground_gterm_a @ Xs @ I2 )
                  = ( nth_Ground_gterm_a @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_46_nth__equalityE,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s5321119603959729319term_a @ Xs )
            = ( size_s5321119603959729319term_a @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Xs ) )
               => ( ( nth_li6172171122907299772term_a @ Xs @ I2 )
                  = ( nth_li6172171122907299772term_a @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_47_nth__equalityE,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
               => ( ( nth_nat @ Xs @ I2 )
                  = ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_48_nth__equalityE,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( Xs = Ys )
     => ~ ( ( ( size_s281034491502894169term_a @ Xs )
            = ( size_s281034491502894169term_a @ Ys ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Xs ) )
               => ( ( nth_Pr5943965302187034734term_a @ Xs @ I2 )
                  = ( nth_Pr5943965302187034734term_a @ Ys @ I2 ) ) ) ) ) ).

% nth_equalityE
thf(fact_49_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > ground_gmctxt_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: ground_gmctxt_a] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs2: list_Ground_gmctxt_a] :
            ( ( ( size_s962548663205356226ctxt_a @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Ground_gmctxt_a @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_50_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > ground_gterm_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: ground_gterm_a] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs2: list_Ground_gterm_a] :
            ( ( ( size_s1893836384466666913term_a @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Ground_gterm_a @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_51_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > list_Ground_gterm_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: list_Ground_gterm_a] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs2: list_l4678906206682822203term_a] :
            ( ( ( size_s5321119603959729319term_a @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_li6172171122907299772term_a @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_52_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: nat] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_53_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc1525979690750518759term_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X4: produc1525979690750518759term_a] : ( P @ I3 @ X4 ) ) )
      = ( ? [Xs2: list_P2433084232546440685term_a] :
            ( ( ( size_s281034491502894169term_a @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Pr5943965302187034734term_a @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_54_neq__if__length__neq,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
       != ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_55_neq__if__length__neq,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
       != ( size_s1893836384466666913term_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_56_neq__if__length__neq,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
       != ( size_s5321119603959729319term_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_57_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_58_neq__if__length__neq,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( ( size_s281034491502894169term_a @ Xs )
       != ( size_s281034491502894169term_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_59_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_Ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_60_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_Ground_gterm_a] :
      ( ( size_s1893836384466666913term_a @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_61_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_l4678906206682822203term_a] :
      ( ( size_s5321119603959729319term_a @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_62_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_63_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P2433084232546440685term_a] :
      ( ( size_s281034491502894169term_a @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_64_length__induct,axiom,
    ! [P: list_Ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a] :
      ( ! [Xs3: list_Ground_gmctxt_a] :
          ( ! [Ys2: list_Ground_gmctxt_a] :
              ( ( ord_less_nat @ ( size_s962548663205356226ctxt_a @ Ys2 ) @ ( size_s962548663205356226ctxt_a @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_65_length__induct,axiom,
    ! [P: list_Ground_gterm_a > $o,Xs: list_Ground_gterm_a] :
      ( ! [Xs3: list_Ground_gterm_a] :
          ( ! [Ys2: list_Ground_gterm_a] :
              ( ( ord_less_nat @ ( size_s1893836384466666913term_a @ Ys2 ) @ ( size_s1893836384466666913term_a @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_66_length__induct,axiom,
    ! [P: list_l4678906206682822203term_a > $o,Xs: list_l4678906206682822203term_a] :
      ( ! [Xs3: list_l4678906206682822203term_a] :
          ( ! [Ys2: list_l4678906206682822203term_a] :
              ( ( ord_less_nat @ ( size_s5321119603959729319term_a @ Ys2 ) @ ( size_s5321119603959729319term_a @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_67_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_68_length__induct,axiom,
    ! [P: list_P2433084232546440685term_a > $o,Xs: list_P2433084232546440685term_a] :
      ( ! [Xs3: list_P2433084232546440685term_a] :
          ( ! [Ys2: list_P2433084232546440685term_a] :
              ( ( ord_less_nat @ ( size_s281034491502894169term_a @ Ys2 ) @ ( size_s281034491502894169term_a @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_69_Ex__list__of__length__P,axiom,
    ! [N: nat,P: ground_gmctxt_a > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X5: ground_gmctxt_a] : ( P @ X5 @ I ) )
     => ? [Xs3: list_Ground_gmctxt_a] :
          ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_Ground_gmctxt_a @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_70_Ex__list__of__length__P,axiom,
    ! [N: nat,P: ground_gterm_a > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X5: ground_gterm_a] : ( P @ X5 @ I ) )
     => ? [Xs3: list_Ground_gterm_a] :
          ( ( ( size_s1893836384466666913term_a @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_71_Ex__list__of__length__P,axiom,
    ! [N: nat,P: list_Ground_gterm_a > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X5: list_Ground_gterm_a] : ( P @ X5 @ I ) )
     => ? [Xs3: list_l4678906206682822203term_a] :
          ( ( ( size_s5321119603959729319term_a @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_72_Ex__list__of__length__P,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X5: nat] : ( P @ X5 @ I ) )
     => ? [Xs3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_nat @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_73_Ex__list__of__length__P,axiom,
    ! [N: nat,P: produc1525979690750518759term_a > nat > $o] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ N )
         => ? [X5: produc1525979690750518759term_a] : ( P @ X5 @ I ) )
     => ? [Xs3: list_P2433084232546440685term_a] :
          ( ( ( size_s281034491502894169term_a @ Xs3 )
            = N )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N )
             => ( P @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I2 ) @ I2 ) ) ) ) ).

% Ex_list_of_length_P
thf(fact_74_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_Ground_gmctxt_a,Z: list_Ground_gmctxt_a] : ( Y = Z ) )
    = ( ^ [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
          ( ( ( size_s962548663205356226ctxt_a @ Xs2 )
            = ( size_s962548663205356226ctxt_a @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs2 ) )
             => ( ( nth_Ground_gmctxt_a @ Xs2 @ I3 )
                = ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_75_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_Ground_gterm_a,Z: list_Ground_gterm_a] : ( Y = Z ) )
    = ( ^ [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a] :
          ( ( ( size_s1893836384466666913term_a @ Xs2 )
            = ( size_s1893836384466666913term_a @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs2 ) )
             => ( ( nth_Ground_gterm_a @ Xs2 @ I3 )
                = ( nth_Ground_gterm_a @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_76_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_l4678906206682822203term_a,Z: list_l4678906206682822203term_a] : ( Y = Z ) )
    = ( ^ [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a] :
          ( ( ( size_s5321119603959729319term_a @ Xs2 )
            = ( size_s5321119603959729319term_a @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Xs2 ) )
             => ( ( nth_li6172171122907299772term_a @ Xs2 @ I3 )
                = ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_77_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_78_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_P2433084232546440685term_a,Z: list_P2433084232546440685term_a] : ( Y = Z ) )
    = ( ^ [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a] :
          ( ( ( size_s281034491502894169term_a @ Xs2 )
            = ( size_s281034491502894169term_a @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs2 ) )
             => ( ( nth_Pr5943965302187034734term_a @ Xs2 @ I3 )
                = ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_79_gmctxtex__onp__listE,axiom,
    ! [Ts: list_Ground_gterm_a,Ss: list_Ground_gterm_a,Q: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
         => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) @ ( contex3086340098647126761_onp_a @ Q @ R ) ) )
     => ( ( ( size_s1893836384466666913term_a @ Ss )
          = ( size_s1893836384466666913term_a @ Ts ) )
       => ~ ! [Ds2: list_Ground_gmctxt_a] :
              ( ( ( size_s1893836384466666913term_a @ Ts )
                = ( size_s962548663205356226ctxt_a @ Ds2 ) )
             => ! [Sss: list_l4678906206682822203term_a] :
                  ( ( ( size_s962548663205356226ctxt_a @ Ds2 )
                    = ( size_s5321119603959729319term_a @ Sss ) )
                 => ! [Tss: list_l4678906206682822203term_a] :
                      ( ( ( size_s5321119603959729319term_a @ Sss )
                        = ( size_s5321119603959729319term_a @ Tss ) )
                     => ( ! [I2: nat] :
                            ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Tss ) )
                           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Sss @ I2 ) )
                              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Tss @ I2 ) ) ) )
                       => ( ! [X5: ground_gmctxt_a] :
                              ( ( member3306386364193653941ctxt_a @ X5 @ ( set_Ground_gmctxt_a2 @ Ds2 ) )
                             => ( Q @ X5 ) )
                         => ( ! [I2: nat] :
                                ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Tss ) )
                               => ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss @ I2 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds2 @ I2 ) @ ( nth_li6172171122907299772term_a @ Sss @ I2 ) ) ) )
                           => ( ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Tss ) )
                                 => ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts @ I2 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds2 @ I2 ) @ ( nth_li6172171122907299772term_a @ Tss @ I2 ) ) ) )
                             => ~ ! [I2: nat] :
                                    ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Tss ) ) )
                                   => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Sss ) @ I2 ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Tss ) @ I2 ) ) @ R ) ) ) ) ) ) ) ) ) ) ) ).

% gmctxtex_onp_listE
thf(fact_80_permut__sound,axiom,
    ! [I4: nat,As: list_Ground_gmctxt_a,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ As ) )
     => ( ( nth_Ground_gmctxt_a @ ( missin4491563000646178374ctxt_a @ As @ F ) @ I4 )
        = ( nth_Ground_gmctxt_a @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_81_permut__sound,axiom,
    ! [I4: nat,As: list_Ground_gterm_a,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ As ) )
     => ( ( nth_Ground_gterm_a @ ( missin5607496018787941767term_a @ As @ F ) @ I4 )
        = ( nth_Ground_gterm_a @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_82_permut__sound,axiom,
    ! [I4: nat,As: list_l4678906206682822203term_a,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ As ) )
     => ( ( nth_li6172171122907299772term_a @ ( missin103505523753413389term_a @ As @ F ) @ I4 )
        = ( nth_li6172171122907299772term_a @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_83_permut__sound,axiom,
    ! [I4: nat,As: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
     => ( ( nth_nat @ ( missing_permut_nat @ As @ F ) @ I4 )
        = ( nth_nat @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_84_permut__sound,axiom,
    ! [I4: nat,As: list_P2433084232546440685term_a,F: nat > nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ As ) )
     => ( ( nth_Pr5943965302187034734term_a @ ( missin4968302876397823935term_a @ As @ F ) @ I4 )
        = ( nth_Pr5943965302187034734term_a @ As @ ( F @ I4 ) ) ) ) ).

% permut_sound
thf(fact_85_old_Oprod_Oinject,axiom,
    ! [A: ground_gmctxt_a,B: list_Ground_gterm_a,A2: ground_gmctxt_a,B2: list_Ground_gterm_a] :
      ( ( ( produc4720362732934906942term_a @ A @ B )
        = ( produc4720362732934906942term_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_86_old_Oprod_Oinject,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,A2: ground_gterm_a,B2: ground_gterm_a] :
      ( ( ( produc7419039783125785943term_a @ A @ B )
        = ( produc7419039783125785943term_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_87_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_88_prod_Oinject,axiom,
    ! [X1: ground_gmctxt_a,X22: list_Ground_gterm_a,Y1: ground_gmctxt_a,Y2: list_Ground_gterm_a] :
      ( ( ( produc4720362732934906942term_a @ X1 @ X22 )
        = ( produc4720362732934906942term_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_89_prod_Oinject,axiom,
    ! [X1: ground_gterm_a,X22: ground_gterm_a,Y1: ground_gterm_a,Y2: ground_gterm_a] :
      ( ( ( produc7419039783125785943term_a @ X1 @ X22 )
        = ( produc7419039783125785943term_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_90_prod_Oinject,axiom,
    ! [X1: nat > nat,X22: nat,Y1: nat > nat,Y2: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X22 )
        = ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_91_gmctxtex__onpE,axiom,
    ! [S: ground_gterm_a,T: ground_gterm_a,P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
     => ~ ! [C2: ground_gmctxt_a,Us2: list_Ground_gterm_a] :
            ( ( S
              = ( ground_fill_gholes_a @ C2 @ Us2 ) )
           => ! [Vs2: list_Ground_gterm_a] :
                ( ( T
                  = ( ground_fill_gholes_a @ C2 @ Vs2 ) )
               => ( ( ( ground_num_gholes_a @ C2 )
                    = ( size_s1893836384466666913term_a @ Us2 ) )
                 => ( ( ( size_s1893836384466666913term_a @ Us2 )
                      = ( size_s1893836384466666913term_a @ Vs2 ) )
                   => ( ( P @ C2 )
                     => ~ ! [I2: nat] :
                            ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Vs2 ) )
                           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Us2 @ I2 ) @ ( nth_Ground_gterm_a @ Vs2 @ I2 ) ) @ R ) ) ) ) ) ) ) ) ).

% gmctxtex_onpE
thf(fact_92_permut__aux__sound,axiom,
    ! [I4: nat,As: list_Ground_gmctxt_a,F: nat > nat,Bs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ As ) )
     => ( ( nth_Ground_gmctxt_a @ ( missin1343833451449182556ctxt_a @ As @ F @ Bs ) @ I4 )
        = ( nth_Ground_gmctxt_a @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_93_permut__aux__sound,axiom,
    ! [I4: nat,As: list_Ground_gterm_a,F: nat > nat,Bs: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ As ) )
     => ( ( nth_Ground_gterm_a @ ( missin7316750521486066929term_a @ As @ F @ Bs ) @ I4 )
        = ( nth_Ground_gterm_a @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_94_permut__aux__sound,axiom,
    ! [I4: nat,As: list_l4678906206682822203term_a,F: nat > nat,Bs: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ As ) )
     => ( ( nth_li6172171122907299772term_a @ ( missin8070190275936138615term_a @ As @ F @ Bs ) @ I4 )
        = ( nth_li6172171122907299772term_a @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_95_permut__aux__sound,axiom,
    ! [I4: nat,As: list_nat,F: nat > nat,Bs: list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ As ) )
     => ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F @ Bs ) @ I4 )
        = ( nth_nat @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_96_permut__aux__sound,axiom,
    ! [I4: nat,As: list_P2433084232546440685term_a,F: nat > nat,Bs: list_P2433084232546440685term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ As ) )
     => ( ( nth_Pr5943965302187034734term_a @ ( missin3681154605564148265term_a @ As @ F @ Bs ) @ I4 )
        = ( nth_Pr5943965302187034734term_a @ Bs @ ( F @ I4 ) ) ) ) ).

% permut_aux_sound
thf(fact_97_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys @ I ) ) @ R ) )
       => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_98_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys @ I ) ) @ R ) )
       => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( trancl1902929335784340279term_a @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_99_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys ) )
           => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) @ R ) )
       => ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( trancl5818250397338112189term_a @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_100_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_101_Utils_Otrancl__list_Obase,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s281034491502894169term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys ) )
           => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ I ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I ) ) @ R ) )
       => ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( trancl6567088178672586095term_a @ R ) ) ) ) ).

% Utils.trancl_list.base
thf(fact_102_remdups__sort_I2_J,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2 @ ( missin6101193410121742181rt_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% remdups_sort(2)
thf(fact_103_eq__gfill_Ointros,axiom,
    ! [T: ground_gterm_a,D: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
      ( ( T
        = ( ground_fill_gholes_a @ D @ Ss ) )
     => ( ( ( ground_num_gholes_a @ D )
          = ( size_s1893836384466666913term_a @ Ss ) )
       => ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss ) ) ) ) ).

% eq_gfill.intros
thf(fact_104_eq__gfill_Osimps,axiom,
    ( ground_eq_gfill_a
    = ( ^ [A1: ground_gterm_a,A22: produc2459281416120113548term_a] :
        ? [T2: ground_gterm_a,D2: ground_gmctxt_a,Ss2: list_Ground_gterm_a] :
          ( ( A1 = T2 )
          & ( A22
            = ( produc4720362732934906942term_a @ D2 @ Ss2 ) )
          & ( T2
            = ( ground_fill_gholes_a @ D2 @ Ss2 ) )
          & ( ( ground_num_gholes_a @ D2 )
            = ( size_s1893836384466666913term_a @ Ss2 ) ) ) ) ) ).

% eq_gfill.simps
thf(fact_105_eq__gfill_Ocases,axiom,
    ! [A12: ground_gterm_a,A23: produc2459281416120113548term_a] :
      ( ( ground_eq_gfill_a @ A12 @ A23 )
     => ~ ! [D3: ground_gmctxt_a,Ss3: list_Ground_gterm_a] :
            ( ( A23
              = ( produc4720362732934906942term_a @ D3 @ Ss3 ) )
           => ( ( A12
                = ( ground_fill_gholes_a @ D3 @ Ss3 ) )
             => ( ( ground_num_gholes_a @ D3 )
               != ( size_s1893836384466666913term_a @ Ss3 ) ) ) ) ) ).

% eq_gfill.cases
thf(fact_106_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l1578326660578982846ctxt_a,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J2: nat,K2: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J2 @ K2 ) )
              & ( ord_less_nat @ J1 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ J2 @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K2 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I1 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I22 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_107_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l4678906206682822203term_a,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J2: nat,K2: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J2 @ K2 ) )
              & ( ord_less_nat @ J1 @ ( size_s5321119603959729319term_a @ Xs ) )
              & ( ord_less_nat @ J2 @ ( size_s5321119603959729319term_a @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K2 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ J2 ) ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I1 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I22 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_108_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l5172476077732501185term_a,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J2: nat,K2: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J2 @ K2 ) )
              & ( ord_less_nat @ J1 @ ( size_s3474408278181496621term_a @ Xs ) )
              & ( ord_less_nat @ J2 @ ( size_s3474408278181496621term_a @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K2 @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ J2 ) ) )
              & ( ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Xs ) @ I1 )
                = ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Xs ) @ I22 )
                = ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Xs @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_109_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_list_nat,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J2: nat,K2: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J2 @ K2 ) )
              & ( ord_less_nat @ J1 @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K2 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J2 ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I1 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I22 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_110_nth__concat__diff,axiom,
    ! [I1: nat,Xs: list_l8493803889202805107term_a,I22: nat] :
      ( ( ord_less_nat @ I1 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ( ord_less_nat @ I22 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
       => ( ( I1 != I22 )
         => ? [J1: nat,K1: nat,J2: nat,K2: nat] :
              ( ( ( product_Pair_nat_nat @ J1 @ K1 )
               != ( product_Pair_nat_nat @ J2 @ K2 ) )
              & ( ord_less_nat @ J1 @ ( size_s7283615635009460703term_a @ Xs ) )
              & ( ord_less_nat @ J2 @ ( size_s7283615635009460703term_a @ Xs ) )
              & ( ord_less_nat @ K1 @ ( size_s281034491502894169term_a @ ( nth_li7906007133205842164term_a @ Xs @ J1 ) ) )
              & ( ord_less_nat @ K2 @ ( size_s281034491502894169term_a @ ( nth_li7906007133205842164term_a @ Xs @ J2 ) ) )
              & ( ( nth_Pr5943965302187034734term_a @ ( concat2365498962271017768term_a @ Xs ) @ I1 )
                = ( nth_Pr5943965302187034734term_a @ ( nth_li7906007133205842164term_a @ Xs @ J1 ) @ K1 ) )
              & ( ( nth_Pr5943965302187034734term_a @ ( concat2365498962271017768term_a @ Xs ) @ I22 )
                = ( nth_Pr5943965302187034734term_a @ ( nth_li7906007133205842164term_a @ Xs @ J2 ) @ K2 ) ) ) ) ) ) ).

% nth_concat_diff
thf(fact_111_eqgfE_I1_J,axiom,
    ! [T: ground_gterm_a,D: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
      ( ( ground_eq_gfill_a @ T @ ( produc4720362732934906942term_a @ D @ Ss ) )
     => ( T
        = ( ground_fill_gholes_a @ D @ Ss ) ) ) ).

% eqgfE(1)
thf(fact_112_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2459281416120113548term_a] :
      ~ ! [A3: ground_gmctxt_a,B3: list_Ground_gterm_a] :
          ( Y3
         != ( produc4720362732934906942term_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_113_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1525979690750518759term_a] :
      ~ ! [A3: ground_gterm_a,B3: ground_gterm_a] :
          ( Y3
         != ( produc7419039783125785943term_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_114_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8199716216217303280at_nat] :
      ~ ! [A3: nat > nat,B3: nat] :
          ( Y3
         != ( produc72220940542539688at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_115_surj__pair,axiom,
    ! [P2: produc2459281416120113548term_a] :
    ? [X: ground_gmctxt_a,Y4: list_Ground_gterm_a] :
      ( P2
      = ( produc4720362732934906942term_a @ X @ Y4 ) ) ).

% surj_pair
thf(fact_116_surj__pair,axiom,
    ! [P2: produc1525979690750518759term_a] :
    ? [X: ground_gterm_a,Y4: ground_gterm_a] :
      ( P2
      = ( produc7419039783125785943term_a @ X @ Y4 ) ) ).

% surj_pair
thf(fact_117_surj__pair,axiom,
    ! [P2: produc8199716216217303280at_nat] :
    ? [X: nat > nat,Y4: nat] :
      ( P2
      = ( produc72220940542539688at_nat @ X @ Y4 ) ) ).

% surj_pair
thf(fact_118_prod__cases,axiom,
    ! [P: produc2459281416120113548term_a > $o,P2: produc2459281416120113548term_a] :
      ( ! [A3: ground_gmctxt_a,B3: list_Ground_gterm_a] : ( P @ ( produc4720362732934906942term_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_119_prod__cases,axiom,
    ! [P: produc1525979690750518759term_a > $o,P2: produc1525979690750518759term_a] :
      ( ! [A3: ground_gterm_a,B3: ground_gterm_a] : ( P @ ( produc7419039783125785943term_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_120_prod__cases,axiom,
    ! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
      ( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_121_Pair__inject,axiom,
    ! [A: ground_gmctxt_a,B: list_Ground_gterm_a,A2: ground_gmctxt_a,B2: list_Ground_gterm_a] :
      ( ( ( produc4720362732934906942term_a @ A @ B )
        = ( produc4720362732934906942term_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_122_Pair__inject,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,A2: ground_gterm_a,B2: ground_gterm_a] :
      ( ( ( produc7419039783125785943term_a @ A @ B )
        = ( produc7419039783125785943term_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_123_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_124_mem__Collect__eq,axiom,
    ! [A: produc1525979690750518759term_a,P: produc1525979690750518759term_a > $o] :
      ( ( member4351891425000572176term_a @ A @ ( collec6713184107424615762term_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
    ! [A: list_Ground_gterm_a,P: list_Ground_gterm_a > $o] :
      ( ( member5011579089780872286term_a @ A @ ( collec4295779454384174240term_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
    ! [A: ground_gterm_a,P: ground_gterm_a > $o] :
      ( ( member5261315023172567896term_a @ A @ ( collec3522850604242857114term_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
    ! [A: ground_gmctxt_a,P: ground_gmctxt_a > $o] :
      ( ( member3306386364193653941ctxt_a @ A @ ( collec8774204703398931443ctxt_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_129_Collect__mem__eq,axiom,
    ! [A4: set_Pr8282532413760598727term_a] :
      ( ( collec6713184107424615762term_a
        @ ^ [X3: produc1525979690750518759term_a] : ( member4351891425000572176term_a @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
    ! [A4: set_li8588095516285831957term_a] :
      ( ( collec4295779454384174240term_a
        @ ^ [X3: list_Ground_gterm_a] : ( member5011579089780872286term_a @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
    ! [A4: set_Ground_gterm_a] :
      ( ( collec3522850604242857114term_a
        @ ^ [X3: ground_gterm_a] : ( member5261315023172567896term_a @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
    ! [A4: set_Ground_gmctxt_a] :
      ( ( collec8774204703398931443ctxt_a
        @ ^ [X3: ground_gmctxt_a] : ( member3306386364193653941ctxt_a @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_134_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_135_eq__length__concat__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_136_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_137_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l4678906206682822203term_a] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_138_eq__length__concat__nth,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_list_nat] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_139_eq__length__concat__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
       => ( ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) )
          = ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_140_eq__length__concat__nth,axiom,
    ! [Xs: list_l5172476077732501185term_a,Ys: list_list_nat] :
      ( ( ( size_s3474408278181496621term_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Xs ) )
           => ( ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) )
          = ( size_size_list_nat @ ( concat_nat @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_141_eq__length__concat__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l5172476077732501185term_a] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3474408278181496621term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Ys @ I ) ) ) )
       => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
          = ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_142_eq__length__concat__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l4678906206682822203term_a] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
       => ( ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) )
          = ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_143_eq__length__concat__nth,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
       => ( ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) )
          = ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) ) ) ) ) ).

% eq_length_concat_nth
thf(fact_144_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_list_nat,P: nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I2 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_145_concat__nth__nthI,axiom,
    ! [Ss: list_l1578326660578982846ctxt_a,Ts: list_list_nat,P: ground_gmctxt_a > nat > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) )
                 => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ss ) @ I2 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_146_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_l1578326660578982846ctxt_a,P: nat > ground_gmctxt_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s6994882312647770194ctxt_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I ) @ J ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_147_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_l4678906206682822203term_a,P: nat > ground_gterm_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s5321119603959729319term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I ) )
              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I2 ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_148_concat__nth__nthI,axiom,
    ! [Ss: list_l4678906206682822203term_a,Ts: list_list_nat,P: ground_gterm_a > nat > $o] :
      ( ( ( size_s5321119603959729319term_a @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ss @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) )
                 => ( P @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ss @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ss ) @ I2 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_149_concat__nth__nthI,axiom,
    ! [Ss: list_l1578326660578982846ctxt_a,Ts: list_l1578326660578982846ctxt_a,P: ground_gmctxt_a > ground_gmctxt_a > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Ss )
        = ( size_s6994882312647770194ctxt_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ts ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) ) )
                 => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) @ J ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ss ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_150_concat__nth__nthI,axiom,
    ! [Ss: list_l5172476077732501185term_a,Ts: list_list_nat,P: list_Ground_gterm_a > nat > $o] :
      ( ( ( size_s3474408278181496621term_a @ Ss )
        = ( size_s3023201423986296836st_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
           => ( ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Ss @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Ts @ I ) ) )
                 => ( P @ ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Ss @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( concat_nat @ Ts ) ) )
             => ( P @ ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Ss ) @ I2 ) @ ( nth_nat @ ( concat_nat @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_151_concat__nth__nthI,axiom,
    ! [Ss: list_list_nat,Ts: list_l5172476077732501185term_a,P: nat > list_Ground_gterm_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Ss )
        = ( size_s3474408278181496621term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Ts ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Ss @ I ) )
              = ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Ts @ I ) ) )
                 => ( P @ ( nth_nat @ ( nth_list_nat @ Ss @ I ) @ J ) @ ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Ts ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Ss ) @ I2 ) @ ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_152_concat__nth__nthI,axiom,
    ! [Ss: list_l1578326660578982846ctxt_a,Ts: list_l4678906206682822203term_a,P: ground_gmctxt_a > ground_gterm_a > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Ss )
        = ( size_s5321119603959729319term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ts ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) )
              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) )
                 => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Ts ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ss ) @ I2 ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_153_concat__nth__nthI,axiom,
    ! [Ss: list_l4678906206682822203term_a,Ts: list_l1578326660578982846ctxt_a,P: ground_gterm_a > ground_gmctxt_a > $o] :
      ( ( ( size_s5321119603959729319term_a @ Ss )
        = ( size_s6994882312647770194ctxt_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ts ) )
           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ss @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) ) ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Ts ) )
             => ! [J: nat] :
                  ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) ) )
                 => ( P @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ss @ I ) @ J ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ts @ I ) @ J ) ) ) )
         => ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) ) )
             => ( P @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ss ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ts ) @ I2 ) ) ) ) ) ) ).

% concat_nth_nthI
thf(fact_154_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat,P: nat > nat > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K3 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_155_concat__all__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat,P: ground_gmctxt_a > nat > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ K3 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_156_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a,P: nat > ground_gmctxt_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I ) @ J ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K3 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_157_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l4678906206682822203term_a,P: nat > ground_gterm_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K3 ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_158_concat__all__nth,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_list_nat,P: ground_gterm_a > nat > $o] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) ) )
               => ( P @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
             => ( P @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ K3 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_159_concat__all__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,P: ground_gmctxt_a > ground_gmctxt_a > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) @ J ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ K3 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_160_concat__all__nth,axiom,
    ! [Xs: list_l5172476077732501185term_a,Ys: list_list_nat,P: list_Ground_gterm_a > nat > $o] :
      ( ( ( size_s3474408278181496621term_a @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Xs ) )
           => ( ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ I ) )
              = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ I ) ) )
               => ( P @ ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Xs @ I ) @ J ) @ ( nth_nat @ ( nth_list_nat @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
             => ( P @ ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Xs ) @ K3 ) @ ( nth_nat @ ( concat_nat @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_161_concat__all__nth,axiom,
    ! [Xs: list_list_nat,Ys: list_l5172476077732501185term_a,P: nat > list_Ground_gterm_a > $o] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
        = ( size_s3474408278181496621term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
              = ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) )
               => ( P @ ( nth_nat @ ( nth_list_nat @ Xs @ I ) @ J ) @ ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
             => ( P @ ( nth_nat @ ( concat_nat @ Xs ) @ K3 ) @ ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_162_concat__all__nth,axiom,
    ! [Xs: list_l1578326660578982846ctxt_a,Ys: list_l4678906206682822203term_a,P: ground_gmctxt_a > ground_gterm_a > $o] :
      ( ( ( size_s6994882312647770194ctxt_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
           => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
              = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
             => ( P @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ K3 ) @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_163_concat__all__nth,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l1578326660578982846ctxt_a,P: ground_gterm_a > ground_gmctxt_a > $o] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
              = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
       => ( ! [I: nat,J: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) ) )
               => ( P @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ J ) @ ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) @ J ) ) ) )
         => ! [K3: nat] :
              ( ( ord_less_nat @ K3 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
             => ( P @ ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ K3 ) @ ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ K3 ) ) ) ) ) ) ).

% concat_all_nth
thf(fact_164_nth__concat__split,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ? [J: nat,K4: nat] :
          ( ( ord_less_nat @ J @ ( size_s6994882312647770194ctxt_a @ Xs ) )
          & ( ord_less_nat @ K4 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) )
          & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I4 )
            = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) @ K4 ) ) ) ) ).

% nth_concat_split
thf(fact_165_nth__concat__split,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ? [J: nat,K4: nat] :
          ( ( ord_less_nat @ J @ ( size_s5321119603959729319term_a @ Xs ) )
          & ( ord_less_nat @ K4 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) ) )
          & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I4 )
            = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) @ K4 ) ) ) ) ).

% nth_concat_split
thf(fact_166_nth__concat__split,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ? [J: nat,K4: nat] :
          ( ( ord_less_nat @ J @ ( size_s3474408278181496621term_a @ Xs ) )
          & ( ord_less_nat @ K4 @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ J ) ) )
          & ( ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Xs ) @ I4 )
            = ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Xs @ J ) @ K4 ) ) ) ) ).

% nth_concat_split
thf(fact_167_nth__concat__split,axiom,
    ! [I4: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ? [J: nat,K4: nat] :
          ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
          & ( ord_less_nat @ K4 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J ) ) )
          & ( ( nth_nat @ ( concat_nat @ Xs ) @ I4 )
            = ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ K4 ) ) ) ) ).

% nth_concat_split
thf(fact_168_nth__concat__split,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ? [J: nat,K4: nat] :
          ( ( ord_less_nat @ J @ ( size_s7283615635009460703term_a @ Xs ) )
          & ( ord_less_nat @ K4 @ ( size_s281034491502894169term_a @ ( nth_li7906007133205842164term_a @ Xs @ J ) ) )
          & ( ( nth_Pr5943965302187034734term_a @ ( concat2365498962271017768term_a @ Xs ) @ I4 )
            = ( nth_Pr5943965302187034734term_a @ ( nth_li7906007133205842164term_a @ Xs @ J ) @ K4 ) ) ) ) ).

% nth_concat_split
thf(fact_169_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ K4 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_170_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I4 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_171_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s5321119603959729319term_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) )
                = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s5321119603959729319term_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I4 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_172_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Ys )
          = ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ K4 ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ I4 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_173_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_list_nat,Ys: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Ys )
          = ( size_s3023201423986296836st_nat @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ J ) ) )
              & ( ( nth_nat @ ( concat_nat @ Xs ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ K4 ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys ) @ I4 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_174_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Ys )
          = ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I4 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ I4 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_175_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Ys )
          = ( size_s6994882312647770194ctxt_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s6994882312647770194ctxt_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I4 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys ) @ I4 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_176_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Ys )
          = ( size_s5321119603959729319term_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) )
                = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s5321119603959729319term_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I4 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Ys ) @ I4 )
                = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_177_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Ys )
          = ( size_s5321119603959729319term_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) )
                = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s5321119603959729319term_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I4 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Ys ) @ I4 )
                = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_178_nth__concat__two__lists,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,Ys: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Ys )
          = ( size_s3474408278181496621term_a @ Xs ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) )
                = ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ I ) ) ) )
         => ? [J: nat,K4: nat] :
              ( ( ord_less_nat @ J @ ( size_s3474408278181496621term_a @ Xs ) )
              & ( ord_less_nat @ K4 @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ J ) ) )
              & ( ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Xs ) @ I4 )
                = ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Xs @ J ) @ K4 ) )
              & ( ( nth_nat @ ( concat_nat @ Ys ) @ I4 )
                = ( nth_nat @ ( nth_list_nat @ Ys @ J ) @ K4 ) ) ) ) ) ) ).

% nth_concat_two_lists
thf(fact_179_fill__gholes__gmctxt__sound,axiom,
    ! [Ds: list_Ground_gmctxt_a,C: ground_gmctxt_a,Sss2: list_l4678906206682822203term_a,Ts: list_Ground_gterm_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Ds )
        = ( ground_num_gholes_a @ C ) )
     => ( ( ( size_s5321119603959729319term_a @ Sss2 )
          = ( ground_num_gholes_a @ C ) )
       => ( ( ( size_s1893836384466666913term_a @ Ts )
            = ( ground_num_gholes_a @ C ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
               => ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds @ I ) @ ( nth_li6172171122907299772term_a @ Sss2 @ I ) ) ) )
           => ( ground_eq_gfill_a @ ( ground_fill_gholes_a @ C @ Ts ) @ ( produc4720362732934906942term_a @ ( ground2530127064035342964ctxt_a @ C @ Ds ) @ ( concat1399541994378716528term_a @ Sss2 ) ) ) ) ) ) ) ).

% fill_gholes_gmctxt_sound
thf(fact_180_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_list_nat,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_181_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
         => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_182_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
         => ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_183_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_list_nat,Ys: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_184_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_list_nat,Ys: list_l4678906206682822203term_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( size_s3023201423986296836st_nat @ Xs )
          = ( size_s5321119603959729319term_a @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
             => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I ) )
                = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
         => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_185_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
         => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_186_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,Ys: list_l4678906206682822203term_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Xs )
          = ( size_s5321119603959729319term_a @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6994882312647770194ctxt_a @ Xs ) )
             => ( ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ I ) )
                = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
         => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_187_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Xs )
          = ( size_s6994882312647770194ctxt_a @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
                = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ys @ I ) ) ) )
         => ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_188_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Xs )
          = ( size_s5321119603959729319term_a @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Xs ) )
             => ( ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) )
                = ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) ) )
         => ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_189_concat__index__split__unique,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,Ys: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( ( size_s3474408278181496621term_a @ Xs )
          = ( size_s3023201423986296836st_nat @ Ys ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s3474408278181496621term_a @ Xs ) )
             => ( ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ I ) )
                = ( size_size_list_nat @ ( nth_list_nat @ Ys @ I ) ) ) )
         => ( ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs )
            = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Ys ) ) ) ) ) ).

% concat_index_split_unique
thf(fact_190_Utils_Otrancl__list_Ocases,axiom,
    ! [A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A12 @ A23 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
     => ( ( ( ( size_s962548663205356226ctxt_a @ A12 )
            = ( size_s962548663205356226ctxt_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ A23 ) )
               => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ A12 @ I2 ) @ ( nth_Ground_gmctxt_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_Ground_gmctxt_a,I: nat,Z2: ground_gmctxt_a] :
              ( ( A23
                = ( list_u7950812632479372336ctxt_a @ Ys4 @ I @ Z2 ) )
             => ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A12 @ Ys4 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                 => ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_191_Utils_Otrancl__list_Ocases,axiom,
    ! [A12: list_Ground_gterm_a,A23: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A12 @ A23 ) @ ( trancl1902929335784340279term_a @ R ) )
     => ( ( ( ( size_s1893836384466666913term_a @ A12 )
            = ( size_s1893836384466666913term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ A23 ) )
               => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ A12 @ I2 ) @ ( nth_Ground_gterm_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_Ground_gterm_a,I: nat,Z2: ground_gterm_a] :
              ( ( A23
                = ( list_u6582344683876525341term_a @ Ys4 @ I @ Z2 ) )
             => ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A12 @ Ys4 ) @ ( trancl1902929335784340279term_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys4 ) )
                 => ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_192_Utils_Otrancl__list_Ocases,axiom,
    ! [A12: list_l4678906206682822203term_a,A23: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ A12 @ A23 ) @ ( trancl5818250397338112189term_a @ R ) )
     => ( ( ( ( size_s5321119603959729319term_a @ A12 )
            = ( size_s5321119603959729319term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ A23 ) )
               => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ A12 @ I2 ) @ ( nth_li6172171122907299772term_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_l4678906206682822203term_a,I: nat,Z2: list_Ground_gterm_a] :
              ( ( A23
                = ( list_u412135983231242147term_a @ Ys4 @ I @ Z2 ) )
             => ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ A12 @ Ys4 ) @ ( trancl5818250397338112189term_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys4 ) )
                 => ~ ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_193_Utils_Otrancl__list_Ocases,axiom,
    ! [A12: list_nat,A23: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( trancl_list_nat @ R ) )
     => ( ( ( ( size_size_list_nat @ A12 )
            = ( size_size_list_nat @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ A23 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A12 @ I2 ) @ ( nth_nat @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_nat,I: nat,Z2: nat] :
              ( ( A23
                = ( list_update_nat @ Ys4 @ I @ Z2 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ Ys4 ) @ ( trancl_list_nat @ R ) )
               => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_194_Utils_Otrancl__list_Ocases,axiom,
    ! [A12: list_P2433084232546440685term_a,A23: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ A12 @ A23 ) @ ( trancl6567088178672586095term_a @ R ) )
     => ( ( ( ( size_s281034491502894169term_a @ A12 )
            = ( size_s281034491502894169term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ A23 ) )
               => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ A12 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_P2433084232546440685term_a,I: nat,Z2: produc1525979690750518759term_a] :
              ( ( A23
                = ( list_u8102571067576484949term_a @ Ys4 @ I @ Z2 ) )
             => ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ A12 @ Ys4 ) @ ( trancl6567088178672586095term_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys4 ) )
                 => ~ ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Utils.trancl_list.cases
thf(fact_195_Utils_Otrancl__list_Osimps,axiom,
    ! [A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A12 @ A23 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
      = ( ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s962548663205356226ctxt_a @ Xs2 )
              = ( size_s962548663205356226ctxt_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
               => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs2 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,I3: nat,Z3: ground_gmctxt_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u7950812632479372336ctxt_a @ Ys3 @ I3 @ Z3 ) )
            & ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs2 @ Ys3 ) @ ( trancl6028151697025156246ctxt_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
            & ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_196_Utils_Otrancl__list_Osimps,axiom,
    ! [A12: list_Ground_gterm_a,A23: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A12 @ A23 ) @ ( trancl1902929335784340279term_a @ R ) )
      = ( ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s1893836384466666913term_a @ Xs2 )
              = ( size_s1893836384466666913term_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
               => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs2 @ I3 ) @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a,I3: nat,Z3: ground_gterm_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u6582344683876525341term_a @ Ys3 @ I3 @ Z3 ) )
            & ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs2 @ Ys3 ) @ ( trancl1902929335784340279term_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
            & ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_197_Utils_Otrancl__list_Osimps,axiom,
    ! [A12: list_l4678906206682822203term_a,A23: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ A12 @ A23 ) @ ( trancl5818250397338112189term_a @ R ) )
      = ( ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s5321119603959729319term_a @ Xs2 )
              = ( size_s5321119603959729319term_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
               => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs2 @ I3 ) @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a,I3: nat,Z3: list_Ground_gterm_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u412135983231242147term_a @ Ys3 @ I3 @ Z3 ) )
            & ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs2 @ Ys3 ) @ ( trancl5818250397338112189term_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
            & ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_198_Utils_Otrancl__list_Osimps,axiom,
    ! [A12: list_nat,A23: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( trancl_list_nat @ R ) )
      = ( ? [Xs2: list_nat,Ys3: list_nat] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_size_list_nat @ Xs2 )
              = ( size_size_list_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( trancl_list_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_199_Utils_Otrancl__list_Osimps,axiom,
    ! [A12: list_P2433084232546440685term_a,A23: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ A12 @ A23 ) @ ( trancl6567088178672586095term_a @ R ) )
      = ( ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s281034491502894169term_a @ Xs2 )
              = ( size_s281034491502894169term_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
               => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs2 @ I3 ) @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a,I3: nat,Z3: produc1525979690750518759term_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u8102571067576484949term_a @ Ys3 @ I3 @ Z3 ) )
            & ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs2 @ Ys3 ) @ ( trancl6567088178672586095term_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
            & ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Utils.trancl_list.simps
thf(fact_200_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,I4: nat,Z4: ground_gmctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( trancl6028151697025156246ctxt_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ I4 @ Z4 ) ) @ ( trancl6028151697025156246ctxt_a @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_201_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,I4: nat,Z4: ground_gterm_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( trancl1902929335784340279term_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ ( list_u6582344683876525341term_a @ Ys @ I4 @ Z4 ) ) @ ( trancl1902929335784340279term_a @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_202_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a,I4: nat,Z4: list_Ground_gterm_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( trancl5818250397338112189term_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Ys ) )
       => ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ ( list_u412135983231242147term_a @ Ys @ I4 @ Z4 ) ) @ ( trancl5818250397338112189term_a @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_203_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,I4: nat,Z4: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( list_update_nat @ Ys @ I4 @ Z4 ) ) @ ( trancl_list_nat @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_204_Utils_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a,I4: nat,Z4: produc1525979690750518759term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( trancl6567088178672586095term_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Ys ) )
       => ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ ( list_u8102571067576484949term_a @ Ys @ I4 @ Z4 ) ) @ ( trancl6567088178672586095term_a @ R ) ) ) ) ) ).

% Utils.trancl_list.list_trancl
thf(fact_205_Utils_Otrancl__list__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( trancl6028151697025156246ctxt_a @ R ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ) ).

% Utils.trancl_list_conv
thf(fact_206_Utils_Otrancl__list__conv,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( trancl1902929335784340279term_a @ R ) )
      = ( ( ( size_s1893836384466666913term_a @ Xs )
          = ( size_s1893836384466666913term_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I3 ) @ ( nth_Ground_gterm_a @ Ys @ I3 ) ) @ ( transi8854406246868658384term_a @ R ) ) ) ) ) ).

% Utils.trancl_list_conv
thf(fact_207_Utils_Otrancl__list__conv,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( trancl5818250397338112189term_a @ R ) )
      = ( ( ( size_s5321119603959729319term_a @ Xs )
          = ( size_s5321119603959729319term_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys ) )
           => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs @ I3 ) @ ( nth_li6172171122907299772term_a @ Ys @ I3 ) ) @ ( transi554849227584806358term_a @ R ) ) ) ) ) ).

% Utils.trancl_list_conv
thf(fact_208_Utils_Otrancl__list__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% Utils.trancl_list_conv
thf(fact_209_Utils_Otrancl__list__conv,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( trancl6567088178672586095term_a @ R ) )
      = ( ( ( size_s281034491502894169term_a @ Xs )
          = ( size_s281034491502894169term_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys ) )
           => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ I3 ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I3 ) ) @ ( transi4994575220883992200term_a @ R ) ) ) ) ) ).

% Utils.trancl_list_conv
thf(fact_210_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_211_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,R2: set_Pr9093710254965148079_a_nat] :
      ( ( member9152701222192952216st_nat @ ( produc5273301679375258151st_nat @ Xs @ Ys ) @ ( listre5839456775373167073_a_nat @ R2 ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member1226896217267473912_a_nat @ ( produc2030840239341956231_a_nat @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_212_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_nat,R2: set_Pr4872037744881986020_a_nat] :
      ( ( member6774170118143593819st_nat @ ( produc7210411988535939518st_nat @ Xs @ Ys ) @ ( listre4290348974220565582_a_nat @ R2 ) )
      = ( ( ( size_s1893836384466666913term_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
           => ( member2899317942255798213_a_nat @ ( produc6032213142089859112_a_nat @ ( nth_Ground_gterm_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_213_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gmctxt_a,R2: set_Pr1619835084168866991ctxt_a] :
      ( ( member5348271269840303512ctxt_a @ ( produc781480777803334439ctxt_a @ Xs @ Ys ) @ ( listre6913154111349222369ctxt_a @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member2885886537299685624ctxt_a @ ( produc3104537575318011527ctxt_a @ ( nth_nat @ Xs @ N2 ) @ ( nth_Ground_gmctxt_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_214_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gterm_a,R2: set_Pr1502840878978300170term_a] :
      ( ( member4459061637085242849term_a @ ( produc8125750555744107516term_a @ Xs @ Ys ) @ ( listre2955210640777335084term_a @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s1893836384466666913term_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member6622040750283622891term_a @ ( produc4697074808646628614term_a @ ( nth_nat @ Xs @ N2 ) @ ( nth_Ground_gterm_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_215_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R2: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_216_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( listre9013468577358163503ctxt_a @ R2 ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ ( nth_Ground_gmctxt_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_217_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a,R2: set_Pr3844680646802797116term_a] :
      ( ( member5883382257082020883term_a @ ( produc2492016761743805998term_a @ Xs @ Ys ) @ ( listre7705070462779104094term_a @ R2 ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s1893836384466666913term_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member1041919251622114333term_a @ ( produc7893155741350732600term_a @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ ( nth_Ground_gterm_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_218_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a,R2: set_Pr828580167737221682ctxt_a] :
      ( ( member6380717558036200361ctxt_a @ ( produc8294886102066967308ctxt_a @ Xs @ Ys ) @ ( listre5837551683383141020ctxt_a @ R2 ) )
      = ( ( ( size_s1893836384466666913term_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
           => ( member3925204351666028307ctxt_a @ ( produc6025636961954769526ctxt_a @ ( nth_Ground_gterm_a @ Xs @ N2 ) @ ( nth_Ground_gmctxt_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_219_listrel__iff__nth,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( listre9082503044497914801term_a @ R2 ) )
      = ( ( ( size_s1893836384466666913term_a @ Xs )
          = ( size_s1893836384466666913term_a @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ N2 ) @ ( nth_Ground_gterm_a @ Ys @ N2 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_220_eqgf__GMFunE,axiom,
    ! [S: ground_gterm_a,F: a,Cs: list_Ground_gmctxt_a,Ss: list_Ground_gterm_a] :
      ( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ ( ground_GMFun_a @ F @ Cs ) @ Ss ) )
     => ~ ! [Ts2: list_Ground_gterm_a] :
            ( ( S
              = ( ground_GFun_a @ F @ Ts2 ) )
           => ( ( ( size_s1893836384466666913term_a @ Ts2 )
                = ( size_s962548663205356226ctxt_a @ Cs ) )
             => ! [Sss: list_l4678906206682822203term_a] :
                  ( ( ( size_s5321119603959729319term_a @ Sss )
                    = ( size_s962548663205356226ctxt_a @ Cs ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Cs ) )
                       => ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Cs @ I2 ) @ ( nth_li6172171122907299772term_a @ Sss @ I2 ) ) ) )
                   => ( Ss
                     != ( concat1399541994378716528term_a @ Sss ) ) ) ) ) ) ) ).

% eqgf_GMFunE
thf(fact_221_eqgf__GMFunI,axiom,
    ! [Cs: list_Ground_gmctxt_a,Ss: list_Ground_gterm_a,Ts: list_l4678906206682822203term_a,F: a] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs ) )
         => ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Cs @ I ) @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs )
          = ( size_s1893836384466666913term_a @ Ss ) )
       => ( ( ( size_s1893836384466666913term_a @ Ss )
            = ( size_s5321119603959729319term_a @ Ts ) )
         => ( ground_eq_gfill_a @ ( ground_GFun_a @ F @ Ss ) @ ( produc4720362732934906942term_a @ ( ground_GMFun_a @ F @ Cs ) @ ( concat1399541994378716528term_a @ Ts ) ) ) ) ) ) ).

% eqgf_GMFunI
thf(fact_222_length__list__update,axiom,
    ! [Xs: list_Ground_gmctxt_a,I4: nat,X2: ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_list_update
thf(fact_223_length__list__update,axiom,
    ! [Xs: list_Ground_gterm_a,I4: nat,X2: ground_gterm_a] :
      ( ( size_s1893836384466666913term_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) )
      = ( size_s1893836384466666913term_a @ Xs ) ) ).

% length_list_update
thf(fact_224_length__list__update,axiom,
    ! [Xs: list_l4678906206682822203term_a,I4: nat,X2: list_Ground_gterm_a] :
      ( ( size_s5321119603959729319term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) )
      = ( size_s5321119603959729319term_a @ Xs ) ) ).

% length_list_update
thf(fact_225_length__list__update,axiom,
    ! [Xs: list_nat,I4: nat,X2: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_226_length__list__update,axiom,
    ! [Xs: list_P2433084232546440685term_a,I4: nat,X2: produc1525979690750518759term_a] :
      ( ( size_s281034491502894169term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) )
      = ( size_s281034491502894169term_a @ Xs ) ) ).

% length_list_update
thf(fact_227_nth__list__update__neq,axiom,
    ! [I4: nat,J3: nat,Xs: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( I4 != J3 )
     => ( ( nth_Ground_gterm_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) @ J3 )
        = ( nth_Ground_gterm_a @ Xs @ J3 ) ) ) ).

% nth_list_update_neq
thf(fact_228_nth__list__update__neq,axiom,
    ! [I4: nat,J3: nat,Xs: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( I4 != J3 )
     => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) @ J3 )
        = ( nth_Ground_gmctxt_a @ Xs @ J3 ) ) ) ).

% nth_list_update_neq
thf(fact_229_nth__list__update__neq,axiom,
    ! [I4: nat,J3: nat,Xs: list_l4678906206682822203term_a,X2: list_Ground_gterm_a] :
      ( ( I4 != J3 )
     => ( ( nth_li6172171122907299772term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) @ J3 )
        = ( nth_li6172171122907299772term_a @ Xs @ J3 ) ) ) ).

% nth_list_update_neq
thf(fact_230_nth__list__update__neq,axiom,
    ! [I4: nat,J3: nat,Xs: list_nat,X2: nat] :
      ( ( I4 != J3 )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J3 )
        = ( nth_nat @ Xs @ J3 ) ) ) ).

% nth_list_update_neq
thf(fact_231_nth__list__update__neq,axiom,
    ! [I4: nat,J3: nat,Xs: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a] :
      ( ( I4 != J3 )
     => ( ( nth_Pr5943965302187034734term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) @ J3 )
        = ( nth_Pr5943965302187034734term_a @ Xs @ J3 ) ) ) ).

% nth_list_update_neq
thf(fact_232_list__update__id,axiom,
    ! [Xs: list_Ground_gterm_a,I4: nat] :
      ( ( list_u6582344683876525341term_a @ Xs @ I4 @ ( nth_Ground_gterm_a @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_233_list__update__id,axiom,
    ! [Xs: list_Ground_gmctxt_a,I4: nat] :
      ( ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ ( nth_Ground_gmctxt_a @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_234_list__update__id,axiom,
    ! [Xs: list_l4678906206682822203term_a,I4: nat] :
      ( ( list_u412135983231242147term_a @ Xs @ I4 @ ( nth_li6172171122907299772term_a @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_235_list__update__id,axiom,
    ! [Xs: list_nat,I4: nat] :
      ( ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_236_list__update__id,axiom,
    ! [Xs: list_P2433084232546440685term_a,I4: nat] :
      ( ( list_u8102571067576484949term_a @ Xs @ I4 @ ( nth_Pr5943965302187034734term_a @ Xs @ I4 ) )
      = Xs ) ).

% list_update_id
thf(fact_237_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) @ I4 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_238_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( nth_Ground_gterm_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) @ I4 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_239_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,X2: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ( nth_li6172171122907299772term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) @ I4 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_240_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ I4 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_241_nth__list__update__eq,axiom,
    ! [I4: nat,Xs: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ( nth_Pr5943965302187034734term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) @ I4 )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_242_set__swap,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,J3: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( set_Ground_gmctxt_a2 @ ( list_u7950812632479372336ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ ( nth_Ground_gmctxt_a @ Xs @ J3 ) ) @ J3 @ ( nth_Ground_gmctxt_a @ Xs @ I4 ) ) )
          = ( set_Ground_gmctxt_a2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_243_set__swap,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,J3: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_s1893836384466666913term_a @ Xs ) )
       => ( ( set_Ground_gterm_a2 @ ( list_u6582344683876525341term_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ ( nth_Ground_gterm_a @ Xs @ J3 ) ) @ J3 @ ( nth_Ground_gterm_a @ Xs @ I4 ) ) )
          = ( set_Ground_gterm_a2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_244_set__swap,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,J3: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_s5321119603959729319term_a @ Xs ) )
       => ( ( set_li7546925467519454922term_a @ ( list_u412135983231242147term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ ( nth_li6172171122907299772term_a @ Xs @ J3 ) ) @ J3 @ ( nth_li6172171122907299772term_a @ Xs @ I4 ) ) )
          = ( set_li7546925467519454922term_a @ Xs ) ) ) ) ).

% set_swap
thf(fact_245_set__swap,axiom,
    ! [I4: nat,Xs: list_nat,J3: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ ( nth_nat @ Xs @ J3 ) ) @ J3 @ ( nth_nat @ Xs @ I4 ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_246_set__swap,axiom,
    ! [I4: nat,Xs: list_P2433084232546440685term_a,J3: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ( ord_less_nat @ J3 @ ( size_s281034491502894169term_a @ Xs ) )
       => ( ( set_Pr5405221470436175740term_a @ ( list_u8102571067576484949term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ ( nth_Pr5943965302187034734term_a @ Xs @ J3 ) ) @ J3 @ ( nth_Pr5943965302187034734term_a @ Xs @ I4 ) ) )
          = ( set_Pr5405221470436175740term_a @ Xs ) ) ) ) ).

% set_swap
thf(fact_247_trancl__map,axiom,
    ! [R2: set_Pr8282532413760598727term_a,F: ground_gterm_a > ground_gterm_a,S: set_Pr8282532413760598727term_a,X2: ground_gterm_a,Y3: ground_gterm_a] :
      ( ! [X: ground_gterm_a,Y4: ground_gterm_a] :
          ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y4 ) @ R2 )
         => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X ) @ ( F @ Y4 ) ) @ S ) )
     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ ( transi8854406246868658384term_a @ R2 ) )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( F @ X2 ) @ ( F @ Y3 ) ) @ ( transi8854406246868658384term_a @ S ) ) ) ) ).

% trancl_map
thf(fact_248_fill__gholes__gmctxt__MFun,axiom,
    ! [Cs: list_Ground_gmctxt_a,Ts: list_Ground_gmctxt_a,Ss: list_l1578326660578982846ctxt_a,F: a] :
      ( ( ( size_s962548663205356226ctxt_a @ Cs )
        = ( size_s962548663205356226ctxt_a @ Ts ) )
     => ( ( ( size_s6994882312647770194ctxt_a @ Ss )
          = ( size_s962548663205356226ctxt_a @ Ts ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ts ) )
             => ( ( ( ground_num_gholes_a @ ( nth_Ground_gmctxt_a @ Cs @ I ) )
                  = ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) ) )
                & ( ( ground2530127064035342964ctxt_a @ ( nth_Ground_gmctxt_a @ Cs @ I ) @ ( nth_li6678718213161987943ctxt_a @ Ss @ I ) )
                  = ( nth_Ground_gmctxt_a @ Ts @ I ) ) ) )
         => ( ( ground2530127064035342964ctxt_a @ ( ground_GMFun_a @ F @ Cs ) @ ( concat3993834769610107037ctxt_a @ Ss ) )
            = ( ground_GMFun_a @ F @ Ts ) ) ) ) ) ).

% fill_gholes_gmctxt_MFun
thf(fact_249_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_250_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,R2: set_Pr9093710254965148079_a_nat] :
      ( ( member9152701222192952216st_nat @ ( produc5273301679375258151st_nat @ Xs @ Ys ) @ ( listre5839456775373167073_a_nat @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_251_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_nat,R2: set_Pr4872037744881986020_a_nat] :
      ( ( member6774170118143593819st_nat @ ( produc7210411988535939518st_nat @ Xs @ Ys ) @ ( listre4290348974220565582_a_nat @ R2 ) )
     => ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_252_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gmctxt_a,R2: set_Pr1619835084168866991ctxt_a] :
      ( ( member5348271269840303512ctxt_a @ ( produc781480777803334439ctxt_a @ Xs @ Ys ) @ ( listre6913154111349222369ctxt_a @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_253_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gterm_a,R2: set_Pr1502840878978300170term_a] :
      ( ( member4459061637085242849term_a @ ( produc8125750555744107516term_a @ Xs @ Ys ) @ ( listre2955210640777335084term_a @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_254_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( listre9013468577358163503ctxt_a @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_255_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a,R2: set_Pr3844680646802797116term_a] :
      ( ( member5883382257082020883term_a @ ( produc2492016761743805998term_a @ Xs @ Ys ) @ ( listre7705070462779104094term_a @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_256_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a,R2: set_Pr828580167737221682ctxt_a] :
      ( ( member6380717558036200361ctxt_a @ ( produc8294886102066967308ctxt_a @ Xs @ Ys ) @ ( listre5837551683383141020ctxt_a @ R2 ) )
     => ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_257_listrel__eq__len,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( listre9082503044497914801term_a @ R2 ) )
     => ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_258_listrel__eq__len,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_nat,R2: set_Pr3996311460061108074_a_nat] :
      ( ( member9077793357388750945st_nat @ ( produc6536778741003241412st_nat @ Xs @ Ys ) @ ( listre5455280455255510740_a_nat @ R2 ) )
     => ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_259_gmctxtex__onp__neq__rootE,axiom,
    ! [F: a,Ss: list_Ground_gterm_a,G: a,Ts: list_Ground_gterm_a,P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
     => ( ( F != G )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ R ) ) ) ).

% gmctxtex_onp_neq_rootE
thf(fact_260_set__update__memI,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ ( list_u7950812632479372336ctxt_a @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_261_set__update__memI,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ ( list_u6582344683876525341term_a @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_262_set__update__memI,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a,X2: list_Ground_gterm_a] :
      ( ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( member5011579089780872286term_a @ X2 @ ( set_li7546925467519454922term_a @ ( list_u412135983231242147term_a @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_263_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_264_set__update__memI,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a] :
      ( ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
     => ( member4351891425000572176term_a @ X2 @ ( set_Pr5405221470436175740term_a @ ( list_u8102571067576484949term_a @ Xs @ N @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_265_parallel__list__update,axiom,
    ! [N: nat,R2: ground_gmctxt_a > ground_gmctxt_a > $o,P2: list_Ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ! [Xs3: list_Ground_gmctxt_a,I: nat,Y4: ground_gmctxt_a] :
          ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R2 @ ( nth_Ground_gmctxt_a @ Xs3 @ I ) @ Y4 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_u7950812632479372336ctxt_a @ Xs3 @ I @ Y4 ) ) ) ) ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s962548663205356226ctxt_a @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R2 @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_266_parallel__list__update,axiom,
    ! [N: nat,R2: ground_gterm_a > ground_gterm_a > $o,P2: list_Ground_gterm_a > $o,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ! [Xs3: list_Ground_gterm_a,I: nat,Y4: ground_gterm_a] :
          ( ( ( size_s1893836384466666913term_a @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R2 @ ( nth_Ground_gterm_a @ Xs3 @ I ) @ Y4 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_u6582344683876525341term_a @ Xs3 @ I @ Y4 ) ) ) ) ) )
     => ( ( ( size_s1893836384466666913term_a @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s1893836384466666913term_a @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R2 @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_267_parallel__list__update,axiom,
    ! [N: nat,R2: list_Ground_gterm_a > list_Ground_gterm_a > $o,P2: list_l4678906206682822203term_a > $o,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ! [Xs3: list_l4678906206682822203term_a,I: nat,Y4: list_Ground_gterm_a] :
          ( ( ( size_s5321119603959729319term_a @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R2 @ ( nth_li6172171122907299772term_a @ Xs3 @ I ) @ Y4 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_u412135983231242147term_a @ Xs3 @ I @ Y4 ) ) ) ) ) )
     => ( ( ( size_s5321119603959729319term_a @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s5321119603959729319term_a @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R2 @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_268_parallel__list__update,axiom,
    ! [N: nat,R2: nat > nat > $o,P2: list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ! [Xs3: list_nat,I: nat,Y4: nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R2 @ ( nth_nat @ Xs3 @ I ) @ Y4 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_update_nat @ Xs3 @ I @ Y4 ) ) ) ) ) )
     => ( ( ( size_size_list_nat @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_size_list_nat @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R2 @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_269_parallel__list__update,axiom,
    ! [N: nat,R2: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,P2: list_P2433084232546440685term_a > $o,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ! [Xs3: list_P2433084232546440685term_a,I: nat,Y4: produc1525979690750518759term_a] :
          ( ( ( size_s281034491502894169term_a @ Xs3 )
            = N )
         => ( ( ord_less_nat @ I @ N )
           => ( ( R2 @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I ) @ Y4 )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( list_u8102571067576484949term_a @ Xs3 @ I @ Y4 ) ) ) ) ) )
     => ( ( ( size_s281034491502894169term_a @ Xs )
          = N )
       => ( ( P2 @ Xs )
         => ( ( ( size_s281034491502894169term_a @ Ys )
              = N )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ I @ N )
                 => ( R2 @ ( nth_Pr5943965302187034734term_a @ Xs @ I ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I ) ) )
             => ( P2 @ Ys ) ) ) ) ) ) ).

% parallel_list_update
thf(fact_270_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 )
          = Xs )
        = ( ( nth_Ground_gmctxt_a @ Xs @ I4 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_271_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 )
          = Xs )
        = ( ( nth_Ground_gterm_a @ Xs @ I4 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_272_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,X2: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ( ( list_u412135983231242147term_a @ Xs @ I4 @ X2 )
          = Xs )
        = ( ( nth_li6172171122907299772term_a @ Xs @ I4 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_273_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_nat,X2: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I4 @ X2 )
          = Xs )
        = ( ( nth_nat @ Xs @ I4 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_274_list__update__same__conv,axiom,
    ! [I4: nat,Xs: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ( ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 )
          = Xs )
        = ( ( nth_Pr5943965302187034734term_a @ Xs @ I4 )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_275_nth__list__update,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,J3: nat,X2: ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ( I4 = J3 )
         => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) @ J3 )
            = X2 ) )
        & ( ( I4 != J3 )
         => ( ( nth_Ground_gmctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) @ J3 )
            = ( nth_Ground_gmctxt_a @ Xs @ J3 ) ) ) ) ) ).

% nth_list_update
thf(fact_276_nth__list__update,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,J3: nat,X2: ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( ( I4 = J3 )
         => ( ( nth_Ground_gterm_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) @ J3 )
            = X2 ) )
        & ( ( I4 != J3 )
         => ( ( nth_Ground_gterm_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) @ J3 )
            = ( nth_Ground_gterm_a @ Xs @ J3 ) ) ) ) ) ).

% nth_list_update
thf(fact_277_nth__list__update,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,J3: nat,X2: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ( ( I4 = J3 )
         => ( ( nth_li6172171122907299772term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) @ J3 )
            = X2 ) )
        & ( ( I4 != J3 )
         => ( ( nth_li6172171122907299772term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) @ J3 )
            = ( nth_li6172171122907299772term_a @ Xs @ J3 ) ) ) ) ) ).

% nth_list_update
thf(fact_278_nth__list__update,axiom,
    ! [I4: nat,Xs: list_nat,J3: nat,X2: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I4 = J3 )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J3 )
            = X2 ) )
        & ( ( I4 != J3 )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ J3 )
            = ( nth_nat @ Xs @ J3 ) ) ) ) ) ).

% nth_list_update
thf(fact_279_nth__list__update,axiom,
    ! [I4: nat,Xs: list_P2433084232546440685term_a,J3: nat,X2: produc1525979690750518759term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ( ( I4 = J3 )
         => ( ( nth_Pr5943965302187034734term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) @ J3 )
            = X2 ) )
        & ( ( I4 != J3 )
         => ( ( nth_Pr5943965302187034734term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) @ J3 )
            = ( nth_Pr5943965302187034734term_a @ Xs @ J3 ) ) ) ) ) ).

% nth_list_update
thf(fact_280_gmctxtex__onp__neq__lengthE,axiom,
    ! [F: a,Ss: list_Ground_gterm_a,G: a,Ts: list_Ground_gterm_a,P: ground_gmctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ ( contex3086340098647126761_onp_a @ P @ R ) )
     => ( ( ( size_s1893836384466666913term_a @ Ss )
         != ( size_s1893836384466666913term_a @ Ts ) )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ R ) ) ) ).

% gmctxtex_onp_neq_lengthE
thf(fact_281_Ground__Closure_Otrancl__list__induct,axiom,
    ! [Ss: list_Ground_gmctxt_a,Ts: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ Ss )
        = ( size_s962548663205356226ctxt_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ts ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ss @ I ) @ ( nth_Ground_gmctxt_a @ Ts @ I ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
       => ( ! [Xs3: list_Ground_gmctxt_a,Ys4: list_Ground_gmctxt_a] :
              ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
                = ( size_s962548663205356226ctxt_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                   => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_Ground_gmctxt_a,Ys4: list_Ground_gmctxt_a,I: nat,Z2: ground_gmctxt_a] :
                ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
                  = ( size_s962548663205356226ctxt_a @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                     => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ Ia ) @ ( nth_Ground_gmctxt_a @ Ys4 @ Ia ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                     => ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys4 @ I ) @ Z2 ) @ R )
                       => ( P @ Xs3 @ ( list_u7950812632479372336ctxt_a @ Ys4 @ I @ Z2 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% Ground_Closure.trancl_list_induct
thf(fact_282_Ground__Closure_Otrancl__list__induct,axiom,
    ! [Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,P: list_Ground_gterm_a > list_Ground_gterm_a > $o] :
      ( ( ( size_s1893836384466666913term_a @ Ss )
        = ( size_s1893836384466666913term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) @ ( transi8854406246868658384term_a @ R ) ) )
       => ( ! [Xs3: list_Ground_gterm_a,Ys4: list_Ground_gterm_a] :
              ( ( ( size_s1893836384466666913term_a @ Xs3 )
                = ( size_s1893836384466666913term_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ys4 ) )
                   => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) @ ( nth_Ground_gterm_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_Ground_gterm_a,Ys4: list_Ground_gterm_a,I: nat,Z2: ground_gterm_a] :
                ( ( ( size_s1893836384466666913term_a @ Xs3 )
                  = ( size_s1893836384466666913term_a @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s1893836384466666913term_a @ Ys4 ) )
                     => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ Ia ) @ ( nth_Ground_gterm_a @ Ys4 @ Ia ) ) @ ( transi8854406246868658384term_a @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys4 ) )
                     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys4 @ I ) @ Z2 ) @ R )
                       => ( P @ Xs3 @ ( list_u6582344683876525341term_a @ Ys4 @ I @ Z2 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% Ground_Closure.trancl_list_induct
thf(fact_283_Ground__Closure_Otrancl__list__induct,axiom,
    ! [Ss: list_l4678906206682822203term_a,Ts: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a,P: list_l4678906206682822203term_a > list_l4678906206682822203term_a > $o] :
      ( ( ( size_s5321119603959729319term_a @ Ss )
        = ( size_s5321119603959729319term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ts ) )
           => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ss @ I ) @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) @ ( transi554849227584806358term_a @ R ) ) )
       => ( ! [Xs3: list_l4678906206682822203term_a,Ys4: list_l4678906206682822203term_a] :
              ( ( ( size_s5321119603959729319term_a @ Xs3 )
                = ( size_s5321119603959729319term_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Ys4 ) )
                   => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) @ ( nth_li6172171122907299772term_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_l4678906206682822203term_a,Ys4: list_l4678906206682822203term_a,I: nat,Z2: list_Ground_gterm_a] :
                ( ( ( size_s5321119603959729319term_a @ Xs3 )
                  = ( size_s5321119603959729319term_a @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s5321119603959729319term_a @ Ys4 ) )
                     => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ Ia ) @ ( nth_li6172171122907299772term_a @ Ys4 @ Ia ) ) @ ( transi554849227584806358term_a @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys4 ) )
                     => ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys4 @ I ) @ Z2 ) @ R )
                       => ( P @ Xs3 @ ( list_u412135983231242147term_a @ Ys4 @ I @ Z2 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% Ground_Closure.trancl_list_induct
thf(fact_284_Ground__Closure_Otrancl__list__induct,axiom,
    ! [Ss: list_nat,Ts: list_nat,R: set_Pr1261947904930325089at_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Ss )
        = ( size_size_list_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ss @ I ) @ ( nth_nat @ Ts @ I ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
       => ( ! [Xs3: list_nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_nat,Ys4: list_nat,I: nat,Z2: nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_size_list_nat @ Ys4 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ Ia ) @ ( nth_nat @ Ys4 @ Ia ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I ) @ Z2 ) @ R )
                       => ( P @ Xs3 @ ( list_update_nat @ Ys4 @ I @ Z2 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% Ground_Closure.trancl_list_induct
thf(fact_285_Ground__Closure_Otrancl__list__induct,axiom,
    ! [Ss: list_P2433084232546440685term_a,Ts: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a,P: list_P2433084232546440685term_a > list_P2433084232546440685term_a > $o] :
      ( ( ( size_s281034491502894169term_a @ Ss )
        = ( size_s281034491502894169term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ts ) )
           => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ss @ I ) @ ( nth_Pr5943965302187034734term_a @ Ts @ I ) ) @ ( transi4994575220883992200term_a @ R ) ) )
       => ( ! [Xs3: list_P2433084232546440685term_a,Ys4: list_P2433084232546440685term_a] :
              ( ( ( size_s281034491502894169term_a @ Xs3 )
                = ( size_s281034491502894169term_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Ys4 ) )
                   => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_P2433084232546440685term_a,Ys4: list_P2433084232546440685term_a,I: nat,Z2: produc1525979690750518759term_a] :
                ( ( ( size_s281034491502894169term_a @ Xs3 )
                  = ( size_s281034491502894169term_a @ Ys4 ) )
               => ( ! [Ia: nat] :
                      ( ( ord_less_nat @ Ia @ ( size_s281034491502894169term_a @ Ys4 ) )
                     => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs3 @ Ia ) @ ( nth_Pr5943965302187034734term_a @ Ys4 @ Ia ) ) @ ( transi4994575220883992200term_a @ R ) ) )
                 => ( ( P @ Xs3 @ Ys4 )
                   => ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys4 ) )
                     => ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I ) @ Z2 ) @ R )
                       => ( P @ Xs3 @ ( list_u8102571067576484949term_a @ Ys4 @ I @ Z2 ) ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% Ground_Closure.trancl_list_induct
thf(fact_286_trancl__list__all__step__induct,axiom,
    ! [Ss: list_Ground_gmctxt_a,Ts: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,P: list_Ground_gmctxt_a > list_Ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ Ss )
        = ( size_s962548663205356226ctxt_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ts ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ss @ I ) @ ( nth_Ground_gmctxt_a @ Ts @ I ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
       => ( ! [Xs3: list_Ground_gmctxt_a,Ys4: list_Ground_gmctxt_a] :
              ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
                = ( size_s962548663205356226ctxt_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                   => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_Ground_gmctxt_a,Ys4: list_Ground_gmctxt_a,Zs: list_Ground_gmctxt_a] :
                ( ( ( size_s962548663205356226ctxt_a @ Xs3 )
                  = ( size_s962548663205356226ctxt_a @ Ys4 ) )
               => ( ( ( size_s962548663205356226ctxt_a @ Ys4 )
                    = ( size_s962548663205356226ctxt_a @ Zs ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Zs ) )
                       => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs3 @ I2 ) @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) )
                   => ( ! [I2: nat] :
                          ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ Zs ) )
                         => ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys4 @ I2 ) @ ( nth_Ground_gmctxt_a @ Zs @ I2 ) ) @ R )
                            | ( ( nth_Ground_gmctxt_a @ Ys4 @ I2 )
                              = ( nth_Ground_gmctxt_a @ Zs @ I2 ) ) ) )
                     => ( ( P @ Xs3 @ Ys4 )
                       => ( P @ Xs3 @ Zs ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_all_step_induct
thf(fact_287_trancl__list__all__step__induct,axiom,
    ! [Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,P: list_Ground_gterm_a > list_Ground_gterm_a > $o] :
      ( ( ( size_s1893836384466666913term_a @ Ss )
        = ( size_s1893836384466666913term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) @ ( transi8854406246868658384term_a @ R ) ) )
       => ( ! [Xs3: list_Ground_gterm_a,Ys4: list_Ground_gterm_a] :
              ( ( ( size_s1893836384466666913term_a @ Xs3 )
                = ( size_s1893836384466666913term_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ys4 ) )
                   => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) @ ( nth_Ground_gterm_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_Ground_gterm_a,Ys4: list_Ground_gterm_a,Zs: list_Ground_gterm_a] :
                ( ( ( size_s1893836384466666913term_a @ Xs3 )
                  = ( size_s1893836384466666913term_a @ Ys4 ) )
               => ( ( ( size_s1893836384466666913term_a @ Ys4 )
                    = ( size_s1893836384466666913term_a @ Zs ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Zs ) )
                       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs3 @ I2 ) @ ( nth_Ground_gterm_a @ Ys4 @ I2 ) ) @ ( transi8854406246868658384term_a @ R ) ) )
                   => ( ! [I2: nat] :
                          ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Zs ) )
                         => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys4 @ I2 ) @ ( nth_Ground_gterm_a @ Zs @ I2 ) ) @ R )
                            | ( ( nth_Ground_gterm_a @ Ys4 @ I2 )
                              = ( nth_Ground_gterm_a @ Zs @ I2 ) ) ) )
                     => ( ( P @ Xs3 @ Ys4 )
                       => ( P @ Xs3 @ Zs ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_all_step_induct
thf(fact_288_trancl__list__all__step__induct,axiom,
    ! [Ss: list_l4678906206682822203term_a,Ts: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a,P: list_l4678906206682822203term_a > list_l4678906206682822203term_a > $o] :
      ( ( ( size_s5321119603959729319term_a @ Ss )
        = ( size_s5321119603959729319term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ts ) )
           => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ss @ I ) @ ( nth_li6172171122907299772term_a @ Ts @ I ) ) @ ( transi554849227584806358term_a @ R ) ) )
       => ( ! [Xs3: list_l4678906206682822203term_a,Ys4: list_l4678906206682822203term_a] :
              ( ( ( size_s5321119603959729319term_a @ Xs3 )
                = ( size_s5321119603959729319term_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Ys4 ) )
                   => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) @ ( nth_li6172171122907299772term_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_l4678906206682822203term_a,Ys4: list_l4678906206682822203term_a,Zs: list_l4678906206682822203term_a] :
                ( ( ( size_s5321119603959729319term_a @ Xs3 )
                  = ( size_s5321119603959729319term_a @ Ys4 ) )
               => ( ( ( size_s5321119603959729319term_a @ Ys4 )
                    = ( size_s5321119603959729319term_a @ Zs ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Zs ) )
                       => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs3 @ I2 ) @ ( nth_li6172171122907299772term_a @ Ys4 @ I2 ) ) @ ( transi554849227584806358term_a @ R ) ) )
                   => ( ! [I2: nat] :
                          ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ Zs ) )
                         => ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys4 @ I2 ) @ ( nth_li6172171122907299772term_a @ Zs @ I2 ) ) @ R )
                            | ( ( nth_li6172171122907299772term_a @ Ys4 @ I2 )
                              = ( nth_li6172171122907299772term_a @ Zs @ I2 ) ) ) )
                     => ( ( P @ Xs3 @ Ys4 )
                       => ( P @ Xs3 @ Zs ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_all_step_induct
thf(fact_289_trancl__list__all__step__induct,axiom,
    ! [Ss: list_nat,Ts: list_nat,R: set_Pr1261947904930325089at_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Ss )
        = ( size_size_list_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ss @ I ) @ ( nth_nat @ Ts @ I ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
       => ( ! [Xs3: list_nat,Ys4: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_nat,Ys4: list_nat,Zs: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys4 ) )
               => ( ( ( size_size_list_nat @ Ys4 )
                    = ( size_size_list_nat @ Zs ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Zs ) )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Ys4 @ I2 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) )
                   => ( ! [I2: nat] :
                          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Zs ) )
                         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I2 ) @ ( nth_nat @ Zs @ I2 ) ) @ R )
                            | ( ( nth_nat @ Ys4 @ I2 )
                              = ( nth_nat @ Zs @ I2 ) ) ) )
                     => ( ( P @ Xs3 @ Ys4 )
                       => ( P @ Xs3 @ Zs ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_all_step_induct
thf(fact_290_trancl__list__all__step__induct,axiom,
    ! [Ss: list_P2433084232546440685term_a,Ts: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a,P: list_P2433084232546440685term_a > list_P2433084232546440685term_a > $o] :
      ( ( ( size_s281034491502894169term_a @ Ss )
        = ( size_s281034491502894169term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ts ) )
           => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ss @ I ) @ ( nth_Pr5943965302187034734term_a @ Ts @ I ) ) @ ( transi4994575220883992200term_a @ R ) ) )
       => ( ! [Xs3: list_P2433084232546440685term_a,Ys4: list_P2433084232546440685term_a] :
              ( ( ( size_s281034491502894169term_a @ Xs3 )
                = ( size_s281034491502894169term_a @ Ys4 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Ys4 ) )
                   => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I2 ) ) @ R ) )
               => ( P @ Xs3 @ Ys4 ) ) )
         => ( ! [Xs3: list_P2433084232546440685term_a,Ys4: list_P2433084232546440685term_a,Zs: list_P2433084232546440685term_a] :
                ( ( ( size_s281034491502894169term_a @ Xs3 )
                  = ( size_s281034491502894169term_a @ Ys4 ) )
               => ( ( ( size_s281034491502894169term_a @ Ys4 )
                    = ( size_s281034491502894169term_a @ Zs ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Zs ) )
                       => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs3 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I2 ) ) @ ( transi4994575220883992200term_a @ R ) ) )
                   => ( ! [I2: nat] :
                          ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ Zs ) )
                         => ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ Zs @ I2 ) ) @ R )
                            | ( ( nth_Pr5943965302187034734term_a @ Ys4 @ I2 )
                              = ( nth_Pr5943965302187034734term_a @ Zs @ I2 ) ) ) )
                     => ( ( P @ Xs3 @ Ys4 )
                       => ( P @ Xs3 @ Zs ) ) ) ) ) )
           => ( P @ Ss @ Ts ) ) ) ) ) ).

% trancl_list_all_step_induct
thf(fact_291_gterm__idx__induct,axiom,
    ! [P: ground_gterm_a > $o,T: ground_gterm_a] :
      ( ! [F2: a,Ts2: list_Ground_gterm_a] :
          ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
             => ( P @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) )
         => ( P @ ( ground_GFun_a @ F2 @ Ts2 ) ) )
     => ( P @ T ) ) ).

% gterm_idx_induct
thf(fact_292_gctxtex__onp__neq__lengthE,axiom,
    ! [F: a,Ss: list_Ground_gterm_a,G: a,Ts: list_Ground_gterm_a,P: ground_gctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ ( contex6947914739494133532_onp_a @ P @ R ) )
     => ( ( ( size_s1893836384466666913term_a @ Ss )
         != ( size_s1893836384466666913term_a @ Ts ) )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ R ) ) ) ).

% gctxtex_onp_neq_lengthE
thf(fact_293_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_Ground_gmctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) @ I4 )
          = ( nth_Ground_gmctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_294_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_Ground_gterm_a @ ( concat1399541994378716528term_a @ Xs ) @ I4 )
          = ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_295_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_li6172171122907299772term_a @ ( concat5214662034885773942term_a @ Xs ) @ I4 )
          = ( nth_li6172171122907299772term_a @ ( nth_li9096660926740921666term_a @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_296_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I4: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_nat @ ( concat_nat @ Xs ) @ I4 )
          = ( nth_nat @ ( nth_list_nat @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_297_concat__index__split__less__length__concat_I4_J,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ( ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ( nth_Pr5943965302187034734term_a @ ( concat2365498962271017768term_a @ Xs ) @ I4 )
          = ( nth_Pr5943965302187034734term_a @ ( nth_li7906007133205842164term_a @ Xs @ M ) @ N ) ) ) ) ).

% concat_index_split_less_length_concat(4)
thf(fact_298_all__ctxt__closed__def,axiom,
    ( ground2327783556906289298at_nat
    = ( ^ [F3: set_Pr9093778441882193744at_nat,R3: set_Pr8144746507299050889at_nat] :
        ! [F4: nat > nat,Ts3: list_G4348582881148899572at_nat,Ss2: list_G4348582881148899572at_nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F4 @ ( size_s4398795127965361032at_nat @ Ss2 ) ) @ F3 )
         => ( ( ( size_s4398795127965361032at_nat @ Ss2 )
              = ( size_s4398795127965361032at_nat @ Ts3 ) )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s4398795127965361032at_nat @ Ts3 ) )
                 => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( nth_Gr572344633313914269at_nat @ Ss2 @ I3 ) @ ( nth_Gr572344633313914269at_nat @ Ts3 @ I3 ) ) @ R3 ) )
             => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( ground_GFun_nat_nat @ F4 @ Ss2 ) @ ( ground_GFun_nat_nat @ F4 @ Ts3 ) ) @ R3 ) ) ) ) ) ) ).

% all_ctxt_closed_def
thf(fact_299_all__ctxt__closed__def,axiom,
    ( ground3158306204687651307osed_a
    = ( ^ [F3: set_Pr4934435412358123699_a_nat,R3: set_Pr8282532413760598727term_a] :
        ! [F4: a,Ts3: list_Ground_gterm_a,Ss2: list_Ground_gterm_a] :
          ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F4 @ ( size_s1893836384466666913term_a @ Ss2 ) ) @ F3 )
         => ( ( ( size_s1893836384466666913term_a @ Ss2 )
              = ( size_s1893836384466666913term_a @ Ts3 ) )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ts3 ) )
                 => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss2 @ I3 ) @ ( nth_Ground_gterm_a @ Ts3 @ I3 ) ) @ R3 ) )
             => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F4 @ Ss2 ) @ ( ground_GFun_a @ F4 @ Ts3 ) ) @ R3 ) ) ) ) ) ) ).

% all_ctxt_closed_def
thf(fact_300_all__ctxt__closedI,axiom,
    ! [F5: set_Pr9093778441882193744at_nat,R2: set_Pr8144746507299050889at_nat] :
      ( ! [F2: nat > nat,Ss3: list_G4348582881148899572at_nat,Ts2: list_G4348582881148899572at_nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F2 @ ( size_s4398795127965361032at_nat @ Ss3 ) ) @ F5 )
         => ( ( ( size_s4398795127965361032at_nat @ Ss3 )
              = ( size_s4398795127965361032at_nat @ Ts2 ) )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s4398795127965361032at_nat @ Ts2 ) )
                 => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( nth_Gr572344633313914269at_nat @ Ss3 @ I2 ) @ ( nth_Gr572344633313914269at_nat @ Ts2 @ I2 ) ) @ R2 ) )
             => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( ground_GFun_nat_nat @ F2 @ Ss3 ) @ ( ground_GFun_nat_nat @ F2 @ Ts2 ) ) @ R2 ) ) ) )
     => ( ground2327783556906289298at_nat @ F5 @ R2 ) ) ).

% all_ctxt_closedI
thf(fact_301_all__ctxt__closedI,axiom,
    ! [F5: set_Pr4934435412358123699_a_nat,R2: set_Pr8282532413760598727term_a] :
      ( ! [F2: a,Ss3: list_Ground_gterm_a,Ts2: list_Ground_gterm_a] :
          ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s1893836384466666913term_a @ Ss3 ) ) @ F5 )
         => ( ( ( size_s1893836384466666913term_a @ Ss3 )
              = ( size_s1893836384466666913term_a @ Ts2 ) )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
                 => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss3 @ I2 ) @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) @ R2 ) )
             => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F2 @ Ss3 ) @ ( ground_GFun_a @ F2 @ Ts2 ) ) @ R2 ) ) ) )
     => ( ground3158306204687651307osed_a @ F5 @ R2 ) ) ).

% all_ctxt_closedI
thf(fact_302_all__ctxt__closedD,axiom,
    ! [F6: set_Pr9093778441882193744at_nat,R2: set_Pr8144746507299050889at_nat,F: nat > nat,Ss: list_G4348582881148899572at_nat,Ts: list_G4348582881148899572at_nat] :
      ( ( ground2327783556906289298at_nat @ F6 @ R2 )
     => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F @ ( size_s4398795127965361032at_nat @ Ss ) ) @ F6 )
       => ( ( ( size_s4398795127965361032at_nat @ Ss )
            = ( size_s4398795127965361032at_nat @ Ts ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s4398795127965361032at_nat @ Ts ) )
               => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( nth_Gr572344633313914269at_nat @ Ss @ I ) @ ( nth_Gr572344633313914269at_nat @ Ts @ I ) ) @ R2 ) )
           => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( ground_GFun_nat_nat @ F @ Ss ) @ ( ground_GFun_nat_nat @ F @ Ts ) ) @ R2 ) ) ) ) ) ).

% all_ctxt_closedD
thf(fact_303_all__ctxt__closedD,axiom,
    ! [F6: set_Pr4934435412358123699_a_nat,R2: set_Pr8282532413760598727term_a,F: a,Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a] :
      ( ( ground3158306204687651307osed_a @ F6 @ R2 )
     => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s1893836384466666913term_a @ Ss ) ) @ F6 )
       => ( ( ( size_s1893836384466666913term_a @ Ss )
            = ( size_s1893836384466666913term_a @ Ts ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
               => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) @ R2 ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ F @ Ts ) ) @ R2 ) ) ) ) ) ).

% all_ctxt_closedD
thf(fact_304_less__eq__gmctxt_Oind,axiom,
    ! [Cs2: list_Ground_gmctxt_a,Ds3: list_Ground_gmctxt_a,F: a] :
      ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
        = ( size_s962548663205356226ctxt_a @ Ds3 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs2 ) )
           => ( ground6477899500129135038ctxt_a @ ( nth_Ground_gmctxt_a @ Cs2 @ I ) @ ( nth_Ground_gmctxt_a @ Ds3 @ I ) ) )
       => ( ground6477899500129135038ctxt_a @ ( ground_GMFun_a @ F @ Cs2 ) @ ( ground_GMFun_a @ F @ Ds3 ) ) ) ) ).

% less_eq_gmctxt.ind
thf(fact_305_length__pos__if__in__set,axiom,
    ! [X2: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_306_length__pos__if__in__set,axiom,
    ! [X2: ground_gterm_a,Xs: list_Ground_gterm_a] :
      ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s1893836384466666913term_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_307_length__pos__if__in__set,axiom,
    ! [X2: list_Ground_gterm_a,Xs: list_l4678906206682822203term_a] :
      ( ( member5011579089780872286term_a @ X2 @ ( set_li7546925467519454922term_a @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5321119603959729319term_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_308_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_309_length__pos__if__in__set,axiom,
    ! [X2: produc1525979690750518759term_a,Xs: list_P2433084232546440685term_a] :
      ( ( member4351891425000572176term_a @ X2 @ ( set_Pr5405221470436175740term_a @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s281034491502894169term_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_310_gctxtex__onp__neq__rootE,axiom,
    ! [F: a,Ss: list_Ground_gterm_a,G: a,Ts: list_Ground_gterm_a,P: ground_gctxt_a > $o,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ ( contex6947914739494133532_onp_a @ P @ R ) )
     => ( ( F != G )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ G @ Ts ) ) @ R ) ) ) ).

% gctxtex_onp_neq_rootE
thf(fact_311_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_312_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_313_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_314_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I4: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_315_concat__index__split__less__length__concat_I3_J,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ( ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ ( nth_li7906007133205842164term_a @ Xs @ M ) ) ) ) ) ).

% concat_index_split_less_length_concat(3)
thf(fact_316_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s6994882312647770194ctxt_a @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_317_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s5321119603959729319term_a @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_318_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s3474408278181496621term_a @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_319_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I4: nat,Xs: list_list_nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_320_concat__index__split__less__length__concat_I2_J,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,M: nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ( ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs )
          = ( product_Pair_nat_nat @ M @ N ) )
       => ( ord_less_nat @ M @ ( size_s7283615635009460703term_a @ Xs ) ) ) ) ).

% concat_index_split_less_length_concat(2)
thf(fact_321_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_322_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_323_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_324_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_325_Ground__Closure_Otrancl__list_Ocases,axiom,
    ! [A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A12 @ A23 ) @ ( ground4295926921573292569ctxt_a @ R ) )
     => ( ( ( ( size_s962548663205356226ctxt_a @ A12 )
            = ( size_s962548663205356226ctxt_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ A23 ) )
               => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ A12 @ I2 ) @ ( nth_Ground_gmctxt_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_Ground_gmctxt_a,I: nat,Z2: ground_gmctxt_a] :
              ( ( A23
                = ( list_u7950812632479372336ctxt_a @ Ys4 @ I @ Z2 ) )
             => ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A12 @ Ys4 ) @ ( ground4295926921573292569ctxt_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                 => ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Ground_Closure.trancl_list.cases
thf(fact_326_Ground__Closure_Otrancl__list_Ocases,axiom,
    ! [A12: list_Ground_gterm_a,A23: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A12 @ A23 ) @ ( ground4492235380174378228term_a @ R ) )
     => ( ( ( ( size_s1893836384466666913term_a @ A12 )
            = ( size_s1893836384466666913term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ A23 ) )
               => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ A12 @ I2 ) @ ( nth_Ground_gterm_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_Ground_gterm_a,I: nat,Z2: ground_gterm_a] :
              ( ( A23
                = ( list_u6582344683876525341term_a @ Ys4 @ I @ Z2 ) )
             => ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A12 @ Ys4 ) @ ( ground4492235380174378228term_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys4 ) )
                 => ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Ground_Closure.trancl_list.cases
thf(fact_327_Ground__Closure_Otrancl__list_Ocases,axiom,
    ! [A12: list_l4678906206682822203term_a,A23: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ A12 @ A23 ) @ ( ground1737800716954789882term_a @ R ) )
     => ( ( ( ( size_s5321119603959729319term_a @ A12 )
            = ( size_s5321119603959729319term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ A23 ) )
               => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ A12 @ I2 ) @ ( nth_li6172171122907299772term_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_l4678906206682822203term_a,I: nat,Z2: list_Ground_gterm_a] :
              ( ( A23
                = ( list_u412135983231242147term_a @ Ys4 @ I @ Z2 ) )
             => ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ A12 @ Ys4 ) @ ( ground1737800716954789882term_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys4 ) )
                 => ~ ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Ground_Closure.trancl_list.cases
thf(fact_328_Ground__Closure_Otrancl__list_Ocases,axiom,
    ! [A12: list_nat,A23: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( ground4984107042736062923st_nat @ R ) )
     => ( ( ( ( size_size_list_nat @ A12 )
            = ( size_size_list_nat @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ A23 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A12 @ I2 ) @ ( nth_nat @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_nat,I: nat,Z2: nat] :
              ( ( A23
                = ( list_update_nat @ Ys4 @ I @ Z2 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ Ys4 ) @ ( ground4984107042736062923st_nat @ R ) )
               => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Ground_Closure.trancl_list.cases
thf(fact_329_Ground__Closure_Otrancl__list_Ocases,axiom,
    ! [A12: list_P2433084232546440685term_a,A23: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ A12 @ A23 ) @ ( ground583886336375474348term_a @ R ) )
     => ( ( ( ( size_s281034491502894169term_a @ A12 )
            = ( size_s281034491502894169term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ A23 ) )
               => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ A12 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ A23 @ I2 ) ) @ R ) ) )
       => ~ ! [Ys4: list_P2433084232546440685term_a,I: nat,Z2: produc1525979690750518759term_a] :
              ( ( A23
                = ( list_u8102571067576484949term_a @ Ys4 @ I @ Z2 ) )
             => ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ A12 @ Ys4 ) @ ( ground583886336375474348term_a @ R ) )
               => ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys4 ) )
                 => ~ ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I ) @ Z2 ) @ R ) ) ) ) ) ) ).

% Ground_Closure.trancl_list.cases
thf(fact_330_Ground__Closure_Otrancl__list_Osimps,axiom,
    ! [A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ A12 @ A23 ) @ ( ground4295926921573292569ctxt_a @ R ) )
      = ( ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s962548663205356226ctxt_a @ Xs2 )
              = ( size_s962548663205356226ctxt_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
               => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs2 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,I3: nat,Z3: ground_gmctxt_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u7950812632479372336ctxt_a @ Ys3 @ I3 @ Z3 ) )
            & ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs2 @ Ys3 ) @ ( ground4295926921573292569ctxt_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
            & ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Ground_Closure.trancl_list.simps
thf(fact_331_Ground__Closure_Otrancl__list_Osimps,axiom,
    ! [A12: list_Ground_gterm_a,A23: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A12 @ A23 ) @ ( ground4492235380174378228term_a @ R ) )
      = ( ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s1893836384466666913term_a @ Xs2 )
              = ( size_s1893836384466666913term_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
               => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs2 @ I3 ) @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a,I3: nat,Z3: ground_gterm_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u6582344683876525341term_a @ Ys3 @ I3 @ Z3 ) )
            & ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs2 @ Ys3 ) @ ( ground4492235380174378228term_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
            & ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Ground_Closure.trancl_list.simps
thf(fact_332_Ground__Closure_Otrancl__list_Osimps,axiom,
    ! [A12: list_l4678906206682822203term_a,A23: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ A12 @ A23 ) @ ( ground1737800716954789882term_a @ R ) )
      = ( ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s5321119603959729319term_a @ Xs2 )
              = ( size_s5321119603959729319term_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
               => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs2 @ I3 ) @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a,I3: nat,Z3: list_Ground_gterm_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u412135983231242147term_a @ Ys3 @ I3 @ Z3 ) )
            & ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs2 @ Ys3 ) @ ( ground1737800716954789882term_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
            & ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Ground_Closure.trancl_list.simps
thf(fact_333_Ground__Closure_Otrancl__list_Osimps,axiom,
    ! [A12: list_nat,A23: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( ground4984107042736062923st_nat @ R ) )
      = ( ? [Xs2: list_nat,Ys3: list_nat] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_size_list_nat @ Xs2 )
              = ( size_size_list_nat @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( ground4984107042736062923st_nat @ R ) )
            & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Ground_Closure.trancl_list.simps
thf(fact_334_Ground__Closure_Otrancl__list_Osimps,axiom,
    ! [A12: list_P2433084232546440685term_a,A23: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ A12 @ A23 ) @ ( ground583886336375474348term_a @ R ) )
      = ( ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a] :
            ( ( A12 = Xs2 )
            & ( A23 = Ys3 )
            & ( ( size_s281034491502894169term_a @ Xs2 )
              = ( size_s281034491502894169term_a @ Ys3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
               => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs2 @ I3 ) @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) ) @ R ) ) )
        | ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a,I3: nat,Z3: produc1525979690750518759term_a] :
            ( ( A12 = Xs2 )
            & ( A23
              = ( list_u8102571067576484949term_a @ Ys3 @ I3 @ Z3 ) )
            & ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs2 @ Ys3 ) @ ( ground583886336375474348term_a @ R ) )
            & ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
            & ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) @ Z3 ) @ R ) ) ) ) ).

% Ground_Closure.trancl_list.simps
thf(fact_335_Ground__Closure_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a,I4: nat,Z4: ground_gmctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( ground4295926921573292569ctxt_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ I4 @ Z4 ) ) @ ( ground4295926921573292569ctxt_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list.list_trancl
thf(fact_336_Ground__Closure_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a,I4: nat,Z4: ground_gterm_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( ground4492235380174378228term_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ ( list_u6582344683876525341term_a @ Ys @ I4 @ Z4 ) ) @ ( ground4492235380174378228term_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list.list_trancl
thf(fact_337_Ground__Closure_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a,I4: nat,Z4: list_Ground_gterm_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( ground1737800716954789882term_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Ys ) )
       => ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ ( list_u412135983231242147term_a @ Ys @ I4 @ Z4 ) ) @ ( ground1737800716954789882term_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list.list_trancl
thf(fact_338_Ground__Closure_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,I4: nat,Z4: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( ground4984107042736062923st_nat @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( list_update_nat @ Ys @ I4 @ Z4 ) ) @ ( ground4984107042736062923st_nat @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list.list_trancl
thf(fact_339_Ground__Closure_Otrancl__list_Olist__trancl,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a,I4: nat,Z4: produc1525979690750518759term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( ground583886336375474348term_a @ R ) )
     => ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Ys ) )
       => ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Ys @ I4 ) @ Z4 ) @ R )
         => ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ ( list_u8102571067576484949term_a @ Ys @ I4 @ Z4 ) ) @ ( ground583886336375474348term_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list.list_trancl
thf(fact_340_Ground__Closure_Otrancl__list__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( ground4295926921573292569ctxt_a @ R ) )
      = ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys @ I3 ) ) @ ( transi3225167780237802813ctxt_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list_conv
thf(fact_341_Ground__Closure_Otrancl__list__conv,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( ground4492235380174378228term_a @ R ) )
      = ( ( ( size_s1893836384466666913term_a @ Xs )
          = ( size_s1893836384466666913term_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I3 ) @ ( nth_Ground_gterm_a @ Ys @ I3 ) ) @ ( transi8854406246868658384term_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list_conv
thf(fact_342_Ground__Closure_Otrancl__list__conv,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( ground1737800716954789882term_a @ R ) )
      = ( ( ( size_s5321119603959729319term_a @ Xs )
          = ( size_s5321119603959729319term_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys ) )
           => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs @ I3 ) @ ( nth_li6172171122907299772term_a @ Ys @ I3 ) ) @ ( transi554849227584806358term_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list_conv
thf(fact_343_Ground__Closure_Otrancl__list__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( ground4984107042736062923st_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list_conv
thf(fact_344_Ground__Closure_Otrancl__list__conv,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( ground583886336375474348term_a @ R ) )
      = ( ( ( size_s281034491502894169term_a @ Xs )
          = ( size_s281034491502894169term_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys ) )
           => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ I3 ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I3 ) ) @ ( transi4994575220883992200term_a @ R ) ) ) ) ) ).

% Ground_Closure.trancl_list_conv
thf(fact_345_groot_Osimps,axiom,
    ! [F: nat > nat,Ts: list_G4348582881148899572at_nat] :
      ( ( ground_groot_nat_nat @ ( ground_GFun_nat_nat @ F @ Ts ) )
      = ( produc72220940542539688at_nat @ F @ ( size_s4398795127965361032at_nat @ Ts ) ) ) ).

% groot.simps
thf(fact_346_groot_Osimps,axiom,
    ! [F: a,Ts: list_Ground_gterm_a] :
      ( ( ground_groot_a @ ( ground_GFun_a @ F @ Ts ) )
      = ( product_Pair_a_nat @ F @ ( size_s1893836384466666913term_a @ Ts ) ) ) ).

% groot.simps
thf(fact_347_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_348_size__neq__size__imp__neq,axiom,
    ! [X2: list_Ground_gmctxt_a,Y3: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ X2 )
       != ( size_s962548663205356226ctxt_a @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_349_size__neq__size__imp__neq,axiom,
    ! [X2: list_Ground_gterm_a,Y3: list_Ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ X2 )
       != ( size_s1893836384466666913term_a @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_350_size__neq__size__imp__neq,axiom,
    ! [X2: list_l4678906206682822203term_a,Y3: list_l4678906206682822203term_a] :
      ( ( ( size_s5321119603959729319term_a @ X2 )
       != ( size_s5321119603959729319term_a @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_351_size__neq__size__imp__neq,axiom,
    ! [X2: char,Y3: char] :
      ( ( ( size_size_char @ X2 )
       != ( size_size_char @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_352_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X2 )
       != ( size_size_list_nat @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_353_size__neq__size__imp__neq,axiom,
    ! [X2: list_P2433084232546440685term_a,Y3: list_P2433084232546440685term_a] :
      ( ( ( size_s281034491502894169term_a @ X2 )
       != ( size_s281034491502894169term_a @ Y3 ) )
     => ( X2 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_354_linorder__neqE__nat,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_355_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_356_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_357_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_358_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_359_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_360_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_361_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_362_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_363_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_364_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_365_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_366_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_367_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_368_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_369_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_370_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_371_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_372_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_373_Ground__Closure_Otrancl__list_Obase,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: set_Pr8705490787052945405ctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys @ I ) ) @ R ) )
       => ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( ground4295926921573292569ctxt_a @ R ) ) ) ) ).

% Ground_Closure.trancl_list.base
thf(fact_374_Ground__Closure_Otrancl__list_Obase,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: set_Pr8282532413760598727term_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys @ I ) ) @ R ) )
       => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( ground4492235380174378228term_a @ R ) ) ) ) ).

% Ground_Closure.trancl_list.base
thf(fact_375_Ground__Closure_Otrancl__list_Obase,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: set_Pr2711430308268100935term_a] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys ) )
           => ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) @ R ) )
       => ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( ground1737800716954789882term_a @ R ) ) ) ) ).

% Ground_Closure.trancl_list.base
thf(fact_376_Ground__Closure_Otrancl__list_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( ground4984107042736062923st_nat @ R ) ) ) ) ).

% Ground_Closure.trancl_list.base
thf(fact_377_Ground__Closure_Otrancl__list_Obase,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: set_Pr5156918327056800967term_a] :
      ( ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s281034491502894169term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys ) )
           => ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ I ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I ) ) @ R ) )
       => ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( ground583886336375474348term_a @ R ) ) ) ) ).

% Ground_Closure.trancl_list.base
thf(fact_378_groot_Oelims,axiom,
    ! [X2: ground_gterm_nat_nat,Y3: produc8199716216217303280at_nat] :
      ( ( ( ground_groot_nat_nat @ X2 )
        = Y3 )
     => ~ ! [F2: nat > nat,Ts2: list_G4348582881148899572at_nat] :
            ( ( X2
              = ( ground_GFun_nat_nat @ F2 @ Ts2 ) )
           => ( Y3
             != ( produc72220940542539688at_nat @ F2 @ ( size_s4398795127965361032at_nat @ Ts2 ) ) ) ) ) ).

% groot.elims
thf(fact_379_groot_Oelims,axiom,
    ! [X2: ground_gterm_a,Y3: product_prod_a_nat] :
      ( ( ( ground_groot_a @ X2 )
        = Y3 )
     => ~ ! [F2: a,Ts2: list_Ground_gterm_a] :
            ( ( X2
              = ( ground_GFun_a @ F2 @ Ts2 ) )
           => ( Y3
             != ( product_Pair_a_nat @ F2 @ ( size_s1893836384466666913term_a @ Ts2 ) ) ) ) ) ).

% groot.elims
thf(fact_380_gmctxt__cl_Osimps,axiom,
    ! [A12: ground_gterm_nat_nat,A23: ground_gterm_nat_nat,F5: set_Pr9093778441882193744at_nat,R: set_Pr8144746507299050889at_nat] :
      ( ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ A12 @ A23 ) @ ( ground1030676277180445380at_nat @ F5 @ R ) )
      = ( ? [S2: ground_gterm_nat_nat,T2: ground_gterm_nat_nat] :
            ( ( A12 = S2 )
            & ( A23 = T2 )
            & ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ S2 @ T2 ) @ R ) )
        | ? [Ss2: list_G4348582881148899572at_nat,Ts3: list_G4348582881148899572at_nat,F4: nat > nat] :
            ( ( A12
              = ( ground_GFun_nat_nat @ F4 @ Ss2 ) )
            & ( A23
              = ( ground_GFun_nat_nat @ F4 @ Ts3 ) )
            & ( ( size_s4398795127965361032at_nat @ Ss2 )
              = ( size_s4398795127965361032at_nat @ Ts3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s4398795127965361032at_nat @ Ts3 ) )
               => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( nth_Gr572344633313914269at_nat @ Ss2 @ I3 ) @ ( nth_Gr572344633313914269at_nat @ Ts3 @ I3 ) ) @ ( ground1030676277180445380at_nat @ F5 @ R ) ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F4 @ ( size_s4398795127965361032at_nat @ Ss2 ) ) @ F5 ) ) ) ) ).

% gmctxt_cl.simps
thf(fact_381_gmctxt__cl_Osimps,axiom,
    ! [A12: ground_gterm_a,A23: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( ground_gmctxt_cl_a @ F5 @ R ) )
      = ( ? [S2: ground_gterm_a,T2: ground_gterm_a] :
            ( ( A12 = S2 )
            & ( A23 = T2 )
            & ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S2 @ T2 ) @ R ) )
        | ? [Ss2: list_Ground_gterm_a,Ts3: list_Ground_gterm_a,F4: a] :
            ( ( A12
              = ( ground_GFun_a @ F4 @ Ss2 ) )
            & ( A23
              = ( ground_GFun_a @ F4 @ Ts3 ) )
            & ( ( size_s1893836384466666913term_a @ Ss2 )
              = ( size_s1893836384466666913term_a @ Ts3 ) )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ts3 ) )
               => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss2 @ I3 ) @ ( nth_Ground_gterm_a @ Ts3 @ I3 ) ) @ ( ground_gmctxt_cl_a @ F5 @ R ) ) )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F4 @ ( size_s1893836384466666913term_a @ Ss2 ) ) @ F5 ) ) ) ) ).

% gmctxt_cl.simps
thf(fact_382_gmctxt__cl_Ocases,axiom,
    ! [A12: ground_gterm_nat_nat,A23: ground_gterm_nat_nat,F5: set_Pr9093778441882193744at_nat,R: set_Pr8144746507299050889at_nat] :
      ( ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ A12 @ A23 ) @ ( ground1030676277180445380at_nat @ F5 @ R ) )
     => ( ~ ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ A12 @ A23 ) @ R )
       => ~ ! [Ss3: list_G4348582881148899572at_nat,Ts2: list_G4348582881148899572at_nat,F2: nat > nat] :
              ( ( A12
                = ( ground_GFun_nat_nat @ F2 @ Ss3 ) )
             => ( ( A23
                  = ( ground_GFun_nat_nat @ F2 @ Ts2 ) )
               => ( ( ( size_s4398795127965361032at_nat @ Ss3 )
                    = ( size_s4398795127965361032at_nat @ Ts2 ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s4398795127965361032at_nat @ Ts2 ) )
                       => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( nth_Gr572344633313914269at_nat @ Ss3 @ I2 ) @ ( nth_Gr572344633313914269at_nat @ Ts2 @ I2 ) ) @ ( ground1030676277180445380at_nat @ F5 @ R ) ) )
                   => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F2 @ ( size_s4398795127965361032at_nat @ Ss3 ) ) @ F5 ) ) ) ) ) ) ) ).

% gmctxt_cl.cases
thf(fact_383_gmctxt__cl_Ocases,axiom,
    ! [A12: ground_gterm_a,A23: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat,R: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( ground_gmctxt_cl_a @ F5 @ R ) )
     => ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ R )
       => ~ ! [Ss3: list_Ground_gterm_a,Ts2: list_Ground_gterm_a,F2: a] :
              ( ( A12
                = ( ground_GFun_a @ F2 @ Ss3 ) )
             => ( ( A23
                  = ( ground_GFun_a @ F2 @ Ts2 ) )
               => ( ( ( size_s1893836384466666913term_a @ Ss3 )
                    = ( size_s1893836384466666913term_a @ Ts2 ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
                       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss3 @ I2 ) @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) @ ( ground_gmctxt_cl_a @ F5 @ R ) ) )
                   => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ ( size_s1893836384466666913term_a @ Ss3 ) ) @ F5 ) ) ) ) ) ) ) ).

% gmctxt_cl.cases
thf(fact_384_gmctxt__cl_Ostep,axiom,
    ! [Ss: list_G4348582881148899572at_nat,Ts: list_G4348582881148899572at_nat,F5: set_Pr9093778441882193744at_nat,R: set_Pr8144746507299050889at_nat,F: nat > nat] :
      ( ( ( size_s4398795127965361032at_nat @ Ss )
        = ( size_s4398795127965361032at_nat @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s4398795127965361032at_nat @ Ts ) )
           => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( nth_Gr572344633313914269at_nat @ Ss @ I ) @ ( nth_Gr572344633313914269at_nat @ Ts @ I ) ) @ ( ground1030676277180445380at_nat @ F5 @ R ) ) )
       => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F @ ( size_s4398795127965361032at_nat @ Ss ) ) @ F5 )
         => ( member7111719770234083026at_nat @ ( produc156577712479908705at_nat @ ( ground_GFun_nat_nat @ F @ Ss ) @ ( ground_GFun_nat_nat @ F @ Ts ) ) @ ( ground1030676277180445380at_nat @ F5 @ R ) ) ) ) ) ).

% gmctxt_cl.step
thf(fact_385_gmctxt__cl_Ostep,axiom,
    ! [Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a,F5: set_Pr4934435412358123699_a_nat,R: set_Pr8282532413760598727term_a,F: a] :
      ( ( ( size_s1893836384466666913term_a @ Ss )
        = ( size_s1893836384466666913term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
           => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) @ ( ground_gmctxt_cl_a @ F5 @ R ) ) )
       => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ ( size_s1893836384466666913term_a @ Ss ) ) @ F5 )
         => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ F @ Ts ) ) @ ( ground_gmctxt_cl_a @ F5 @ R ) ) ) ) ) ).

% gmctxt_cl.step
thf(fact_386_comp__gmctxtp_OGMFun,axiom,
    ! [F: a,G: a,Cs: list_Ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
      ( ( F = G )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs )
          = ( size_s962548663205356226ctxt_a @ Ds ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
             => ( ground4927988881900525590txtp_a @ ( nth_Ground_gmctxt_a @ Cs @ I ) @ ( nth_Ground_gmctxt_a @ Ds @ I ) ) )
         => ( ground4927988881900525590txtp_a @ ( ground_GMFun_a @ F @ Cs ) @ ( ground_GMFun_a @ G @ Ds ) ) ) ) ) ).

% comp_gmctxtp.GMFun
thf(fact_387_concat__index__split__sound__fst__arg,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s6994882312647770194ctxt_a @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_388_concat__index__split__sound__fst__arg,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s5321119603959729319term_a @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_389_concat__index__split__sound__fst__arg,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s3474408278181496621term_a @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_390_concat__index__split__sound__fst__arg,axiom,
    ! [I4: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_391_concat__index__split__sound__fst__arg,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s7283615635009460703term_a @ Xs ) ) ) ).

% concat_index_split_sound_fst_arg
thf(fact_392_gmctxt__clp_Osimps,axiom,
    ( ground_gmctxt_clp_a
    = ( ^ [F7: a > nat > $o,R4: ground_gterm_a > ground_gterm_a > $o,A1: ground_gterm_a,A22: ground_gterm_a] :
          ( ? [S2: ground_gterm_a,T2: ground_gterm_a] :
              ( ( A1 = S2 )
              & ( A22 = T2 )
              & ( R4 @ S2 @ T2 ) )
          | ? [Ss2: list_Ground_gterm_a,Ts3: list_Ground_gterm_a,F4: a] :
              ( ( A1
                = ( ground_GFun_a @ F4 @ Ss2 ) )
              & ( A22
                = ( ground_GFun_a @ F4 @ Ts3 ) )
              & ( ( size_s1893836384466666913term_a @ Ss2 )
                = ( size_s1893836384466666913term_a @ Ts3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ts3 ) )
                 => ( ground_gmctxt_clp_a @ F7 @ R4 @ ( nth_Ground_gterm_a @ Ss2 @ I3 ) @ ( nth_Ground_gterm_a @ Ts3 @ I3 ) ) )
              & ( F7 @ F4 @ ( size_s1893836384466666913term_a @ Ss2 ) ) ) ) ) ) ).

% gmctxt_clp.simps
thf(fact_393_gmctxt__clp_Ocases,axiom,
    ! [F5: a > nat > $o,R: ground_gterm_a > ground_gterm_a > $o,A12: ground_gterm_a,A23: ground_gterm_a] :
      ( ( ground_gmctxt_clp_a @ F5 @ R @ A12 @ A23 )
     => ( ~ ( R @ A12 @ A23 )
       => ~ ! [Ss3: list_Ground_gterm_a,Ts2: list_Ground_gterm_a,F2: a] :
              ( ( A12
                = ( ground_GFun_a @ F2 @ Ss3 ) )
             => ( ( A23
                  = ( ground_GFun_a @ F2 @ Ts2 ) )
               => ( ( ( size_s1893836384466666913term_a @ Ss3 )
                    = ( size_s1893836384466666913term_a @ Ts2 ) )
                 => ( ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ts2 ) )
                       => ( ground_gmctxt_clp_a @ F5 @ R @ ( nth_Ground_gterm_a @ Ss3 @ I2 ) @ ( nth_Ground_gterm_a @ Ts2 @ I2 ) ) )
                   => ~ ( F5 @ F2 @ ( size_s1893836384466666913term_a @ Ss3 ) ) ) ) ) ) ) ) ).

% gmctxt_clp.cases
thf(fact_394_fst__eqD,axiom,
    ! [X2: ground_gmctxt_a,Y3: list_Ground_gterm_a,A: ground_gmctxt_a] :
      ( ( ( produc8428836298496521042term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_395_fst__eqD,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gterm_a,A: ground_gterm_a] :
      ( ( ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_396_fst__eqD,axiom,
    ! [X2: nat > nat,Y3: nat,A: nat > nat] :
      ( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) )
        = A )
     => ( X2 = A ) ) ).

% fst_eqD
thf(fact_397_fst__conv,axiom,
    ! [X1: ground_gmctxt_a,X22: list_Ground_gterm_a] :
      ( ( produc8428836298496521042term_a @ ( produc4720362732934906942term_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_398_fst__conv,axiom,
    ! [X1: ground_gterm_a,X22: ground_gterm_a] :
      ( ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_399_fst__conv,axiom,
    ! [X1: nat > nat,X22: nat] :
      ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_400_gmctxt__cl_Obase,axiom,
    ! [S: ground_gterm_a,T: ground_gterm_a,R: set_Pr8282532413760598727term_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ R )
     => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ T ) @ ( ground_gmctxt_cl_a @ F5 @ R ) ) ) ).

% gmctxt_cl.base
thf(fact_401_gmctxt__clp_Ostep,axiom,
    ! [Ss: list_Ground_gterm_a,Ts: list_Ground_gterm_a,F5: a > nat > $o,R: ground_gterm_a > ground_gterm_a > $o,F: a] :
      ( ( ( size_s1893836384466666913term_a @ Ss )
        = ( size_s1893836384466666913term_a @ Ts ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ts ) )
           => ( ground_gmctxt_clp_a @ F5 @ R @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( nth_Ground_gterm_a @ Ts @ I ) ) )
       => ( ( F5 @ F @ ( size_s1893836384466666913term_a @ Ss ) )
         => ( ground_gmctxt_clp_a @ F5 @ R @ ( ground_GFun_a @ F @ Ss ) @ ( ground_GFun_a @ F @ Ts ) ) ) ) ) ).

% gmctxt_clp.step
thf(fact_402_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_403_concat__index__split__sound__snd__arg,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_404_concat__index__split__sound__snd__arg,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_405_concat__index__split__sound__snd__arg,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ ( product_fst_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_406_concat__index__split__sound__snd__arg,axiom,
    ! [I4: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_407_concat__index__split__sound__snd__arg,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) @ ( size_s281034491502894169term_a @ ( nth_li7906007133205842164term_a @ Xs @ ( product_fst_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ zero_zero_nat @ I4 ) @ Xs ) ) ) ) ) ) ).

% concat_index_split_sound_snd_arg
thf(fact_408_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s6994882312647770194ctxt_a @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_409_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s5321119603959729319term_a @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_410_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s3474408278181496621term_a @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_411_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I4: nat,Xs: list_list_nat,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_412_concat__index__split__sound__fst__arg__aux,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_fst_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) @ ( plus_plus_nat @ ( size_s7283615635009460703term_a @ Xs ) @ O_idx ) ) ) ).

% concat_index_split_sound_fst_arg_aux
thf(fact_413_concat__index__split__mono__first__arg,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_414_concat__index__split__mono__first__arg,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_415_concat__index__split__mono__first__arg,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_416_concat__index__split__mono__first__arg,axiom,
    ! [I4: nat,Xs: list_list_nat,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_417_concat__index__split__mono__first__arg,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,O_idx: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ord_less_eq_nat @ O_idx @ ( product_fst_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ O_idx @ I4 ) @ Xs ) ) ) ) ).

% concat_index_split_mono_first_arg
thf(fact_418__092_060T_062_092_060_094sub_062Gp_Oind,axiom,
    ! [F5: a > nat > $o,F: a,N: nat,Ss: list_Ground_gterm_a] :
      ( ( F5 @ F @ N )
     => ( ( ( size_s1893836384466666913term_a @ Ss )
          = N )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ss ) )
             => ( ground_T_Gp_a @ F5 @ ( nth_Ground_gterm_a @ Ss @ I ) ) )
         => ( ground_T_Gp_a @ F5 @ ( ground_GFun_a @ F @ Ss ) ) ) ) ) ).

% \<T>\<^sub>Gp.ind
thf(fact_419_trancl_Ocases,axiom,
    ! [A12: ground_gterm_a,A23: ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ R2 )
       => ~ ! [B3: ground_gterm_a] :
              ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ B3 ) @ ( transi8854406246868658384term_a @ R2 ) )
             => ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B3 @ A23 ) @ R2 ) ) ) ) ).

% trancl.cases
thf(fact_420_trancl_Osimps,axiom,
    ! [A12: ground_gterm_a,A23: ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A12 @ A23 ) @ ( transi8854406246868658384term_a @ R2 ) )
      = ( ? [A5: ground_gterm_a,B4: ground_gterm_a] :
            ( ( A12 = A5 )
            & ( A23 = B4 )
            & ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A5 @ B4 ) @ R2 ) )
        | ? [A5: ground_gterm_a,B4: ground_gterm_a,C3: ground_gterm_a] :
            ( ( A12 = A5 )
            & ( A23 = C3 )
            & ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A5 @ B4 ) @ ( transi8854406246868658384term_a @ R2 ) )
            & ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B4 @ C3 ) @ R2 ) ) ) ) ).

% trancl.simps
thf(fact_421_trancl_Or__into__trancl,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R2 )
     => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R2 ) ) ) ).

% trancl.r_into_trancl
thf(fact_422_add__right__cancel,axiom,
    ! [B: nat,A: nat,C4: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C4 @ A ) )
      = ( B = C4 ) ) ).

% add_right_cancel
thf(fact_423_add__left__cancel,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C4 ) )
      = ( B = C4 ) ) ).

% add_left_cancel
thf(fact_424_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_425_add__le__cancel__right,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_426_add__le__cancel__left,axiom,
    ! [C4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_427_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_428_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_429_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_430_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_431_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_432_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y3 )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_433_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y3 ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_434_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_435_add__less__cancel__right,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_436_add__less__cancel__left,axiom,
    ! [C4: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_437_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_438_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_439_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_440_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_441_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_442_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_443_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_444_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_445_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_446_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_447_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_448_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_449_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_450_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_451_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_452_prod_Ocollapse,axiom,
    ! [Prod: produc2459281416120113548term_a] :
      ( ( produc4720362732934906942term_a @ ( produc8428836298496521042term_a @ Prod ) @ ( produc7861257790526860176term_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_453_prod_Ocollapse,axiom,
    ! [Prod: produc1525979690750518759term_a] :
      ( ( produc7419039783125785943term_a @ ( produc5097907526672912195term_a @ Prod ) @ ( produc5323275678763266181term_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_454_prod_Ocollapse,axiom,
    ! [Prod: produc8199716216217303280at_nat] :
      ( ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_455_list__update__beyond,axiom,
    ! [Xs: list_Ground_gmctxt_a,I4: nat,X2: ground_gmctxt_a] :
      ( ( ord_less_eq_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ I4 )
     => ( ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_456_list__update__beyond,axiom,
    ! [Xs: list_Ground_gterm_a,I4: nat,X2: ground_gterm_a] :
      ( ( ord_less_eq_nat @ ( size_s1893836384466666913term_a @ Xs ) @ I4 )
     => ( ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_457_list__update__beyond,axiom,
    ! [Xs: list_l4678906206682822203term_a,I4: nat,X2: list_Ground_gterm_a] :
      ( ( ord_less_eq_nat @ ( size_s5321119603959729319term_a @ Xs ) @ I4 )
     => ( ( list_u412135983231242147term_a @ Xs @ I4 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_458_list__update__beyond,axiom,
    ! [Xs: list_nat,I4: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I4 )
     => ( ( list_update_nat @ Xs @ I4 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_459_list__update__beyond,axiom,
    ! [Xs: list_P2433084232546440685term_a,I4: nat,X2: produc1525979690750518759term_a] :
      ( ( ord_less_eq_nat @ ( size_s281034491502894169term_a @ Xs ) @ I4 )
     => ( ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_460_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X: nat] :
            ( ( P @ X )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_461_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K5: nat] :
          ( N2
          = ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).

% nat_le_iff_add
thf(fact_462_trans__le__add2,axiom,
    ! [I4: nat,J3: nat,M: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ M @ J3 ) ) ) ).

% trans_le_add2
thf(fact_463_trans__le__add1,axiom,
    ! [I4: nat,J3: nat,M: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ord_less_eq_nat @ I4 @ ( plus_plus_nat @ J3 @ M ) ) ) ).

% trans_le_add1
thf(fact_464_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_465_add__le__mono1,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).

% add_le_mono1
thf(fact_466_add__le__mono,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).

% add_le_mono
thf(fact_467_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_468_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_469_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_470_le__trans,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( ord_less_eq_nat @ J3 @ K )
       => ( ord_less_eq_nat @ I4 @ K ) ) ) ).

% le_trans
thf(fact_471_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_472_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_473_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_474_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_475_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_476_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_477_add__le__imp__le__right,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_478_add__le__imp__le__left,axiom,
    ! [C4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_479_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_480_add__right__mono,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) ) ) ).

% add_right_mono
thf(fact_481_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C5: nat] :
            ( B
           != ( plus_plus_nat @ A @ C5 ) ) ) ).

% less_eqE
thf(fact_482_add__left__mono,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) ) ) ).

% add_left_mono
thf(fact_483_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C4: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C4 @ A ) )
     => ( B = C4 ) ) ).

% add_right_imp_eq
thf(fact_484_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C4 ) )
     => ( B = C4 ) ) ).

% add_left_imp_eq
thf(fact_485_add__mono,axiom,
    ! [A: nat,B: nat,C4: nat,D4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C4 @ D4 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).

% add_mono
thf(fact_486_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C4: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C4 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C4 ) ) ) ).

% add.left_commute
thf(fact_487_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_488_add_Oassoc,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C4 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C4 ) ) ) ).

% add.assoc
thf(fact_489_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_490_group__cancel_Oadd1,axiom,
    ! [A4: nat,K: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_491_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J3 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_492_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( I4 = J3 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_493_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J3 )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_494_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( I4 = J3 )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I4 @ K )
        = ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_495_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C4 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C4 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_496_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J3 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_497_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I4 @ J3 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_498_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C4: nat,D4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C4 @ D4 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).

% add_le_less_mono
thf(fact_499_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C4: nat,D4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C4 @ D4 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).

% add_less_le_mono
thf(fact_500_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_501_add__decreasing,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C4 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C4 ) @ B ) ) ) ).

% add_decreasing
thf(fact_502_add__increasing,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C4 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C4 ) ) ) ) ).

% add_increasing
thf(fact_503_add__decreasing2,axiom,
    ! [C4: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C4 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C4 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_504_add__increasing2,axiom,
    ! [C4: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C4 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C4 ) ) ) ) ).

% add_increasing2
thf(fact_505_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_506_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_507_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X2 @ Y3 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_508_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X2 @ Y3 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_509_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C4 ) ) ) ) ).

% add_strict_increasing2
thf(fact_510_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C4 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C4 ) ) ) ) ).

% add_strict_increasing
thf(fact_511_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_512_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_513_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_514_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_515_snd__conv,axiom,
    ! [X1: ground_gmctxt_a,X22: list_Ground_gterm_a] :
      ( ( produc7861257790526860176term_a @ ( produc4720362732934906942term_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_516_snd__conv,axiom,
    ! [X1: ground_gterm_a,X22: ground_gterm_a] :
      ( ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_517_snd__conv,axiom,
    ! [X1: nat > nat,X22: nat] :
      ( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_518_snd__eqD,axiom,
    ! [X2: ground_gmctxt_a,Y3: list_Ground_gterm_a,A: list_Ground_gterm_a] :
      ( ( ( produc7861257790526860176term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_519_snd__eqD,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gterm_a,A: ground_gterm_a] :
      ( ( ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_520_snd__eqD,axiom,
    ! [X2: nat > nat,Y3: nat,A: nat] :
      ( ( ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_521_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_522_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_523_add__less__imp__less__right,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_524_add__less__imp__less__left,axiom,
    ! [C4: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_525_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) ) ) ).

% add_strict_right_mono
thf(fact_526_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_527_add__strict__mono,axiom,
    ! [A: nat,B: nat,C4: nat,D4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C4 @ D4 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ D4 ) ) ) ) ).

% add_strict_mono
thf(fact_528_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I4 @ J3 )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_529_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( I4 = J3 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_530_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I4 @ J3 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_531_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_532_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_533_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_534_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_535_trans__less__add2,axiom,
    ! [I4: nat,J3: nat,M: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ord_less_nat @ I4 @ ( plus_plus_nat @ M @ J3 ) ) ) ).

% trans_less_add2
thf(fact_536_trans__less__add1,axiom,
    ! [I4: nat,J3: nat,M: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ord_less_nat @ I4 @ ( plus_plus_nat @ J3 @ M ) ) ) ).

% trans_less_add1
thf(fact_537_add__less__mono1,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ K ) ) ) ).

% add_less_mono1
thf(fact_538_not__add__less2,axiom,
    ! [J3: nat,I4: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I4 ) @ I4 ) ).

% not_add_less2
thf(fact_539_not__add__less1,axiom,
    ! [I4: nat,J3: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ I4 ) ).

% not_add_less1
thf(fact_540_add__less__mono,axiom,
    ! [I4: nat,J3: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).

% add_less_mono
thf(fact_541_add__lessD1,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K )
     => ( ord_less_nat @ I4 @ K ) ) ).

% add_lessD1
thf(fact_542_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_543_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_544_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_545_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_546_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I4: nat,J3: nat] :
      ( ! [I: nat,J: nat] :
          ( ( ord_less_nat @ I @ J )
         => ( ord_less_nat @ ( F @ I ) @ ( F @ J ) ) )
     => ( ( ord_less_eq_nat @ I4 @ J3 )
       => ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J3 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_547_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_548_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_549_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
          | ( M3 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_550_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_551_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M3 @ N2 )
          & ( M3 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_552_surjective__pairing,axiom,
    ! [T: produc2459281416120113548term_a] :
      ( T
      = ( produc4720362732934906942term_a @ ( produc8428836298496521042term_a @ T ) @ ( produc7861257790526860176term_a @ T ) ) ) ).

% surjective_pairing
thf(fact_553_surjective__pairing,axiom,
    ! [T: produc1525979690750518759term_a] :
      ( T
      = ( produc7419039783125785943term_a @ ( produc5097907526672912195term_a @ T ) @ ( produc5323275678763266181term_a @ T ) ) ) ).

% surjective_pairing
thf(fact_554_surjective__pairing,axiom,
    ! [T: produc8199716216217303280at_nat] :
      ( T
      = ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ T ) @ ( produc1852801350702243542at_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_555_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2459281416120113548term_a] :
      ( Prod
      = ( produc4720362732934906942term_a @ ( produc8428836298496521042term_a @ Prod ) @ ( produc7861257790526860176term_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_556_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1525979690750518759term_a] :
      ( Prod
      = ( produc7419039783125785943term_a @ ( produc5097907526672912195term_a @ Prod ) @ ( produc5323275678763266181term_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_557_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8199716216217303280at_nat] :
      ( Prod
      = ( produc72220940542539688at_nat @ ( produc6156676138143019412at_nat @ Prod ) @ ( produc1852801350702243542at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_558_pos__add__strict,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C4 ) ) ) ) ).

% pos_add_strict
thf(fact_559_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C5: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C5 ) )
           => ( C5 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_560_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_561_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_562_less__imp__add__positive,axiom,
    ! [I4: nat,J3: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ? [K4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K4 )
          & ( ( plus_plus_nat @ I4 @ K4 )
            = J3 ) ) ) ).

% less_imp_add_positive
thf(fact_563_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K4: nat] :
            ( ( ord_less_eq_nat @ K4 @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K4 )
               => ~ ( P @ I2 ) )
            & ( P @ K4 ) ) ) ) ).

% ex_least_nat_le
thf(fact_564_trancl__induct2,axiom,
    ! [Ax: ground_gmctxt_a,Ay: list_Ground_gterm_a,Bx: ground_gmctxt_a,By: list_Ground_gterm_a,R2: set_Pr3704946587519556825term_a,P: ground_gmctxt_a > list_Ground_gterm_a > $o] :
      ( ( member3109151654205658658term_a @ ( produc524587728620742321term_a @ ( produc4720362732934906942term_a @ Ax @ Ay ) @ ( produc4720362732934906942term_a @ Bx @ By ) ) @ ( transi5616345106974596523term_a @ R2 ) )
     => ( ! [A3: ground_gmctxt_a,B3: list_Ground_gterm_a] :
            ( ( member3109151654205658658term_a @ ( produc524587728620742321term_a @ ( produc4720362732934906942term_a @ Ax @ Ay ) @ ( produc4720362732934906942term_a @ A3 @ B3 ) ) @ R2 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: ground_gmctxt_a,B3: list_Ground_gterm_a,Aa: ground_gmctxt_a,Ba: list_Ground_gterm_a] :
              ( ( member3109151654205658658term_a @ ( produc524587728620742321term_a @ ( produc4720362732934906942term_a @ Ax @ Ay ) @ ( produc4720362732934906942term_a @ A3 @ B3 ) ) @ ( transi5616345106974596523term_a @ R2 ) )
             => ( ( member3109151654205658658term_a @ ( produc524587728620742321term_a @ ( produc4720362732934906942term_a @ A3 @ B3 ) @ ( produc4720362732934906942term_a @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_565_trancl__induct2,axiom,
    ! [Ax: ground_gterm_a,Ay: ground_gterm_a,Bx: ground_gterm_a,By: ground_gterm_a,R2: set_Pr5156918327056800967term_a,P: ground_gterm_a > ground_gterm_a > $o] :
      ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ Ax @ Ay ) @ ( produc7419039783125785943term_a @ Bx @ By ) ) @ ( transi4994575220883992200term_a @ R2 ) )
     => ( ! [A3: ground_gterm_a,B3: ground_gterm_a] :
            ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ Ax @ Ay ) @ ( produc7419039783125785943term_a @ A3 @ B3 ) ) @ R2 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: ground_gterm_a,B3: ground_gterm_a,Aa: ground_gterm_a,Ba: ground_gterm_a] :
              ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ Ax @ Ay ) @ ( produc7419039783125785943term_a @ A3 @ B3 ) ) @ ( transi4994575220883992200term_a @ R2 ) )
             => ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( produc7419039783125785943term_a @ A3 @ B3 ) @ ( produc7419039783125785943term_a @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_566_trancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R2: set_Pr7047708467701517959at_nat,P: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3146596849628481425at_nat @ R2 ) )
     => ( ! [A3: nat > nat,B3: nat] :
            ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ R2 )
           => ( P @ A3 @ B3 ) )
       => ( ! [A3: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ ( transi3146596849628481425at_nat @ R2 ) )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R2 )
               => ( ( P @ A3 @ B3 )
                 => ( P @ Aa @ Ba ) ) ) )
         => ( P @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_567_r__into__trancl_H,axiom,
    ! [P2: produc1525979690750518759term_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ P2 @ R2 )
     => ( member4351891425000572176term_a @ P2 @ ( transi8854406246868658384term_a @ R2 ) ) ) ).

% r_into_trancl'
thf(fact_568_converse__trancl__induct,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a,P: ground_gterm_a > $o] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ! [Y4: ground_gterm_a] :
            ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ B ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: ground_gterm_a,Z2: ground_gterm_a] :
              ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z2 ) @ R2 )
             => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Z2 @ B ) @ ( transi8854406246868658384term_a @ R2 ) )
               => ( ( P @ Z2 )
                 => ( P @ Y4 ) ) ) )
         => ( P @ A ) ) ) ) ).

% converse_trancl_induct
thf(fact_569_trancl__trans__induct,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gterm_a,R2: set_Pr8282532413760598727term_a,P: ground_gterm_a > ground_gterm_a > $o] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ! [X: ground_gterm_a,Y4: ground_gterm_a] :
            ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y4 ) @ R2 )
           => ( P @ X @ Y4 ) )
       => ( ! [X: ground_gterm_a,Y4: ground_gterm_a,Z2: ground_gterm_a] :
              ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y4 ) @ ( transi8854406246868658384term_a @ R2 ) )
             => ( ( P @ X @ Y4 )
               => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z2 ) @ ( transi8854406246868658384term_a @ R2 ) )
                 => ( ( P @ Y4 @ Z2 )
                   => ( P @ X @ Z2 ) ) ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% trancl_trans_induct
thf(fact_570_trancl__into__trancl2,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a,C4: ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R2 )
     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C4 ) @ ( transi8854406246868658384term_a @ R2 ) )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C4 ) @ ( transi8854406246868658384term_a @ R2 ) ) ) ) ).

% trancl_into_trancl2
thf(fact_571_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a,C4: ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C4 ) @ R2 )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C4 ) @ ( transi8854406246868658384term_a @ R2 ) ) ) ) ).

% Transitive_Closure.trancl_into_trancl
thf(fact_572_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr8282532413760598727term_a,X2: ground_gterm_a,Y3: ground_gterm_a] :
      ( ! [X: ground_gterm_a] :
          ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ X ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ R2 )
       => ( X2 != Y3 ) ) ) ).

% irrefl_trancl_rD
thf(fact_573_converse__tranclE,axiom,
    ! [X2: ground_gterm_a,Z4: ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Z4 ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Z4 ) @ R2 )
       => ~ ! [Y4: ground_gterm_a] :
              ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y4 ) @ R2 )
             => ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z4 ) @ ( transi8854406246868658384term_a @ R2 ) ) ) ) ) ).

% converse_tranclE
thf(fact_574_r__r__into__trancl,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R5: set_Pr8282532413760598727term_a,C4: ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R5 )
     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ B @ C4 ) @ R5 )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C4 ) @ ( transi8854406246868658384term_a @ R5 ) ) ) ) ).

% r_r_into_trancl
thf(fact_575_trancl__induct,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a,P: ground_gterm_a > $o] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ! [Y4: ground_gterm_a] :
            ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ Y4 ) @ R2 )
           => ( P @ Y4 ) )
       => ( ! [Y4: ground_gterm_a,Z2: ground_gterm_a] :
              ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ Y4 ) @ ( transi8854406246868658384term_a @ R2 ) )
             => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y4 @ Z2 ) @ R2 )
               => ( ( P @ Y4 )
                 => ( P @ Z2 ) ) ) )
         => ( P @ B ) ) ) ) ).

% trancl_induct
thf(fact_576_trancl__trans,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gterm_a,R2: set_Pr8282532413760598727term_a,Z4: ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ Y3 @ Z4 ) @ ( transi8854406246868658384term_a @ R2 ) )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Z4 ) @ ( transi8854406246868658384term_a @ R2 ) ) ) ) ).

% trancl_trans
thf(fact_577_tranclE,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ R2 )
       => ~ ! [C5: ground_gterm_a] :
              ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ C5 ) @ ( transi8854406246868658384term_a @ R2 ) )
             => ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ C5 @ B ) @ R2 ) ) ) ) ).

% tranclE
thf(fact_578_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_579_exI__realizer,axiom,
    ! [P: list_Ground_gterm_a > ground_gmctxt_a > $o,Y3: list_Ground_gterm_a,X2: ground_gmctxt_a] :
      ( ( P @ Y3 @ X2 )
     => ( P @ ( produc7861257790526860176term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) ) @ ( produc8428836298496521042term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_580_exI__realizer,axiom,
    ! [P: ground_gterm_a > ground_gterm_a > $o,Y3: ground_gterm_a,X2: ground_gterm_a] :
      ( ( P @ Y3 @ X2 )
     => ( P @ ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) ) @ ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_581_exI__realizer,axiom,
    ! [P: nat > ( nat > nat ) > $o,Y3: nat,X2: nat > nat] :
      ( ( P @ Y3 @ X2 )
     => ( P @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) ) @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_582_conjI__realizer,axiom,
    ! [P: ground_gmctxt_a > $o,P2: ground_gmctxt_a,Q: list_Ground_gterm_a > $o,Q2: list_Ground_gterm_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc8428836298496521042term_a @ ( produc4720362732934906942term_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7861257790526860176term_a @ ( produc4720362732934906942term_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_583_conjI__realizer,axiom,
    ! [P: ground_gterm_a > $o,P2: ground_gterm_a,Q: ground_gterm_a > $o,Q2: ground_gterm_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc5097907526672912195term_a @ ( produc7419039783125785943term_a @ P2 @ Q2 ) ) )
          & ( Q @ ( produc5323275678763266181term_a @ ( produc7419039783125785943term_a @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_584_conjI__realizer,axiom,
    ! [P: ( nat > nat ) > $o,P2: nat > nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( produc1852801350702243542at_nat @ ( produc72220940542539688at_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_585_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ground_gmctxt_a > list_Ground_gterm_a > $o,X2: ground_gmctxt_a,Y3: list_Ground_gterm_a,A: produc2459281416120113548term_a] :
      ( ( P @ X2 @ Y3 )
     => ( ( A
          = ( produc4720362732934906942term_a @ X2 @ Y3 ) )
       => ( P @ ( produc8428836298496521042term_a @ A ) @ ( produc7861257790526860176term_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_586_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ground_gterm_a > ground_gterm_a > $o,X2: ground_gterm_a,Y3: ground_gterm_a,A: produc1525979690750518759term_a] :
      ( ( P @ X2 @ Y3 )
     => ( ( A
          = ( produc7419039783125785943term_a @ X2 @ Y3 ) )
       => ( P @ ( produc5097907526672912195term_a @ A ) @ ( produc5323275678763266181term_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_587_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( nat > nat ) > nat > $o,X2: nat > nat,Y3: nat,A: produc8199716216217303280at_nat] :
      ( ( P @ X2 @ Y3 )
     => ( ( A
          = ( produc72220940542539688at_nat @ X2 @ Y3 ) )
       => ( P @ ( produc6156676138143019412at_nat @ A ) @ ( produc1852801350702243542at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_588_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ ( size_s962548663205356226ctxt_a @ ( nth_li6678718213161987943ctxt_a @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_589_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_590_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ ( size_s5321119603959729319term_a @ ( nth_li9096660926740921666term_a @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_591_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I4: nat,Xs: list_list_nat,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ ( size_size_list_nat @ ( nth_list_nat @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_592_concat__index__split__sound__snd__arg__aux,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,N: nat] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ord_less_nat @ ( product_snd_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ ( size_s281034491502894169term_a @ ( nth_li7906007133205842164term_a @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) @ N ) ) ) ) ) ).

% concat_index_split_sound_snd_arg_aux
thf(fact_593_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_594_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_595_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_596_add__diff__cancel__left,axiom,
    ! [C4: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C4 @ A ) @ ( plus_plus_nat @ C4 @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_597_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_598_add__diff__cancel__right,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C4 ) @ ( plus_plus_nat @ B @ C4 ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_599_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_600_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_601_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_602_diff__diff__cancel,axiom,
    ! [I4: nat,N: nat] :
      ( ( ord_less_eq_nat @ I4 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I4 ) )
        = I4 ) ) ).

% diff_diff_cancel
thf(fact_603_diff__diff__left,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ K )
      = ( minus_minus_nat @ I4 @ ( plus_plus_nat @ J3 @ K ) ) ) ).

% diff_diff_left
thf(fact_604_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_605_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_606_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_607_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_608_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_609_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_610_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( minus_minus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ) ).

% Nat.diff_diff_right
thf(fact_611_diff__right__commute,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C4 ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C4 ) ) ).

% diff_right_commute
thf(fact_612_diff__commute,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I4 @ K ) @ J3 ) ) ).

% diff_commute
thf(fact_613_add__implies__diff,axiom,
    ! [C4: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C4 @ B )
        = A )
     => ( C4
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_614_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C4 )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C4 ) ) ) ).

% diff_diff_eq
thf(fact_615_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_616_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_617_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_618_less__imp__diff__less,axiom,
    ! [J3: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J3 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_619_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_620_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_621_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_622_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_623_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_624_le__diff__iff_H,axiom,
    ! [A: nat,C4: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C4 )
     => ( ( ord_less_eq_nat @ B @ C4 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C4 @ A ) @ ( minus_minus_nat @ C4 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_625_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_626_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_627_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_628_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_629_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_630_subset__code_I1_J,axiom,
    ! [Xs: list_Ground_gmctxt_a,B5: set_Ground_gmctxt_a] :
      ( ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ B5 )
      = ( ! [X3: ground_gmctxt_a] :
            ( ( member3306386364193653941ctxt_a @ X3 @ ( set_Ground_gmctxt_a2 @ Xs ) )
           => ( member3306386364193653941ctxt_a @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_631_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_632_subset__code_I1_J,axiom,
    ! [Xs: list_l4678906206682822203term_a,B5: set_li8588095516285831957term_a] :
      ( ( ord_le1527559921741259445term_a @ ( set_li7546925467519454922term_a @ Xs ) @ B5 )
      = ( ! [X3: list_Ground_gterm_a] :
            ( ( member5011579089780872286term_a @ X3 @ ( set_li7546925467519454922term_a @ Xs ) )
           => ( member5011579089780872286term_a @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_633_subset__code_I1_J,axiom,
    ! [Xs: list_Ground_gterm_a,B5: set_Ground_gterm_a] :
      ( ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Xs ) @ B5 )
      = ( ! [X3: ground_gterm_a] :
            ( ( member5261315023172567896term_a @ X3 @ ( set_Ground_gterm_a2 @ Xs ) )
           => ( member5261315023172567896term_a @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_634_subset__code_I1_J,axiom,
    ! [Xs: list_P2433084232546440685term_a,B5: set_Pr8282532413760598727term_a] :
      ( ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Xs ) @ B5 )
      = ( ! [X3: produc1525979690750518759term_a] :
            ( ( member4351891425000572176term_a @ X3 @ ( set_Pr5405221470436175740term_a @ Xs ) )
           => ( member4351891425000572176term_a @ X3 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_635_trancl__mono,axiom,
    ! [P2: produc1525979690750518759term_a,R2: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
      ( ( member4351891425000572176term_a @ P2 @ ( transi8854406246868658384term_a @ R2 ) )
     => ( ( ord_le7556488698670074471term_a @ R2 @ S )
       => ( member4351891425000572176term_a @ P2 @ ( transi8854406246868658384term_a @ S ) ) ) ) ).

% trancl_mono
thf(fact_636_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_637_le__add__diff,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C4 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C4 ) @ A ) ) ) ).

% le_add_diff
thf(fact_638_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C4 @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C4 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_639_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C4 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C4 @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_640_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C4 @ B ) @ A )
        = ( plus_plus_nat @ C4 @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_641_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C4 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C4 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_642_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C4 ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C4 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_643_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C4 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C4 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_644_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_645_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C4 )
          = ( B
            = ( plus_plus_nat @ C4 @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_646_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_647_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_648_diff__less__mono,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C4 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C4 ) @ ( minus_minus_nat @ B @ C4 ) ) ) ) ).

% diff_less_mono
thf(fact_649_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_650_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_651_less__diff__conv,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ).

% less_diff_conv
thf(fact_652_le__diff__conv,axiom,
    ! [J3: nat,K: nat,I4: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
      = ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I4 @ K ) ) ) ).

% le_diff_conv
thf(fact_653_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( ord_less_eq_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_654_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I4 @ J3 ) @ K )
        = ( plus_plus_nat @ I4 @ ( minus_minus_nat @ J3 @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_655_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I4 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_656_Nat_Ole__imp__diff__is__add,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( ( minus_minus_nat @ J3 @ I4 )
          = K )
        = ( J3
          = ( plus_plus_nat @ K @ I4 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_657_set__update__subsetI,axiom,
    ! [Xs: list_Ground_gmctxt_a,A4: set_Ground_gmctxt_a,X2: ground_gmctxt_a,I4: nat] :
      ( ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ A4 )
     => ( ( member3306386364193653941ctxt_a @ X2 @ A4 )
       => ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_658_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,X2: nat,I4: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
     => ( ( member_nat @ X2 @ A4 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I4 @ X2 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_659_set__update__subsetI,axiom,
    ! [Xs: list_l4678906206682822203term_a,A4: set_li8588095516285831957term_a,X2: list_Ground_gterm_a,I4: nat] :
      ( ( ord_le1527559921741259445term_a @ ( set_li7546925467519454922term_a @ Xs ) @ A4 )
     => ( ( member5011579089780872286term_a @ X2 @ A4 )
       => ( ord_le1527559921741259445term_a @ ( set_li7546925467519454922term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_660_set__update__subsetI,axiom,
    ! [Xs: list_Ground_gterm_a,A4: set_Ground_gterm_a,X2: ground_gterm_a,I4: nat] :
      ( ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Xs ) @ A4 )
     => ( ( member5261315023172567896term_a @ X2 @ A4 )
       => ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_661_set__update__subsetI,axiom,
    ! [Xs: list_P2433084232546440685term_a,A4: set_Pr8282532413760598727term_a,X2: produc1525979690750518759term_a,I4: nat] :
      ( ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Xs ) @ A4 )
     => ( ( member4351891425000572176term_a @ X2 @ A4 )
       => ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_662_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_663_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_664_less__diff__conv2,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K ) @ I4 )
        = ( ord_less_nat @ J3 @ ( plus_plus_nat @ I4 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_665_less__eq__to__sup__mctxt__args,axiom,
    ! [C: ground_gmctxt_a,D: ground_gmctxt_a] :
      ( ( ord_le1940079914304802430ctxt_a @ C @ D )
     => ~ ! [Ds2: list_Ground_gmctxt_a] :
            ( ( ( ground_num_gholes_a @ C )
              = ( size_s962548663205356226ctxt_a @ Ds2 ) )
           => ( D
             != ( ground2530127064035342964ctxt_a @ C @ Ds2 ) ) ) ) ).

% less_eq_to_sup_mctxt_args
thf(fact_666_fill__gholes__gmctxt__less__eq,axiom,
    ! [C: ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_s962548663205356226ctxt_a @ Ds ) )
     => ( ord_le1940079914304802430ctxt_a @ C @ ( ground2530127064035342964ctxt_a @ C @ Ds ) ) ) ).

% fill_gholes_gmctxt_less_eq
thf(fact_667_subseteq__set__conv__nth,axiom,
    ! [Ss: list_Ground_gmctxt_a,T3: set_Ground_gmctxt_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ss ) )
           => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Ss @ I3 ) @ T3 ) ) )
      = ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Ss ) @ T3 ) ) ).

% subseteq_set_conv_nth
thf(fact_668_subseteq__set__conv__nth,axiom,
    ! [Ss: list_Ground_gterm_a,T3: set_Ground_gterm_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ss ) )
           => ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Ss @ I3 ) @ T3 ) ) )
      = ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Ss ) @ T3 ) ) ).

% subseteq_set_conv_nth
thf(fact_669_subseteq__set__conv__nth,axiom,
    ! [Ss: list_l4678906206682822203term_a,T3: set_li8588095516285831957term_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ss ) )
           => ( member5011579089780872286term_a @ ( nth_li6172171122907299772term_a @ Ss @ I3 ) @ T3 ) ) )
      = ( ord_le1527559921741259445term_a @ ( set_li7546925467519454922term_a @ Ss ) @ T3 ) ) ).

% subseteq_set_conv_nth
thf(fact_670_subseteq__set__conv__nth,axiom,
    ! [Ss: list_nat,T3: set_nat] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ss ) )
           => ( member_nat @ ( nth_nat @ Ss @ I3 ) @ T3 ) ) )
      = ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ T3 ) ) ).

% subseteq_set_conv_nth
thf(fact_671_subseteq__set__conv__nth,axiom,
    ! [Ss: list_P2433084232546440685term_a,T3: set_Pr8282532413760598727term_a] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ss ) )
           => ( member4351891425000572176term_a @ ( nth_Pr5943965302187034734term_a @ Ss @ I3 ) @ T3 ) ) )
      = ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Ss ) @ T3 ) ) ).

% subseteq_set_conv_nth
thf(fact_672_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,A4: set_Ground_gmctxt_a] :
      ( ( ord_le4032861497522155828ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Xs ) )
           => ( member3306386364193653941ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_673_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_Ground_gterm_a,A4: set_Ground_gterm_a] :
      ( ( ord_le2380367239515552431term_a @ ( set_Ground_gterm_a2 @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Xs ) )
           => ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_674_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_l4678906206682822203term_a,A4: set_li8588095516285831957term_a] :
      ( ( ord_le1527559921741259445term_a @ ( set_li7546925467519454922term_a @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Xs ) )
           => ( member5011579089780872286term_a @ ( nth_li6172171122907299772term_a @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_675_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( member_nat @ ( nth_nat @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_676_set__list__subset__eq__nth__conv,axiom,
    ! [Xs: list_P2433084232546440685term_a,A4: set_Pr8282532413760598727term_a] :
      ( ( ord_le7556488698670074471term_a @ ( set_Pr5405221470436175740term_a @ Xs ) @ A4 )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Xs ) )
           => ( member4351891425000572176term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ I3 ) @ A4 ) ) ) ) ).

% set_list_subset_eq_nth_conv
thf(fact_677_less__eq__gmctxt__intros_I2_J,axiom,
    ! [Cs2: list_Ground_gmctxt_a,Ds3: list_Ground_gmctxt_a,F: a] :
      ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
        = ( size_s962548663205356226ctxt_a @ Ds3 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Cs2 ) )
           => ( ord_le1940079914304802430ctxt_a @ ( nth_Ground_gmctxt_a @ Cs2 @ I ) @ ( nth_Ground_gmctxt_a @ Ds3 @ I ) ) )
       => ( ord_le1940079914304802430ctxt_a @ ( ground_GMFun_a @ F @ Cs2 ) @ ( ground_GMFun_a @ F @ Ds3 ) ) ) ) ).

% less_eq_gmctxt_intros(2)
thf(fact_678_in__set__enumerate__eq,axiom,
    ! [P2: produc932717863267311695ctxt_a,N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( member2885886537299685624ctxt_a @ P2 @ ( set_Pr3995427970001988964ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc8631767124948943475ctxt_a @ P2 ) )
        & ( ord_less_nat @ ( produc8631767124948943475ctxt_a @ P2 ) @ ( plus_plus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N ) )
        & ( ( nth_Ground_gmctxt_a @ Xs @ ( minus_minus_nat @ ( produc8631767124948943475ctxt_a @ P2 ) @ N ) )
          = ( produc3552744573514591925ctxt_a @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_679_in__set__enumerate__eq,axiom,
    ! [P2: produc7109479242990851284term_a,N: nat,Xs: list_Ground_gterm_a] :
      ( ( member6622040750283622891term_a @ P2 @ ( set_Pr3203312087732499071term_a @ ( enumer8595146184022354964term_a @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc6672059348947727898term_a @ P2 ) )
        & ( ord_less_nat @ ( produc6672059348947727898term_a @ P2 ) @ ( plus_plus_nat @ ( size_s1893836384466666913term_a @ Xs ) @ N ) )
        & ( ( nth_Ground_gterm_a @ Xs @ ( minus_minus_nat @ ( produc6672059348947727898term_a @ P2 ) @ N ) )
          = ( produc4207926146178914648term_a @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_680_in__set__enumerate__eq,axiom,
    ! [P2: produc7252121546586809946term_a,N: nat,Xs: list_l4678906206682822203term_a] :
      ( ( member825974164662248433term_a @ P2 @ ( set_Pr4194642937221624453term_a @ ( enumer2156928005790035738term_a @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc5587245019348778528term_a @ P2 ) )
        & ( ord_less_nat @ ( produc5587245019348778528term_a @ P2 ) @ ( plus_plus_nat @ ( size_s5321119603959729319term_a @ Xs ) @ N ) )
        & ( ( nth_li6172171122907299772term_a @ Xs @ ( minus_minus_nat @ ( produc5587245019348778528term_a @ P2 ) @ N ) )
          = ( produc8862243304664279646term_a @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_681_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,N: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( product_fst_nat_nat @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_nat @ P2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
        & ( ( nth_nat @ Xs @ ( minus_minus_nat @ ( product_fst_nat_nat @ P2 ) @ N ) )
          = ( product_snd_nat_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_682_in__set__enumerate__eq,axiom,
    ! [P2: produc604664025456906764term_a,N: nat,Xs: list_P2433084232546440685term_a] :
      ( ( member2072927181793131427term_a @ P2 @ ( set_Pr8429021916176931383term_a @ ( enumer7770827359896883148term_a @ N @ Xs ) ) )
      = ( ( ord_less_eq_nat @ N @ ( produc4655883665681864402term_a @ P2 ) )
        & ( ord_less_nat @ ( produc4655883665681864402term_a @ P2 ) @ ( plus_plus_nat @ ( size_s281034491502894169term_a @ Xs ) @ N ) )
        & ( ( nth_Pr5943965302187034734term_a @ Xs @ ( minus_minus_nat @ ( produc4655883665681864402term_a @ P2 ) @ N ) )
          = ( produc5049413271025703696term_a @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_683_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_684_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_Ground_gmctxt_a,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( nth_Pr3641570550162790742ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) @ M )
        = ( produc3104537575318011527ctxt_a @ ( plus_plus_nat @ N @ M ) @ ( nth_Ground_gmctxt_a @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_685_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_Ground_gterm_a,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( nth_Pr5718682320234703373term_a @ ( enumer8595146184022354964term_a @ N @ Xs ) @ M )
        = ( produc4697074808646628614term_a @ ( plus_plus_nat @ N @ M ) @ ( nth_Ground_gterm_a @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_686_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_l4678906206682822203term_a,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ( nth_Pr7045840962053786387term_a @ ( enumer2156928005790035738term_a @ N @ Xs ) @ M )
        = ( produc1055361199944718092term_a @ ( plus_plus_nat @ N @ M ) @ ( nth_li6172171122907299772term_a @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_687_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_688_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_P2433084232546440685term_a,N: nat] :
      ( ( ord_less_nat @ M @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ( nth_Pr8448722032445901509term_a @ ( enumer7770827359896883148term_a @ N @ Xs ) @ M )
        = ( produc543911762419408830term_a @ ( plus_plus_nat @ N @ M ) @ ( nth_Pr5943965302187034734term_a @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_689__092_060T_062_092_060_094sub_062Gp_Ocases,axiom,
    ! [F5: a > nat > $o,A: ground_gterm_a] :
      ( ( ground_T_Gp_a @ F5 @ A )
     => ( ! [A3: a] :
            ( ( A
              = ( ground_GFun_a @ A3 @ nil_Ground_gterm_a ) )
           => ~ ( F5 @ A3 @ zero_zero_nat ) )
       => ~ ! [F2: a,N3: nat,Ss3: list_Ground_gterm_a] :
              ( ( A
                = ( ground_GFun_a @ F2 @ Ss3 ) )
             => ( ( F5 @ F2 @ N3 )
               => ( ( ( size_s1893836384466666913term_a @ Ss3 )
                    = N3 )
                 => ~ ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ss3 ) )
                       => ( ground_T_Gp_a @ F5 @ ( nth_Ground_gterm_a @ Ss3 @ I2 ) ) ) ) ) ) ) ) ).

% \<T>\<^sub>Gp.cases
thf(fact_690__092_060T_062_092_060_094sub_062Gp_Osimps,axiom,
    ( ground_T_Gp_a
    = ( ^ [F7: a > nat > $o,A5: ground_gterm_a] :
          ( ? [B4: a] :
              ( ( A5
                = ( ground_GFun_a @ B4 @ nil_Ground_gterm_a ) )
              & ( F7 @ B4 @ zero_zero_nat ) )
          | ? [F4: a,N2: nat,Ss2: list_Ground_gterm_a] :
              ( ( A5
                = ( ground_GFun_a @ F4 @ Ss2 ) )
              & ( F7 @ F4 @ N2 )
              & ( ( size_s1893836384466666913term_a @ Ss2 )
                = N2 )
              & ! [X3: nat] :
                  ( ( ord_less_nat @ X3 @ ( size_s1893836384466666913term_a @ Ss2 ) )
                 => ( ground_T_Gp_a @ F7 @ ( nth_Ground_gterm_a @ Ss2 @ X3 ) ) ) ) ) ) ) ).

% \<T>\<^sub>Gp.simps
thf(fact_691_concat__eq__Nil__conv,axiom,
    ! [Xss: list_list_nat] :
      ( ( ( concat_nat @ Xss )
        = nil_nat )
      = ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss ) )
           => ( X3 = nil_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_692_concat__eq__Nil__conv,axiom,
    ! [Xss: list_l4678906206682822203term_a] :
      ( ( ( concat1399541994378716528term_a @ Xss )
        = nil_Ground_gterm_a )
      = ( ! [X3: list_Ground_gterm_a] :
            ( ( member5011579089780872286term_a @ X3 @ ( set_li7546925467519454922term_a @ Xss ) )
           => ( X3 = nil_Ground_gterm_a ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_693_Nil__eq__concat__conv,axiom,
    ! [Xss: list_list_nat] :
      ( ( nil_nat
        = ( concat_nat @ Xss ) )
      = ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss ) )
           => ( X3 = nil_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_694_Nil__eq__concat__conv,axiom,
    ! [Xss: list_l4678906206682822203term_a] :
      ( ( nil_Ground_gterm_a
        = ( concat1399541994378716528term_a @ Xss ) )
      = ( ! [X3: list_Ground_gterm_a] :
            ( ( member5011579089780872286term_a @ X3 @ ( set_li7546925467519454922term_a @ Xss ) )
           => ( X3 = nil_Ground_gterm_a ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_695_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X2: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X2 )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_696_length__enumerate,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a] :
      ( ( size_s4225619979278260545ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) )
      = ( size_s962548663205356226ctxt_a @ Xs ) ) ).

% length_enumerate
thf(fact_697_length__enumerate,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a] :
      ( ( size_s2378411291002718200term_a @ ( enumer8595146184022354964term_a @ N @ Xs ) )
      = ( size_s1893836384466666913term_a @ Xs ) ) ).

% length_enumerate
thf(fact_698_length__enumerate,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a] :
      ( ( size_s2253831711320483326term_a @ ( enumer2156928005790035738term_a @ N @ Xs ) )
      = ( size_s5321119603959729319term_a @ Xs ) ) ).

% length_enumerate
thf(fact_699_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_700_length__enumerate,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a] :
      ( ( size_s1935027362139368624term_a @ ( enumer7770827359896883148term_a @ N @ Xs ) )
      = ( size_s281034491502894169term_a @ Xs ) ) ).

% length_enumerate
thf(fact_701_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_702_length__0__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Ground_gmctxt_a ) ) ).

% length_0_conv
thf(fact_703_length__0__conv,axiom,
    ! [Xs: list_Ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Ground_gterm_a ) ) ).

% length_0_conv
thf(fact_704_length__0__conv,axiom,
    ! [Xs: list_l4678906206682822203term_a] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li1555898576520524091term_a ) ) ).

% length_0_conv
thf(fact_705_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_706_length__0__conv,axiom,
    ! [Xs: list_P2433084232546440685term_a] :
      ( ( ( size_s281034491502894169term_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr1701271659634034157term_a ) ) ).

% length_0_conv
thf(fact_707_length__greater__0__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s962548663205356226ctxt_a @ Xs ) )
      = ( Xs != nil_Ground_gmctxt_a ) ) ).

% length_greater_0_conv
thf(fact_708_length__greater__0__conv,axiom,
    ! [Xs: list_Ground_gterm_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s1893836384466666913term_a @ Xs ) )
      = ( Xs != nil_Ground_gterm_a ) ) ).

% length_greater_0_conv
thf(fact_709_length__greater__0__conv,axiom,
    ! [Xs: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s5321119603959729319term_a @ Xs ) )
      = ( Xs != nil_li1555898576520524091term_a ) ) ).

% length_greater_0_conv
thf(fact_710_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_711_length__greater__0__conv,axiom,
    ! [Xs: list_P2433084232546440685term_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s281034491502894169term_a @ Xs ) )
      = ( Xs != nil_Pr1701271659634034157term_a ) ) ).

% length_greater_0_conv
thf(fact_712_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_713_list__update_Osimps_I1_J,axiom,
    ! [I4: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I4 @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_714_list__update__code_I1_J,axiom,
    ! [I4: nat,Y3: nat] :
      ( ( list_update_nat @ nil_nat @ I4 @ Y3 )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_715_permut__aux_Osimps_I1_J,axiom,
    ! [Uu: nat > nat,Uv: list_nat] :
      ( ( missin1888654203714970382ux_nat @ nil_nat @ Uu @ Uv )
      = nil_nat ) ).

% permut_aux.simps(1)
thf(fact_716_list_Osize_I3_J,axiom,
    ( ( size_s962548663205356226ctxt_a @ nil_Ground_gmctxt_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_717_list_Osize_I3_J,axiom,
    ( ( size_s1893836384466666913term_a @ nil_Ground_gterm_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_718_list_Osize_I3_J,axiom,
    ( ( size_s5321119603959729319term_a @ nil_li1555898576520524091term_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_719_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_720_list_Osize_I3_J,axiom,
    ( ( size_s281034491502894169term_a @ nil_Pr1701271659634034157term_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_721_listrel__Nil2,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R2 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_722_listrel__Nil1,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R2 ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_723_listrel_ONil,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R2 ) ) ).

% listrel.Nil
thf(fact_724_fill__gholes__no__holes,axiom,
    ! [C: ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = zero_zero_nat )
     => ( ( ground_fill_gholes_a @ C @ nil_Ground_gterm_a )
        = ( ground2297065866753446215ctxt_a @ C ) ) ) ).

% fill_gholes_no_holes
thf(fact_725__092_060T_062_092_060_094sub_062G_Osimps,axiom,
    ! [A: ground_gterm_nat_nat,F5: set_Pr9093778441882193744at_nat] :
      ( ( member4528447914156422779at_nat @ A @ ( ground_T_G_nat_nat @ F5 ) )
      = ( ? [A5: nat > nat] :
            ( ( A
              = ( ground_GFun_nat_nat @ A5 @ nil_Gr7643157726654171486at_nat ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A5 @ zero_zero_nat ) @ F5 ) )
        | ? [F4: nat > nat,N2: nat,Ss2: list_G4348582881148899572at_nat] :
            ( ( A
              = ( ground_GFun_nat_nat @ F4 @ Ss2 ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F4 @ N2 ) @ F5 )
            & ( ( size_s4398795127965361032at_nat @ Ss2 )
              = N2 )
            & ! [X3: nat] :
                ( ( ord_less_nat @ X3 @ ( size_s4398795127965361032at_nat @ Ss2 ) )
               => ( member4528447914156422779at_nat @ ( nth_Gr572344633313914269at_nat @ Ss2 @ X3 ) @ ( ground_T_G_nat_nat @ F5 ) ) ) ) ) ) ).

% \<T>\<^sub>G.simps
thf(fact_726__092_060T_062_092_060_094sub_062G_Osimps,axiom,
    ! [A: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( member5261315023172567896term_a @ A @ ( ground_T_G_a @ F5 ) )
      = ( ? [A5: a] :
            ( ( A
              = ( ground_GFun_a @ A5 @ nil_Ground_gterm_a ) )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ zero_zero_nat ) @ F5 ) )
        | ? [F4: a,N2: nat,Ss2: list_Ground_gterm_a] :
            ( ( A
              = ( ground_GFun_a @ F4 @ Ss2 ) )
            & ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F4 @ N2 ) @ F5 )
            & ( ( size_s1893836384466666913term_a @ Ss2 )
              = N2 )
            & ! [X3: nat] :
                ( ( ord_less_nat @ X3 @ ( size_s1893836384466666913term_a @ Ss2 ) )
               => ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Ss2 @ X3 ) @ ( ground_T_G_a @ F5 ) ) ) ) ) ) ).

% \<T>\<^sub>G.simps
thf(fact_727__092_060T_062_092_060_094sub_062G_Ocases,axiom,
    ! [A: ground_gterm_nat_nat,F5: set_Pr9093778441882193744at_nat] :
      ( ( member4528447914156422779at_nat @ A @ ( ground_T_G_nat_nat @ F5 ) )
     => ( ! [A3: nat > nat] :
            ( ( A
              = ( ground_GFun_nat_nat @ A3 @ nil_Gr7643157726654171486at_nat ) )
           => ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A3 @ zero_zero_nat ) @ F5 ) )
       => ~ ! [F2: nat > nat,N3: nat,Ss3: list_G4348582881148899572at_nat] :
              ( ( A
                = ( ground_GFun_nat_nat @ F2 @ Ss3 ) )
             => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F2 @ N3 ) @ F5 )
               => ( ( ( size_s4398795127965361032at_nat @ Ss3 )
                    = N3 )
                 => ~ ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s4398795127965361032at_nat @ Ss3 ) )
                       => ( member4528447914156422779at_nat @ ( nth_Gr572344633313914269at_nat @ Ss3 @ I2 ) @ ( ground_T_G_nat_nat @ F5 ) ) ) ) ) ) ) ) ).

% \<T>\<^sub>G.cases
thf(fact_728__092_060T_062_092_060_094sub_062G_Ocases,axiom,
    ! [A: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( member5261315023172567896term_a @ A @ ( ground_T_G_a @ F5 ) )
     => ( ! [A3: a] :
            ( ( A
              = ( ground_GFun_a @ A3 @ nil_Ground_gterm_a ) )
           => ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A3 @ zero_zero_nat ) @ F5 ) )
       => ~ ! [F2: a,N3: nat,Ss3: list_Ground_gterm_a] :
              ( ( A
                = ( ground_GFun_a @ F2 @ Ss3 ) )
             => ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F2 @ N3 ) @ F5 )
               => ( ( ( size_s1893836384466666913term_a @ Ss3 )
                    = N3 )
                 => ~ ! [I2: nat] :
                        ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ Ss3 ) )
                       => ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Ss3 @ I2 ) @ ( ground_T_G_a @ F5 ) ) ) ) ) ) ) ) ).

% \<T>\<^sub>G.cases
thf(fact_729_fill__gholes__gmctxt__no__holes,axiom,
    ! [C: ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = zero_zero_nat )
     => ( ( ground2530127064035342964ctxt_a @ C @ nil_Ground_gmctxt_a )
        = C ) ) ).

% fill_gholes_gmctxt_no_holes
thf(fact_730_all__ctxt__closed__refl__on,axiom,
    ! [F5: set_Pr4934435412358123699_a_nat,R2: set_Pr8282532413760598727term_a,S: ground_gterm_a] :
      ( ( ground3158306204687651307osed_a @ F5 @ R2 )
     => ( ( member5261315023172567896term_a @ S @ ( ground_T_G_a @ F5 ) )
       => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ S @ S ) @ R2 ) ) ) ).

% all_ctxt_closed_refl_on
thf(fact_731__092_060T_062_092_060_094sub_062G_Oconst,axiom,
    ! [A: a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ zero_zero_nat ) @ F5 )
     => ( member5261315023172567896term_a @ ( ground_GFun_a @ A @ nil_Ground_gterm_a ) @ ( ground_T_G_a @ F5 ) ) ) ).

% \<T>\<^sub>G.const
thf(fact_732__092_060T_062_092_060_094sub_062G_Oconst,axiom,
    ! [A: nat > nat,F5: set_Pr9093778441882193744at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ zero_zero_nat ) @ F5 )
     => ( member4528447914156422779at_nat @ ( ground_GFun_nat_nat @ A @ nil_Gr7643157726654171486at_nat ) @ ( ground_T_G_nat_nat @ F5 ) ) ) ).

% \<T>\<^sub>G.const
thf(fact_733__092_060T_062_092_060_094sub_062G_Oind,axiom,
    ! [F: nat > nat,N: nat,F5: set_Pr9093778441882193744at_nat,Ss: list_G4348582881148899572at_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ F @ N ) @ F5 )
     => ( ( ( size_s4398795127965361032at_nat @ Ss )
          = N )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s4398795127965361032at_nat @ Ss ) )
             => ( member4528447914156422779at_nat @ ( nth_Gr572344633313914269at_nat @ Ss @ I ) @ ( ground_T_G_nat_nat @ F5 ) ) )
         => ( member4528447914156422779at_nat @ ( ground_GFun_nat_nat @ F @ Ss ) @ ( ground_T_G_nat_nat @ F5 ) ) ) ) ) ).

% \<T>\<^sub>G.ind
thf(fact_734__092_060T_062_092_060_094sub_062G_Oind,axiom,
    ! [F: a,N: nat,F5: set_Pr4934435412358123699_a_nat,Ss: list_Ground_gterm_a] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ N ) @ F5 )
     => ( ( ( size_s1893836384466666913term_a @ Ss )
          = N )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ss ) )
             => ( member5261315023172567896term_a @ ( nth_Ground_gterm_a @ Ss @ I ) @ ( ground_T_G_a @ F5 ) ) )
         => ( member5261315023172567896term_a @ ( ground_GFun_a @ F @ Ss ) @ ( ground_T_G_a @ F5 ) ) ) ) ) ).

% \<T>\<^sub>G.ind
thf(fact_735_gmctxt__subtgm__at__fill__args__empty__pos,axiom,
    ! [C: ground_gmctxt_a,Ts: list_Ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_s962548663205356226ctxt_a @ Ts ) )
     => ( ( ground6121243438039047541ctxt_a @ nil_nat @ C @ Ts )
        = Ts ) ) ).

% gmctxt_subtgm_at_fill_args_empty_pos
thf(fact_736_gmctxt__subtgm__at__fill__args__empty__pos,axiom,
    ! [C: ground_gmctxt_a,Ts: list_Ground_gterm_a] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_s1893836384466666913term_a @ Ts ) )
     => ( ( ground8448531477516352152term_a @ nil_nat @ C @ Ts )
        = Ts ) ) ).

% gmctxt_subtgm_at_fill_args_empty_pos
thf(fact_737_gmctxt__subtgm__at__fill__args__empty__pos,axiom,
    ! [C: ground_gmctxt_a,Ts: list_l4678906206682822203term_a] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_s5321119603959729319term_a @ Ts ) )
     => ( ( ground8506523905942776734term_a @ nil_nat @ C @ Ts )
        = Ts ) ) ).

% gmctxt_subtgm_at_fill_args_empty_pos
thf(fact_738_gmctxt__subtgm__at__fill__args__empty__pos,axiom,
    ! [C: ground_gmctxt_a,Ts: list_nat] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_size_list_nat @ Ts ) )
     => ( ( ground6209547762876788007_a_nat @ nil_nat @ C @ Ts )
        = Ts ) ) ).

% gmctxt_subtgm_at_fill_args_empty_pos
thf(fact_739_gmctxt__subtgm__at__fill__args__empty__pos,axiom,
    ! [C: ground_gmctxt_a,Ts: list_P2433084232546440685term_a] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_s281034491502894169term_a @ Ts ) )
     => ( ( ground7952841227834989648term_a @ nil_nat @ C @ Ts )
        = Ts ) ) ).

% gmctxt_subtgm_at_fill_args_empty_pos
thf(fact_740_inf__pigeonhole__principle,axiom,
    ! [N: nat,F: nat > nat > $o] :
      ( ! [K4: nat] :
        ? [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
          & ( F @ K4 @ I2 ) )
     => ? [I: nat] :
          ( ( ord_less_nat @ I @ N )
          & ! [K3: nat] :
            ? [K6: nat] :
              ( ( ord_less_eq_nat @ K3 @ K6 )
              & ( F @ K6 @ I ) ) ) ) ).

% inf_pigeonhole_principle
thf(fact_741_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K4: nat] :
          ( ( ord_less_nat @ N @ K4 )
         => ( P @ K4 ) )
     => ( ! [K4: nat] :
            ( ( ord_less_eq_nat @ K4 @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K4 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K4 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_742_subrelI,axiom,
    ! [R2: set_Pr6749809045309873730term_a,S: set_Pr6749809045309873730term_a] :
      ( ! [X: ground_gmctxt_a,Y4: list_Ground_gterm_a] :
          ( ( member8767046467196472611term_a @ ( produc4720362732934906942term_a @ X @ Y4 ) @ R2 )
         => ( member8767046467196472611term_a @ ( produc4720362732934906942term_a @ X @ Y4 ) @ S ) )
     => ( ord_le8148545718494969506term_a @ R2 @ S ) ) ).

% subrelI
thf(fact_743_subrelI,axiom,
    ! [R2: set_Pr8282532413760598727term_a,S: set_Pr8282532413760598727term_a] :
      ( ! [X: ground_gterm_a,Y4: ground_gterm_a] :
          ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y4 ) @ R2 )
         => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y4 ) @ S ) )
     => ( ord_le7556488698670074471term_a @ R2 @ S ) ) ).

% subrelI
thf(fact_744_subrelI,axiom,
    ! [R2: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ! [X: nat > nat,Y4: nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y4 ) @ R2 )
         => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y4 ) @ S ) )
     => ( ord_le3678578370064672496at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_745_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_746_sndI,axiom,
    ! [X2: produc2459281416120113548term_a,Y3: ground_gmctxt_a,Z4: list_Ground_gterm_a] :
      ( ( X2
        = ( produc4720362732934906942term_a @ Y3 @ Z4 ) )
     => ( ( produc7861257790526860176term_a @ X2 )
        = Z4 ) ) ).

% sndI
thf(fact_747_sndI,axiom,
    ! [X2: produc1525979690750518759term_a,Y3: ground_gterm_a,Z4: ground_gterm_a] :
      ( ( X2
        = ( produc7419039783125785943term_a @ Y3 @ Z4 ) )
     => ( ( produc5323275678763266181term_a @ X2 )
        = Z4 ) ) ).

% sndI
thf(fact_748_sndI,axiom,
    ! [X2: produc8199716216217303280at_nat,Y3: nat > nat,Z4: nat] :
      ( ( X2
        = ( produc72220940542539688at_nat @ Y3 @ Z4 ) )
     => ( ( produc1852801350702243542at_nat @ X2 )
        = Z4 ) ) ).

% sndI
thf(fact_749_eq__snd__iff,axiom,
    ! [B: list_Ground_gterm_a,P2: produc2459281416120113548term_a] :
      ( ( B
        = ( produc7861257790526860176term_a @ P2 ) )
      = ( ? [A5: ground_gmctxt_a] :
            ( P2
            = ( produc4720362732934906942term_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_750_eq__snd__iff,axiom,
    ! [B: ground_gterm_a,P2: produc1525979690750518759term_a] :
      ( ( B
        = ( produc5323275678763266181term_a @ P2 ) )
      = ( ? [A5: ground_gterm_a] :
            ( P2
            = ( produc7419039783125785943term_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_751_eq__snd__iff,axiom,
    ! [B: nat,P2: produc8199716216217303280at_nat] :
      ( ( B
        = ( produc1852801350702243542at_nat @ P2 ) )
      = ( ? [A5: nat > nat] :
            ( P2
            = ( produc72220940542539688at_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_752_eq__fst__iff,axiom,
    ! [A: ground_gmctxt_a,P2: produc2459281416120113548term_a] :
      ( ( A
        = ( produc8428836298496521042term_a @ P2 ) )
      = ( ? [B4: list_Ground_gterm_a] :
            ( P2
            = ( produc4720362732934906942term_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_753_eq__fst__iff,axiom,
    ! [A: ground_gterm_a,P2: produc1525979690750518759term_a] :
      ( ( A
        = ( produc5097907526672912195term_a @ P2 ) )
      = ( ? [B4: ground_gterm_a] :
            ( P2
            = ( produc7419039783125785943term_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_754_eq__fst__iff,axiom,
    ! [A: nat > nat,P2: produc8199716216217303280at_nat] :
      ( ( A
        = ( produc6156676138143019412at_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( produc72220940542539688at_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_755_fstI,axiom,
    ! [X2: produc2459281416120113548term_a,Y3: ground_gmctxt_a,Z4: list_Ground_gterm_a] :
      ( ( X2
        = ( produc4720362732934906942term_a @ Y3 @ Z4 ) )
     => ( ( produc8428836298496521042term_a @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_756_fstI,axiom,
    ! [X2: produc1525979690750518759term_a,Y3: ground_gterm_a,Z4: ground_gterm_a] :
      ( ( X2
        = ( produc7419039783125785943term_a @ Y3 @ Z4 ) )
     => ( ( produc5097907526672912195term_a @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_757_fstI,axiom,
    ! [X2: produc8199716216217303280at_nat,Y3: nat > nat,Z4: nat] :
      ( ( X2
        = ( produc72220940542539688at_nat @ Y3 @ Z4 ) )
     => ( ( produc6156676138143019412at_nat @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_758_no__gholes__gmctxt__of__gterm__gterm__of__gmctxt__id,axiom,
    ! [C: ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = zero_zero_nat )
     => ( ( ground582559344758063963term_a @ ( ground2297065866753446215ctxt_a @ C ) )
        = C ) ) ).

% no_gholes_gmctxt_of_gterm_gterm_of_gmctxt_id
thf(fact_759_Ground__Closure_Otrancl__listp_Ocases,axiom,
    ! [R: ground_gmctxt_a > ground_gmctxt_a > $o,A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a] :
      ( ( ground3546414749463070293ctxt_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s962548663205356226ctxt_a @ A12 )
            = ( size_s962548663205356226ctxt_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ A23 ) )
               => ( R @ ( nth_Ground_gmctxt_a @ A12 @ I2 ) @ ( nth_Ground_gmctxt_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_Ground_gmctxt_a,I: nat,Z2: ground_gmctxt_a] :
              ( ( A23
                = ( list_u7950812632479372336ctxt_a @ Ys4 @ I @ Z2 ) )
             => ( ( ground3546414749463070293ctxt_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Ground_gmctxt_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Ground_Closure.trancl_listp.cases
thf(fact_760_Ground__Closure_Otrancl__listp_Ocases,axiom,
    ! [R: ground_gterm_a > ground_gterm_a > $o,A12: list_Ground_gterm_a,A23: list_Ground_gterm_a] :
      ( ( ground1804610477406788024term_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s1893836384466666913term_a @ A12 )
            = ( size_s1893836384466666913term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ A23 ) )
               => ( R @ ( nth_Ground_gterm_a @ A12 @ I2 ) @ ( nth_Ground_gterm_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_Ground_gterm_a,I: nat,Z2: ground_gterm_a] :
              ( ( A23
                = ( list_u6582344683876525341term_a @ Ys4 @ I @ Z2 ) )
             => ( ( ground1804610477406788024term_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Ground_gterm_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Ground_Closure.trancl_listp.cases
thf(fact_761_Ground__Closure_Otrancl__listp_Ocases,axiom,
    ! [R: list_Ground_gterm_a > list_Ground_gterm_a > $o,A12: list_l4678906206682822203term_a,A23: list_l4678906206682822203term_a] :
      ( ( ground8433361165389822654term_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s5321119603959729319term_a @ A12 )
            = ( size_s5321119603959729319term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ A23 ) )
               => ( R @ ( nth_li6172171122907299772term_a @ A12 @ I2 ) @ ( nth_li6172171122907299772term_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_l4678906206682822203term_a,I: nat,Z2: list_Ground_gterm_a] :
              ( ( A23
                = ( list_u412135983231242147term_a @ Ys4 @ I @ Z2 ) )
             => ( ( ground8433361165389822654term_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys4 ) )
                 => ~ ( R @ ( nth_li6172171122907299772term_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Ground_Closure.trancl_listp.cases
thf(fact_762_Ground__Closure_Otrancl__listp_Ocases,axiom,
    ! [R: nat > nat > $o,A12: list_nat,A23: list_nat] :
      ( ( ground2818829557537609223tp_nat @ R @ A12 @ A23 )
     => ( ( ( ( size_size_list_nat @ A12 )
            = ( size_size_list_nat @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ A23 ) )
               => ( R @ ( nth_nat @ A12 @ I2 ) @ ( nth_nat @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_nat,I: nat,Z2: nat] :
              ( ( A23
                = ( list_update_nat @ Ys4 @ I @ Z2 ) )
             => ( ( ground2818829557537609223tp_nat @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( R @ ( nth_nat @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Ground_Closure.trancl_listp.cases
thf(fact_763_Ground__Closure_Otrancl__listp_Ocases,axiom,
    ! [R: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,A12: list_P2433084232546440685term_a,A23: list_P2433084232546440685term_a] :
      ( ( ground1143888467740470128term_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s281034491502894169term_a @ A12 )
            = ( size_s281034491502894169term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ A23 ) )
               => ( R @ ( nth_Pr5943965302187034734term_a @ A12 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_P2433084232546440685term_a,I: nat,Z2: produc1525979690750518759term_a] :
              ( ( A23
                = ( list_u8102571067576484949term_a @ Ys4 @ I @ Z2 ) )
             => ( ( ground1143888467740470128term_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Ground_Closure.trancl_listp.cases
thf(fact_764_Ground__Closure_Otrancl__listp_Osimps,axiom,
    ( ground3546414749463070293ctxt_a
    = ( ^ [R4: ground_gmctxt_a > ground_gmctxt_a > $o,A1: list_Ground_gmctxt_a,A22: list_Ground_gmctxt_a] :
          ( ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s962548663205356226ctxt_a @ Xs2 )
                = ( size_s962548663205356226ctxt_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
                 => ( R4 @ ( nth_Ground_gmctxt_a @ Xs2 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,I3: nat,Z3: ground_gmctxt_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u7950812632479372336ctxt_a @ Ys3 @ I3 @ Z3 ) )
              & ( ground3546414749463070293ctxt_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
              & ( R4 @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.simps
thf(fact_765_Ground__Closure_Otrancl__listp_Osimps,axiom,
    ( ground1804610477406788024term_a
    = ( ^ [R4: ground_gterm_a > ground_gterm_a > $o,A1: list_Ground_gterm_a,A22: list_Ground_gterm_a] :
          ( ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s1893836384466666913term_a @ Xs2 )
                = ( size_s1893836384466666913term_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
                 => ( R4 @ ( nth_Ground_gterm_a @ Xs2 @ I3 ) @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a,I3: nat,Z3: ground_gterm_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u6582344683876525341term_a @ Ys3 @ I3 @ Z3 ) )
              & ( ground1804610477406788024term_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
              & ( R4 @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.simps
thf(fact_766_Ground__Closure_Otrancl__listp_Osimps,axiom,
    ( ground8433361165389822654term_a
    = ( ^ [R4: list_Ground_gterm_a > list_Ground_gterm_a > $o,A1: list_l4678906206682822203term_a,A22: list_l4678906206682822203term_a] :
          ( ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s5321119603959729319term_a @ Xs2 )
                = ( size_s5321119603959729319term_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
                 => ( R4 @ ( nth_li6172171122907299772term_a @ Xs2 @ I3 ) @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a,I3: nat,Z3: list_Ground_gterm_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u412135983231242147term_a @ Ys3 @ I3 @ Z3 ) )
              & ( ground8433361165389822654term_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
              & ( R4 @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.simps
thf(fact_767_Ground__Closure_Otrancl__listp_Osimps,axiom,
    ( ground2818829557537609223tp_nat
    = ( ^ [R4: nat > nat > $o,A1: list_nat,A22: list_nat] :
          ( ? [Xs2: list_nat,Ys3: list_nat] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
                 => ( R4 @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
              & ( ground2818829557537609223tp_nat @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
              & ( R4 @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.simps
thf(fact_768_Ground__Closure_Otrancl__listp_Osimps,axiom,
    ( ground1143888467740470128term_a
    = ( ^ [R4: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,A1: list_P2433084232546440685term_a,A22: list_P2433084232546440685term_a] :
          ( ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s281034491502894169term_a @ Xs2 )
                = ( size_s281034491502894169term_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
                 => ( R4 @ ( nth_Pr5943965302187034734term_a @ Xs2 @ I3 ) @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a,I3: nat,Z3: produc1525979690750518759term_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u8102571067576484949term_a @ Ys3 @ I3 @ Z3 ) )
              & ( ground1143888467740470128term_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
              & ( R4 @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.simps
thf(fact_769_num__holes__mctxt__of__term,axiom,
    ! [T: ground_gterm_a] :
      ( ( ground_num_gholes_a @ ( ground582559344758063963term_a @ T ) )
      = zero_zero_nat ) ).

% num_holes_mctxt_of_term
thf(fact_770_Ground__Closure_Otrancl__listp_Obase,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: ground_gmctxt_a > ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( R @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys @ I ) ) )
       => ( ground3546414749463070293ctxt_a @ R @ Xs @ Ys ) ) ) ).

% Ground_Closure.trancl_listp.base
thf(fact_771_Ground__Closure_Otrancl__listp_Obase,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: ground_gterm_a > ground_gterm_a > $o] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys ) )
           => ( R @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys @ I ) ) )
       => ( ground1804610477406788024term_a @ R @ Xs @ Ys ) ) ) ).

% Ground_Closure.trancl_listp.base
thf(fact_772_Ground__Closure_Otrancl__listp_Obase,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: list_Ground_gterm_a > list_Ground_gterm_a > $o] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys ) )
           => ( R @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) )
       => ( ground8433361165389822654term_a @ R @ Xs @ Ys ) ) ) ).

% Ground_Closure.trancl_listp.base
thf(fact_773_Ground__Closure_Otrancl__listp_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: nat > nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( R @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) )
       => ( ground2818829557537609223tp_nat @ R @ Xs @ Ys ) ) ) ).

% Ground_Closure.trancl_listp.base
thf(fact_774_Ground__Closure_Otrancl__listp_Obase,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: produc1525979690750518759term_a > produc1525979690750518759term_a > $o] :
      ( ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s281034491502894169term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys ) )
           => ( R @ ( nth_Pr5943965302187034734term_a @ Xs @ I ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I ) ) )
       => ( ground1143888467740470128term_a @ R @ Xs @ Ys ) ) ) ).

% Ground_Closure.trancl_listp.base
thf(fact_775_eqgf__less__eq,axiom,
    ! [S: ground_gterm_a,C: ground_gmctxt_a,Ss: list_Ground_gterm_a] :
      ( ( ground_eq_gfill_a @ S @ ( produc4720362732934906942term_a @ C @ Ss ) )
     => ( ord_le1940079914304802430ctxt_a @ C @ ( ground582559344758063963term_a @ S ) ) ) ).

% eqgf_less_eq
thf(fact_776_Ground__Closure_Otrancl__listp_Olist__trancl,axiom,
    ! [R: ground_gmctxt_a > ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,I4: nat,Z4: ground_gmctxt_a] :
      ( ( ground3546414749463070293ctxt_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( R @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) @ Z4 )
         => ( ground3546414749463070293ctxt_a @ R @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.list_trancl
thf(fact_777_Ground__Closure_Otrancl__listp_Olist__trancl,axiom,
    ! [R: ground_gterm_a > ground_gterm_a > $o,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,I4: nat,Z4: ground_gterm_a] :
      ( ( ground1804610477406788024term_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( R @ ( nth_Ground_gterm_a @ Ys @ I4 ) @ Z4 )
         => ( ground1804610477406788024term_a @ R @ Xs @ ( list_u6582344683876525341term_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.list_trancl
thf(fact_778_Ground__Closure_Otrancl__listp_Olist__trancl,axiom,
    ! [R: list_Ground_gterm_a > list_Ground_gterm_a > $o,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,I4: nat,Z4: list_Ground_gterm_a] :
      ( ( ground8433361165389822654term_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Ys ) )
       => ( ( R @ ( nth_li6172171122907299772term_a @ Ys @ I4 ) @ Z4 )
         => ( ground8433361165389822654term_a @ R @ Xs @ ( list_u412135983231242147term_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.list_trancl
thf(fact_779_Ground__Closure_Otrancl__listp_Olist__trancl,axiom,
    ! [R: nat > nat > $o,Xs: list_nat,Ys: list_nat,I4: nat,Z4: nat] :
      ( ( ground2818829557537609223tp_nat @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( R @ ( nth_nat @ Ys @ I4 ) @ Z4 )
         => ( ground2818829557537609223tp_nat @ R @ Xs @ ( list_update_nat @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.list_trancl
thf(fact_780_Ground__Closure_Otrancl__listp_Olist__trancl,axiom,
    ! [R: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,I4: nat,Z4: produc1525979690750518759term_a] :
      ( ( ground1143888467740470128term_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Ys ) )
       => ( ( R @ ( nth_Pr5943965302187034734term_a @ Ys @ I4 ) @ Z4 )
         => ( ground1143888467740470128term_a @ R @ Xs @ ( list_u8102571067576484949term_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Ground_Closure.trancl_listp.list_trancl
thf(fact_781_length__unfill__gholes,axiom,
    ! [C: ground_gmctxt_a,T: ground_gterm_a] :
      ( ( ord_le1940079914304802430ctxt_a @ C @ ( ground582559344758063963term_a @ T ) )
     => ( ( size_s1893836384466666913term_a @ ( ground8214737007138946830oles_a @ C @ T ) )
        = ( ground_num_gholes_a @ C ) ) ) ).

% length_unfill_gholes
thf(fact_782_Utils_Otrancl__listp_Olist__trancl,axiom,
    ! [R: ground_gmctxt_a > ground_gmctxt_a > $o,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,I4: nat,Z4: ground_gmctxt_a] :
      ( ( trancl3815965565430463256ctxt_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( R @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) @ Z4 )
         => ( trancl3815965565430463256ctxt_a @ R @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Utils.trancl_listp.list_trancl
thf(fact_783_Utils_Otrancl__listp_Olist__trancl,axiom,
    ! [R: ground_gterm_a > ground_gterm_a > $o,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,I4: nat,Z4: ground_gterm_a] :
      ( ( trancl3536835252858651701term_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( R @ ( nth_Ground_gterm_a @ Ys @ I4 ) @ Z4 )
         => ( trancl3536835252858651701term_a @ R @ Xs @ ( list_u6582344683876525341term_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Utils.trancl_listp.list_trancl
thf(fact_784_Utils_Otrancl__listp_Olist__trancl,axiom,
    ! [R: list_Ground_gterm_a > list_Ground_gterm_a > $o,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,I4: nat,Z4: list_Ground_gterm_a] :
      ( ( trancl1148691031646339771term_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Ys ) )
       => ( ( R @ ( nth_li6172171122907299772term_a @ Ys @ I4 ) @ Z4 )
         => ( trancl1148691031646339771term_a @ R @ Xs @ ( list_u412135983231242147term_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Utils.trancl_listp.list_trancl
thf(fact_785_Utils_Otrancl__listp_Olist__trancl,axiom,
    ! [R: nat > nat > $o,Xs: list_nat,Ys: list_nat,I4: nat,Z4: nat] :
      ( ( trancl_listp_nat @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( R @ ( nth_nat @ Ys @ I4 ) @ Z4 )
         => ( trancl_listp_nat @ R @ Xs @ ( list_update_nat @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Utils.trancl_listp.list_trancl
thf(fact_786_Utils_Otrancl__listp_Olist__trancl,axiom,
    ! [R: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,I4: nat,Z4: produc1525979690750518759term_a] :
      ( ( trancl1827884997844227949term_a @ R @ Xs @ Ys )
     => ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Ys ) )
       => ( ( R @ ( nth_Pr5943965302187034734term_a @ Ys @ I4 ) @ Z4 )
         => ( trancl1827884997844227949term_a @ R @ Xs @ ( list_u8102571067576484949term_a @ Ys @ I4 @ Z4 ) ) ) ) ) ).

% Utils.trancl_listp.list_trancl
thf(fact_787_Utils_Otrancl__listp_Osimps,axiom,
    ( trancl3815965565430463256ctxt_a
    = ( ^ [R4: ground_gmctxt_a > ground_gmctxt_a > $o,A1: list_Ground_gmctxt_a,A22: list_Ground_gmctxt_a] :
          ( ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s962548663205356226ctxt_a @ Xs2 )
                = ( size_s962548663205356226ctxt_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
                 => ( R4 @ ( nth_Ground_gmctxt_a @ Xs2 @ I3 ) @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_Ground_gmctxt_a,Ys3: list_Ground_gmctxt_a,I3: nat,Z3: ground_gmctxt_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u7950812632479372336ctxt_a @ Ys3 @ I3 @ Z3 ) )
              & ( trancl3815965565430463256ctxt_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s962548663205356226ctxt_a @ Ys3 ) )
              & ( R4 @ ( nth_Ground_gmctxt_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Utils.trancl_listp.simps
thf(fact_788_Utils_Otrancl__listp_Osimps,axiom,
    ( trancl3536835252858651701term_a
    = ( ^ [R4: ground_gterm_a > ground_gterm_a > $o,A1: list_Ground_gterm_a,A22: list_Ground_gterm_a] :
          ( ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s1893836384466666913term_a @ Xs2 )
                = ( size_s1893836384466666913term_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
                 => ( R4 @ ( nth_Ground_gterm_a @ Xs2 @ I3 ) @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_Ground_gterm_a,Ys3: list_Ground_gterm_a,I3: nat,Z3: ground_gterm_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u6582344683876525341term_a @ Ys3 @ I3 @ Z3 ) )
              & ( trancl3536835252858651701term_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s1893836384466666913term_a @ Ys3 ) )
              & ( R4 @ ( nth_Ground_gterm_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Utils.trancl_listp.simps
thf(fact_789_Utils_Otrancl__listp_Osimps,axiom,
    ( trancl1148691031646339771term_a
    = ( ^ [R4: list_Ground_gterm_a > list_Ground_gterm_a > $o,A1: list_l4678906206682822203term_a,A22: list_l4678906206682822203term_a] :
          ( ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s5321119603959729319term_a @ Xs2 )
                = ( size_s5321119603959729319term_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
                 => ( R4 @ ( nth_li6172171122907299772term_a @ Xs2 @ I3 ) @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_l4678906206682822203term_a,Ys3: list_l4678906206682822203term_a,I3: nat,Z3: list_Ground_gterm_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u412135983231242147term_a @ Ys3 @ I3 @ Z3 ) )
              & ( trancl1148691031646339771term_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s5321119603959729319term_a @ Ys3 ) )
              & ( R4 @ ( nth_li6172171122907299772term_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Utils.trancl_listp.simps
thf(fact_790_Utils_Otrancl__listp_Osimps,axiom,
    ( trancl_listp_nat
    = ( ^ [R4: nat > nat > $o,A1: list_nat,A22: list_nat] :
          ( ? [Xs2: list_nat,Ys3: list_nat] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
                 => ( R4 @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_nat,Ys3: list_nat,I3: nat,Z3: nat] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_update_nat @ Ys3 @ I3 @ Z3 ) )
              & ( trancl_listp_nat @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys3 ) )
              & ( R4 @ ( nth_nat @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Utils.trancl_listp.simps
thf(fact_791_Utils_Otrancl__listp_Osimps,axiom,
    ( trancl1827884997844227949term_a
    = ( ^ [R4: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,A1: list_P2433084232546440685term_a,A22: list_P2433084232546440685term_a] :
          ( ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a] :
              ( ( A1 = Xs2 )
              & ( A22 = Ys3 )
              & ( ( size_s281034491502894169term_a @ Xs2 )
                = ( size_s281034491502894169term_a @ Ys3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
                 => ( R4 @ ( nth_Pr5943965302187034734term_a @ Xs2 @ I3 ) @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) ) ) )
          | ? [Xs2: list_P2433084232546440685term_a,Ys3: list_P2433084232546440685term_a,I3: nat,Z3: produc1525979690750518759term_a] :
              ( ( A1 = Xs2 )
              & ( A22
                = ( list_u8102571067576484949term_a @ Ys3 @ I3 @ Z3 ) )
              & ( trancl1827884997844227949term_a @ R4 @ Xs2 @ Ys3 )
              & ( ord_less_nat @ I3 @ ( size_s281034491502894169term_a @ Ys3 ) )
              & ( R4 @ ( nth_Pr5943965302187034734term_a @ Ys3 @ I3 ) @ Z3 ) ) ) ) ) ).

% Utils.trancl_listp.simps
thf(fact_792_Utils_Otrancl__listp_Obase,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R: ground_gmctxt_a > ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys ) )
           => ( R @ ( nth_Ground_gmctxt_a @ Xs @ I ) @ ( nth_Ground_gmctxt_a @ Ys @ I ) ) )
       => ( trancl3815965565430463256ctxt_a @ R @ Xs @ Ys ) ) ) ).

% Utils.trancl_listp.base
thf(fact_793_Utils_Otrancl__listp_Obase,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R: ground_gterm_a > ground_gterm_a > $o] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys ) )
           => ( R @ ( nth_Ground_gterm_a @ Xs @ I ) @ ( nth_Ground_gterm_a @ Ys @ I ) ) )
       => ( trancl3536835252858651701term_a @ R @ Xs @ Ys ) ) ) ).

% Utils.trancl_listp.base
thf(fact_794_Utils_Otrancl__listp_Obase,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R: list_Ground_gterm_a > list_Ground_gterm_a > $o] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys ) )
           => ( R @ ( nth_li6172171122907299772term_a @ Xs @ I ) @ ( nth_li6172171122907299772term_a @ Ys @ I ) ) )
       => ( trancl1148691031646339771term_a @ R @ Xs @ Ys ) ) ) ).

% Utils.trancl_listp.base
thf(fact_795_Utils_Otrancl__listp_Obase,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: nat > nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( R @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) )
       => ( trancl_listp_nat @ R @ Xs @ Ys ) ) ) ).

% Utils.trancl_listp.base
thf(fact_796_Utils_Otrancl__listp_Obase,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R: produc1525979690750518759term_a > produc1525979690750518759term_a > $o] :
      ( ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s281034491502894169term_a @ Ys ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys ) )
           => ( R @ ( nth_Pr5943965302187034734term_a @ Xs @ I ) @ ( nth_Pr5943965302187034734term_a @ Ys @ I ) ) )
       => ( trancl1827884997844227949term_a @ R @ Xs @ Ys ) ) ) ).

% Utils.trancl_listp.base
thf(fact_797_Utils_Otrancl__listp_Ocases,axiom,
    ! [R: ground_gmctxt_a > ground_gmctxt_a > $o,A12: list_Ground_gmctxt_a,A23: list_Ground_gmctxt_a] :
      ( ( trancl3815965565430463256ctxt_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s962548663205356226ctxt_a @ A12 )
            = ( size_s962548663205356226ctxt_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ A23 ) )
               => ( R @ ( nth_Ground_gmctxt_a @ A12 @ I2 ) @ ( nth_Ground_gmctxt_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_Ground_gmctxt_a,I: nat,Z2: ground_gmctxt_a] :
              ( ( A23
                = ( list_u7950812632479372336ctxt_a @ Ys4 @ I @ Z2 ) )
             => ( ( trancl3815965565430463256ctxt_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Ground_gmctxt_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Utils.trancl_listp.cases
thf(fact_798_Utils_Otrancl__listp_Ocases,axiom,
    ! [R: ground_gterm_a > ground_gterm_a > $o,A12: list_Ground_gterm_a,A23: list_Ground_gterm_a] :
      ( ( trancl3536835252858651701term_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s1893836384466666913term_a @ A12 )
            = ( size_s1893836384466666913term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ A23 ) )
               => ( R @ ( nth_Ground_gterm_a @ A12 @ I2 ) @ ( nth_Ground_gterm_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_Ground_gterm_a,I: nat,Z2: ground_gterm_a] :
              ( ( A23
                = ( list_u6582344683876525341term_a @ Ys4 @ I @ Z2 ) )
             => ( ( trancl3536835252858651701term_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Ground_gterm_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Utils.trancl_listp.cases
thf(fact_799_Utils_Otrancl__listp_Ocases,axiom,
    ! [R: list_Ground_gterm_a > list_Ground_gterm_a > $o,A12: list_l4678906206682822203term_a,A23: list_l4678906206682822203term_a] :
      ( ( trancl1148691031646339771term_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s5321119603959729319term_a @ A12 )
            = ( size_s5321119603959729319term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ A23 ) )
               => ( R @ ( nth_li6172171122907299772term_a @ A12 @ I2 ) @ ( nth_li6172171122907299772term_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_l4678906206682822203term_a,I: nat,Z2: list_Ground_gterm_a] :
              ( ( A23
                = ( list_u412135983231242147term_a @ Ys4 @ I @ Z2 ) )
             => ( ( trancl1148691031646339771term_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s5321119603959729319term_a @ Ys4 ) )
                 => ~ ( R @ ( nth_li6172171122907299772term_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Utils.trancl_listp.cases
thf(fact_800_Utils_Otrancl__listp_Ocases,axiom,
    ! [R: nat > nat > $o,A12: list_nat,A23: list_nat] :
      ( ( trancl_listp_nat @ R @ A12 @ A23 )
     => ( ( ( ( size_size_list_nat @ A12 )
            = ( size_size_list_nat @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ A23 ) )
               => ( R @ ( nth_nat @ A12 @ I2 ) @ ( nth_nat @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_nat,I: nat,Z2: nat] :
              ( ( A23
                = ( list_update_nat @ Ys4 @ I @ Z2 ) )
             => ( ( trancl_listp_nat @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys4 ) )
                 => ~ ( R @ ( nth_nat @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Utils.trancl_listp.cases
thf(fact_801_Utils_Otrancl__listp_Ocases,axiom,
    ! [R: produc1525979690750518759term_a > produc1525979690750518759term_a > $o,A12: list_P2433084232546440685term_a,A23: list_P2433084232546440685term_a] :
      ( ( trancl1827884997844227949term_a @ R @ A12 @ A23 )
     => ( ( ( ( size_s281034491502894169term_a @ A12 )
            = ( size_s281034491502894169term_a @ A23 ) )
         => ~ ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s281034491502894169term_a @ A23 ) )
               => ( R @ ( nth_Pr5943965302187034734term_a @ A12 @ I2 ) @ ( nth_Pr5943965302187034734term_a @ A23 @ I2 ) ) ) )
       => ~ ! [Ys4: list_P2433084232546440685term_a,I: nat,Z2: produc1525979690750518759term_a] :
              ( ( A23
                = ( list_u8102571067576484949term_a @ Ys4 @ I @ Z2 ) )
             => ( ( trancl1827884997844227949term_a @ R @ A12 @ Ys4 )
               => ( ( ord_less_nat @ I @ ( size_s281034491502894169term_a @ Ys4 ) )
                 => ~ ( R @ ( nth_Pr5943965302187034734term_a @ Ys4 @ I ) @ Z2 ) ) ) ) ) ) ).

% Utils.trancl_listp.cases
thf(fact_802_listrel1__iff__update,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( listre839271074516595327ctxt_a @ R2 ) )
      = ( ? [Y6: ground_gmctxt_a,N2: nat] :
            ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ N2 ) @ Y6 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( Ys
              = ( list_u7950812632479372336ctxt_a @ Xs @ N2 @ Y6 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_803_listrel1__iff__update,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( listre9104501415344685838term_a @ R2 ) )
      = ( ? [Y6: ground_gterm_a,N2: nat] :
            ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ N2 ) @ Y6 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
            & ( Ys
              = ( list_u6582344683876525341term_a @ Xs @ N2 @ Y6 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_804_listrel1__iff__update,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R2: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( listre1188387051607982356term_a @ R2 ) )
      = ( ? [Y6: list_Ground_gterm_a,N2: nat] :
            ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ ( nth_li6172171122907299772term_a @ Xs @ N2 ) @ Y6 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s5321119603959729319term_a @ Xs ) )
            & ( Ys
              = ( list_u412135983231242147term_a @ Xs @ N2 @ Y6 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_805_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y6: nat,N2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y6 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N2 @ Y6 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_806_listrel1__iff__update,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R2: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( listre5249125925726661062term_a @ R2 ) )
      = ( ? [Y6: produc1525979690750518759term_a,N2: nat] :
            ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ ( nth_Pr5943965302187034734term_a @ Xs @ N2 ) @ Y6 ) @ R2 )
            & ( ord_less_nat @ N2 @ ( size_s281034491502894169term_a @ Xs ) )
            & ( Ys
              = ( list_u8102571067576484949term_a @ Xs @ N2 @ Y6 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_807_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_808_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_809_fill__gholes__gmctxt__sup__mctxt__args,axiom,
    ! [C: ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = ( size_s962548663205356226ctxt_a @ Ds ) )
     => ( ( ground7778985915038999110args_a @ C @ ( ground2530127064035342964ctxt_a @ C @ Ds ) )
        = Ds ) ) ).

% fill_gholes_gmctxt_sup_mctxt_args
thf(fact_810_listrel1__eq__len,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,R2: set_Pr8705490787052945405ctxt_a] :
      ( ( member6597889099603620838ctxt_a @ ( produc4772472886035905909ctxt_a @ Xs @ Ys ) @ ( listre839271074516595327ctxt_a @ R2 ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_811_listrel1__eq__len,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,R2: set_Pr8282532413760598727term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ Xs @ Ys ) @ ( listre9104501415344685838term_a @ R2 ) )
     => ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_812_listrel1__eq__len,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,R2: set_Pr2711430308268100935term_a] :
      ( ( member3396161095182428176term_a @ ( produc6239833450114252375term_a @ Xs @ Ys ) @ ( listre1188387051607982356term_a @ R2 ) )
     => ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_813_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_814_listrel1__eq__len,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,R2: set_Pr5156918327056800967term_a] :
      ( ( member4622865958810230928term_a @ ( produc6894016452101132247term_a @ Xs @ Ys ) @ ( listre5249125925726661062term_a @ R2 ) )
     => ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s281034491502894169term_a @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_815_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R2 ) ) ).

% not_listrel1_Nil
thf(fact_816_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R2 ) ) ).

% not_Nil_listrel1
thf(fact_817_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_818_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_819_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_820_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_821_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_822_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_823_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_nat @ T @ X5 ) ) ).

% minf(7)
thf(fact_824_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_nat @ X5 @ T ) ) ).

% minf(5)
thf(fact_825_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_826_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_827_minf_I2_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z5 )
         => ( ( P @ X )
            = ( P3 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q3 @ X ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P3 @ X5 )
                | ( Q3 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_828_minf_I1_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z5 )
         => ( ( P @ X )
            = ( P3 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q3 @ X ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P3 @ X5 )
                & ( Q3 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_829_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_nat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_830_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_nat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_831_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_832_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_833_pinf_I2_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z5 @ X )
         => ( ( P @ X )
            = ( P3 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q3 @ X ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P3 @ X5 )
                | ( Q3 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_834_pinf_I1_J,axiom,
    ! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z5 @ X )
         => ( ( P @ X )
            = ( P3 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q3 @ X ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P3 @ X5 )
                & ( Q3 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_835_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_836_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_837_order__le__imp__less__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ X2 @ Y3 )
        | ( X2 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_838_linorder__le__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_le_less_linear
thf(fact_839_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C4 ) ) ) ) ).

% order_less_le_subst2
thf(fact_840_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C4 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_841_order__less__imp__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_842_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_843_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_844_linorder__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_845_order__less__imp__triv,axiom,
    ! [X2: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_846_order__less__not__sym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_847_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C4 ) ) ) ) ).

% order_less_subst2
thf(fact_848_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C4 ) ) ) ) ) ).

% order_less_subst1
thf(fact_849_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_850_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C4 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_851_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C4: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C4 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_852_order__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z4: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z4 )
       => ( ord_less_nat @ X2 @ Z4 ) ) ) ).

% order_less_trans
thf(fact_853_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_854_linorder__neq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
      = ( ( ord_less_nat @ X2 @ Y3 )
        | ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_855_order__less__asym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_856_linorder__neqE,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_857_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_858_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_859_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C4: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C4 @ B )
       => ( ord_less_nat @ C4 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_860_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_861_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ord_less_nat @ A @ C4 ) ) ) ).

% order.strict_trans
thf(fact_862_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_863_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X6: nat] : ( P4 @ X6 ) )
    = ( ^ [P5: nat > $o] :
        ? [N2: nat] :
          ( ( P5 @ N2 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ~ ( P5 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_864_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_865_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_866_linorder__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_867_antisym__conv3,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_868_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X )
             => ( P @ Y5 ) )
         => ( P @ X ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_869_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C4 )
       => ( ord_less_nat @ A @ C4 ) ) ) ).

% ord_less_eq_trans
thf(fact_870_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ord_less_nat @ A @ C4 ) ) ) ).

% ord_eq_less_trans
thf(fact_871_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_872_less__imp__neq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_873_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_874_leD,axiom,
    ! [Y3: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).

% leD
thf(fact_875_leI,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% leI
thf(fact_876_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_877_antisym__conv1,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ X2 @ Y3 )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_878_antisym__conv2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_879_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y6 )
          & ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_880_not__le__imp__less,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
     => ( ord_less_nat @ X2 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_881_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_882_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_883_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ord_less_nat @ A @ C4 ) ) ) ).

% order.strict_trans1
thf(fact_884_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C4: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C4 )
       => ( ord_less_nat @ A @ C4 ) ) ) ).

% order.strict_trans2
thf(fact_885_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_886_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_887_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_888_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C4: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C4 @ B )
       => ( ord_less_nat @ C4 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_889_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C4: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C4 @ B )
       => ( ord_less_nat @ C4 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_890_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_891_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_892_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_893_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y6: nat] :
          ( ( ord_less_nat @ X3 @ Y6 )
          | ( X3 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_894_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y6 )
          & ( X3 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_895_linorder__not__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_not_le
thf(fact_896_linorder__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).

% linorder_not_less
thf(fact_897_order__less__imp__le,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_898_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_899_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_900_order__le__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z4: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z4 )
       => ( ord_less_nat @ X2 @ Z4 ) ) ) ).

% order_le_less_trans
thf(fact_901_order__less__le__trans,axiom,
    ! [X2: nat,Y3: nat,Z4: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z4 )
       => ( ord_less_nat @ X2 @ Z4 ) ) ) ).

% order_less_le_trans
thf(fact_902_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_nat @ X @ Y4 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C4 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_903_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C4: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C4 )
       => ( ! [X: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C4 ) ) ) ) ).

% order_le_less_subst2
thf(fact_904_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C5: nat] :
              ( ( ord_less_eq_nat @ A @ C5 )
              & ( ord_less_eq_nat @ C5 @ B )
              & ! [X5: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X5 )
                    & ( ord_less_nat @ X5 @ C5 ) )
                 => ( P @ X5 ) )
              & ! [D6: nat] :
                  ( ! [X: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X )
                        & ( ord_less_nat @ X @ D6 ) )
                     => ( P @ X ) )
                 => ( ord_less_eq_nat @ D6 @ C5 ) ) ) ) ) ) ).

% complete_interval
thf(fact_905_concat__index__split__larger__lists,axiom,
    ! [I4: nat,Xs: list_l1578326660578982846ctxt_a,N: nat,Ys: list_l1578326660578982846ctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ ( concat3993834769610107037ctxt_a @ Xs ) ) )
     => ( ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ ( append272212692390751315ctxt_a @ Xs @ Ys ) )
        = ( concat7860870432694265376ctxt_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_906_concat__index__split__larger__lists,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,N: nat,Ys: list_l4678906206682822203term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ ( concat1399541994378716528term_a @ Xs ) ) )
     => ( ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ ( append488786809011849168term_a @ Xs @ Ys ) )
        = ( concat4415380251885098797term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_907_concat__index__split__larger__lists,axiom,
    ! [I4: nat,Xs: list_l5172476077732501185term_a,N: nat,Ys: list_l5172476077732501185term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ ( concat5214662034885773942term_a @ Xs ) ) )
     => ( ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ ( append7609275726793042774term_a @ Xs @ Ys ) )
        = ( concat5187525703047707059term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_908_concat__index__split__larger__lists,axiom,
    ! [I4: nat,Xs: list_list_nat,N: nat,Ys: list_list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ ( concat_nat @ Xs ) ) )
     => ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ ( append_list_nat @ Xs @ Ys ) )
        = ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_909_concat__index__split__larger__lists,axiom,
    ! [I4: nat,Xs: list_l8493803889202805107term_a,N: nat,Ys: list_l8493803889202805107term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ ( concat2365498962271017768term_a @ Xs ) ) )
     => ( ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ ( append5793913037638878984term_a @ Xs @ Ys ) )
        = ( concat451825319501278821term_a @ ( product_Pair_nat_nat @ N @ I4 ) @ Xs ) ) ) ).

% concat_index_split_larger_lists
thf(fact_910_nth__zip,axiom,
    ! [I4: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I4 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_911_nth__zip,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr1982580230130579030_a_nat @ ( zip_Gr434538988109263783_a_nat @ Xs @ Ys ) @ I4 )
          = ( produc2030840239341956231_a_nat @ ( nth_Ground_gmctxt_a @ Xs @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_912_nth__zip,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,Ys: list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr1995959512206878695_a_nat @ ( zip_Gr5583948589966438664_a_nat @ Xs @ Ys ) @ I4 )
          = ( produc6032213142089859112_a_nat @ ( nth_Ground_gterm_a @ Xs @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_913_nth__zip,axiom,
    ! [I4: nat,Xs: list_nat,Ys: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( nth_Pr3641570550162790742ctxt_a @ ( zip_na1508236324085319079ctxt_a @ Xs @ Ys ) @ I4 )
          = ( produc3104537575318011527ctxt_a @ ( nth_nat @ Xs @ I4 ) @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_914_nth__zip,axiom,
    ! [I4: nat,Xs: list_nat,Ys: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( nth_Pr5718682320234703373term_a @ ( zip_na4248810256523208166term_a @ Xs @ Ys ) @ I4 )
          = ( produc4697074808646628614term_a @ ( nth_nat @ Xs @ I4 ) @ ( nth_Ground_gterm_a @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_915_nth__zip,axiom,
    ! [I4: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I4 @ ( size_s8208510060688613859at_nat @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr4952906083868132727at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I4 )
          = ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ I4 ) @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_916_nth__zip,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( nth_Pr2917159649707247524ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Ys ) @ I4 )
          = ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Xs @ I4 ) @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_917_nth__zip,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( nth_Pr3254615499752982591term_a @ ( zip_Gr4568060137147143192term_a @ Xs @ Ys ) @ I4 )
          = ( produc7893155741350732600term_a @ ( nth_Ground_gmctxt_a @ Xs @ I4 ) @ ( nth_Ground_gterm_a @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_918_nth__zip,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Ys ) )
       => ( ( nth_Pr6137900599796896565ctxt_a @ ( zip_Gr2700541357751180118ctxt_a @ Xs @ Ys ) @ I4 )
          = ( produc6025636961954769526ctxt_a @ ( nth_Ground_gterm_a @ Xs @ I4 ) @ ( nth_Ground_gmctxt_a @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_919_nth__zip,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Ys ) )
       => ( ( nth_Pr5943965302187034734term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) @ I4 )
          = ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ Xs @ I4 ) @ ( nth_Ground_gterm_a @ Ys @ I4 ) ) ) ) ) ).

% nth_zip
thf(fact_920_in__set__product__lists__length,axiom,
    ! [Xs: list_Ground_gmctxt_a,Xss: list_l1578326660578982846ctxt_a] :
      ( ( member2714729351964853061ctxt_a @ Xs @ ( set_li3057194059929822169ctxt_a @ ( produc6102553363301462639ctxt_a @ Xss ) ) )
     => ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s6994882312647770194ctxt_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_921_in__set__product__lists__length,axiom,
    ! [Xs: list_Ground_gterm_a,Xss: list_l4678906206682822203term_a] :
      ( ( member5011579089780872286term_a @ Xs @ ( set_li7546925467519454922term_a @ ( produc9720139385259806term_a @ Xss ) ) )
     => ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s5321119603959729319term_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_922_in__set__product__lists__length,axiom,
    ! [Xs: list_l4678906206682822203term_a,Xss: list_l5172476077732501185term_a] :
      ( ( member6060894471016913124term_a @ Xs @ ( set_li8466561177209531728term_a @ ( produc3538485515374017828term_a @ Xss ) ) )
     => ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_s3474408278181496621term_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_923_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_924_in__set__product__lists__length,axiom,
    ! [Xs: list_P2433084232546440685term_a,Xss: list_l8493803889202805107term_a] :
      ( ( member248923241868414102term_a @ Xs @ ( set_li5508857501565128962term_a @ ( produc3533465718962628054term_a @ Xss ) ) )
     => ( ( size_s281034491502894169term_a @ Xs )
        = ( size_s7283615635009460703term_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_925_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C4: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C4 )
      = ( append_nat @ A @ ( append_nat @ B @ C4 ) ) ) ).

% append.assoc
thf(fact_926_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs2 )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs2 ) ) ) ).

% append_assoc
thf(fact_927_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs2: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs2 @ Xs ) )
      = ( Ys = Zs2 ) ) ).

% append_same_eq
thf(fact_928_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs2 ) )
      = ( Ys = Zs2 ) ) ).

% same_append_eq
thf(fact_929_append__eq__append__conv,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,Us: list_Ground_gmctxt_a,Vs: list_Ground_gmctxt_a] :
      ( ( ( ( size_s962548663205356226ctxt_a @ Xs )
          = ( size_s962548663205356226ctxt_a @ Ys ) )
        | ( ( size_s962548663205356226ctxt_a @ Us )
          = ( size_s962548663205356226ctxt_a @ Vs ) ) )
     => ( ( ( append5238750768297430211ctxt_a @ Xs @ Us )
          = ( append5238750768297430211ctxt_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_930_append__eq__append__conv,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,Us: list_Ground_gterm_a,Vs: list_Ground_gterm_a] :
      ( ( ( ( size_s1893836384466666913term_a @ Xs )
          = ( size_s1893836384466666913term_a @ Ys ) )
        | ( ( size_s1893836384466666913term_a @ Us )
          = ( size_s1893836384466666913term_a @ Vs ) ) )
     => ( ( ( append7798372865478268874term_a @ Xs @ Us )
          = ( append7798372865478268874term_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_931_append__eq__append__conv,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,Us: list_l4678906206682822203term_a,Vs: list_l4678906206682822203term_a] :
      ( ( ( ( size_s5321119603959729319term_a @ Xs )
          = ( size_s5321119603959729319term_a @ Ys ) )
        | ( ( size_s5321119603959729319term_a @ Us )
          = ( size_s5321119603959729319term_a @ Vs ) ) )
     => ( ( ( append488786809011849168term_a @ Xs @ Us )
          = ( append488786809011849168term_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_932_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_933_append__eq__append__conv,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,Us: list_P2433084232546440685term_a,Vs: list_P2433084232546440685term_a] :
      ( ( ( ( size_s281034491502894169term_a @ Xs )
          = ( size_s281034491502894169term_a @ Ys ) )
        | ( ( size_s281034491502894169term_a @ Us )
          = ( size_s281034491502894169term_a @ Vs ) ) )
     => ( ( ( append7909858224561987714term_a @ Xs @ Us )
          = ( append7909858224561987714term_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_934_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_935_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_936_self__append__conv2,axiom,
    ! [Y3: list_nat,Xs: list_nat] :
      ( ( Y3
        = ( append_nat @ Xs @ Y3 ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_937_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_938_self__append__conv,axiom,
    ! [Y3: list_nat,Ys: list_nat] :
      ( ( Y3
        = ( append_nat @ Y3 @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_939_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_940_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_941_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_942_length__append,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ ( size_s962548663205356226ctxt_a @ Ys ) ) ) ).

% length_append
thf(fact_943_length__append,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( size_s1893836384466666913term_a @ ( append7798372865478268874term_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s1893836384466666913term_a @ Xs ) @ ( size_s1893836384466666913term_a @ Ys ) ) ) ).

% length_append
thf(fact_944_length__append,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( size_s5321119603959729319term_a @ ( append488786809011849168term_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s5321119603959729319term_a @ Xs ) @ ( size_s5321119603959729319term_a @ Ys ) ) ) ).

% length_append
thf(fact_945_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_946_length__append,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( size_s281034491502894169term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s281034491502894169term_a @ Xs ) @ ( size_s281034491502894169term_a @ Ys ) ) ) ).

% length_append
thf(fact_947_zip__append,axiom,
    ! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_948_zip__append,axiom,
    ! [Xs: list_Ground_gmctxt_a,Us: list_nat,Ys: list_Ground_gmctxt_a,Vs: list_nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Gr434538988109263783_a_nat @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append584840330893524074_a_nat @ ( zip_Gr434538988109263783_a_nat @ Xs @ Us ) @ ( zip_Gr434538988109263783_a_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_949_zip__append,axiom,
    ! [Xs: list_Ground_gterm_a,Us: list_nat,Ys: list_Ground_gterm_a,Vs: list_nat] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Gr5583948589966438664_a_nat @ ( append7798372865478268874term_a @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append3919659240378768595_a_nat @ ( zip_Gr5583948589966438664_a_nat @ Xs @ Us ) @ ( zip_Gr5583948589966438664_a_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_950_zip__append,axiom,
    ! [Xs: list_nat,Us: list_Ground_gmctxt_a,Ys: list_nat,Vs: list_Ground_gmctxt_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s962548663205356226ctxt_a @ Us ) )
     => ( ( zip_na1508236324085319079ctxt_a @ ( append_nat @ Xs @ Ys ) @ ( append5238750768297430211ctxt_a @ Us @ Vs ) )
        = ( append2243830650925735786ctxt_a @ ( zip_na1508236324085319079ctxt_a @ Xs @ Us ) @ ( zip_na1508236324085319079ctxt_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_951_zip__append,axiom,
    ! [Xs: list_nat,Us: list_Ground_gterm_a,Ys: list_nat,Vs: list_Ground_gterm_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1893836384466666913term_a @ Us ) )
     => ( ( zip_na4248810256523208166term_a @ ( append_nat @ Xs @ Ys ) @ ( append7798372865478268874term_a @ Us @ Vs ) )
        = ( append7642382048406593273term_a @ ( zip_na4248810256523208166term_a @ Xs @ Us ) @ ( zip_na4248810256523208166term_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_952_zip__append,axiom,
    ! [Xs: list_Ground_gmctxt_a,Us: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,Vs: list_Ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Us ) )
     => ( ( zip_Gr6926631713258005493ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ ( append5238750768297430211ctxt_a @ Us @ Vs ) )
        = ( append3687069464929605048ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Us ) @ ( zip_Gr6926631713258005493ctxt_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_953_zip__append,axiom,
    ! [Xs: list_Ground_gmctxt_a,Us: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a,Vs: list_Ground_gterm_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s1893836384466666913term_a @ Us ) )
     => ( ( zip_Gr4568060137147143192term_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ ( append7798372865478268874term_a @ Us @ Vs ) )
        = ( append2676611070818170155term_a @ ( zip_Gr4568060137147143192term_a @ Xs @ Us ) @ ( zip_Gr4568060137147143192term_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_954_zip__append,axiom,
    ! [Xs: list_Ground_gterm_a,Us: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a,Vs: list_Ground_gmctxt_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Us ) )
     => ( ( zip_Gr2700541357751180118ctxt_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ ( append5238750768297430211ctxt_a @ Us @ Vs ) )
        = ( append5559896170862084129ctxt_a @ ( zip_Gr2700541357751180118ctxt_a @ Xs @ Us ) @ ( zip_Gr2700541357751180118ctxt_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_955_zip__append,axiom,
    ! [Xs: list_Ground_gterm_a,Us: list_Ground_gterm_a,Ys: list_Ground_gterm_a,Vs: list_Ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Us ) )
     => ( ( zip_Gr212557436252222583term_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ ( append7798372865478268874term_a @ Us @ Vs ) )
        = ( append7909858224561987714term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Us ) @ ( zip_Gr212557436252222583term_a @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_956_zip__append,axiom,
    ! [Xs: list_l4678906206682822203term_a,Us: list_nat,Ys: list_l4678906206682822203term_a,Vs: list_nat] :
      ( ( ( size_s5321119603959729319term_a @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_li4224161369154485902_a_nat @ ( append488786809011849168term_a @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append441661513692551257_a_nat @ ( zip_li4224161369154485902_a_nat @ Xs @ Us ) @ ( zip_li4224161369154485902_a_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_957_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_958_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_959_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_960_nth__append__length__plus,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,N: nat] :
      ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s962548663205356226ctxt_a @ Xs ) @ N ) )
      = ( nth_Ground_gmctxt_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_961_nth__append__length__plus,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,N: nat] :
      ( ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s1893836384466666913term_a @ Xs ) @ N ) )
      = ( nth_Ground_gterm_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_962_nth__append__length__plus,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,N: nat] :
      ( ( nth_li6172171122907299772term_a @ ( append488786809011849168term_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5321119603959729319term_a @ Xs ) @ N ) )
      = ( nth_li6172171122907299772term_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_963_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_964_nth__append__length__plus,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,N: nat] :
      ( ( nth_Pr5943965302187034734term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s281034491502894169term_a @ Xs ) @ N ) )
      = ( nth_Pr5943965302187034734term_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_965_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_966_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_967_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_968_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs2: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs2 )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs2 @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_969_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs2: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs2 @ Ts ) )
      = ( ? [Us3: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs2 @ Us3 ) )
              & ( ( append_nat @ Us3 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us3 )
                = Zs2 )
              & ( Ys
                = ( append_nat @ Us3 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_970_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( enumer3922341085786344185ctxt_a @ N @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) )
      = ( append2243830650925735786ctxt_a @ ( enumer3922341085786344185ctxt_a @ N @ Xs ) @ ( enumer3922341085786344185ctxt_a @ ( plus_plus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_971_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( enumer8595146184022354964term_a @ N @ ( append7798372865478268874term_a @ Xs @ Ys ) )
      = ( append7642382048406593273term_a @ ( enumer8595146184022354964term_a @ N @ Xs ) @ ( enumer8595146184022354964term_a @ ( plus_plus_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_972_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( enumer2156928005790035738term_a @ N @ ( append488786809011849168term_a @ Xs @ Ys ) )
      = ( append6435179048203901439term_a @ ( enumer2156928005790035738term_a @ N @ Xs ) @ ( enumer2156928005790035738term_a @ ( plus_plus_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_973_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_974_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( enumer7770827359896883148term_a @ N @ ( append7909858224561987714term_a @ Xs @ Ys ) )
      = ( append3972858608879487409term_a @ ( enumer7770827359896883148term_a @ N @ Xs ) @ ( enumer7770827359896883148term_a @ ( plus_plus_nat @ N @ ( size_s281034491502894169term_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_975_zip__update,axiom,
    ! [Xs: list_Ground_gmctxt_a,I4: nat,X2: ground_gmctxt_a,Ys: list_l4678906206682822203term_a,Y3: list_Ground_gterm_a] :
      ( ( zip_Gr7514910105460098334term_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) @ ( list_u412135983231242147term_a @ Ys @ I4 @ Y3 ) )
      = ( list_u3087721638091695518term_a @ ( zip_Gr7514910105460098334term_a @ Xs @ Ys ) @ I4 @ ( produc4720362732934906942term_a @ X2 @ Y3 ) ) ) ).

% zip_update
thf(fact_976_zip__update,axiom,
    ! [Xs: list_Ground_gterm_a,I4: nat,X2: ground_gterm_a,Ys: list_Ground_gterm_a,Y3: ground_gterm_a] :
      ( ( zip_Gr212557436252222583term_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) @ ( list_u6582344683876525341term_a @ Ys @ I4 @ Y3 ) )
      = ( list_u8102571067576484949term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) @ I4 @ ( produc7419039783125785943term_a @ X2 @ Y3 ) ) ) ).

% zip_update
thf(fact_977_zip__update,axiom,
    ! [Xs: list_nat_nat,I4: nat,X2: nat > nat,Ys: list_nat,Y3: nat] :
      ( ( zip_nat_nat_nat @ ( list_update_nat_nat @ Xs @ I4 @ X2 ) @ ( list_update_nat @ Ys @ I4 @ Y3 ) )
      = ( list_u5493879213508619102at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I4 @ ( produc72220940542539688at_nat @ X2 @ Y3 ) ) ) ).

% zip_update
thf(fact_978_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R2 ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R2 ) ) ) ).

% append_listrel1I
thf(fact_979_zip__same,axiom,
    ! [A: ground_gmctxt_a,B: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ A @ B ) @ ( set_Pr5641390345007089586ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Xs ) ) )
      = ( ( member3306386364193653941ctxt_a @ A @ ( set_Ground_gmctxt_a2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_980_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_981_zip__same,axiom,
    ! [A: list_Ground_gterm_a,B: list_Ground_gterm_a,Xs: list_l4678906206682822203term_a] :
      ( ( member3518026002450860944term_a @ ( produc6818685258692100823term_a @ A @ B ) @ ( set_Pr5267327275363854844term_a @ ( zip_li544827491672340983term_a @ Xs @ Xs ) ) )
      = ( ( member5011579089780872286term_a @ A @ ( set_li7546925467519454922term_a @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_982_zip__same,axiom,
    ! [A: produc1525979690750518759term_a,B: produc1525979690750518759term_a,Xs: list_P2433084232546440685term_a] :
      ( ( member4810723946494196496term_a @ ( produc4593192926746741591term_a @ A @ B ) @ ( set_Pr8917639809720154492term_a @ ( zip_Pr3485155535332754039term_a @ Xs @ Xs ) ) )
      = ( ( member4351891425000572176term_a @ A @ ( set_Pr5405221470436175740term_a @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_983_zip__same,axiom,
    ! [A: ground_gterm_a,B: ground_gterm_a,Xs: list_Ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ A @ B ) @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Xs ) ) )
      = ( ( member5261315023172567896term_a @ A @ ( set_Ground_gterm_a2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_984_in__set__zipE,axiom,
    ! [X2: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_985_in__set__zipE,axiom,
    ! [X2: ground_gmctxt_a,Y3: nat,Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( member1226896217267473912_a_nat @ ( produc2030840239341956231_a_nat @ X2 @ Y3 ) @ ( set_Pr2336437649969777252_a_nat @ ( zip_Gr434538988109263783_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_986_in__set__zipE,axiom,
    ! [X2: nat,Y3: ground_gmctxt_a,Xs: list_nat,Ys: list_Ground_gmctxt_a] :
      ( ( member2885886537299685624ctxt_a @ ( produc3104537575318011527ctxt_a @ X2 @ Y3 ) @ ( set_Pr3995427970001988964ctxt_a @ ( zip_na1508236324085319079ctxt_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ~ ( member3306386364193653941ctxt_a @ Y3 @ ( set_Ground_gmctxt_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_987_in__set__zipE,axiom,
    ! [X2: nat,Y3: ground_gterm_a,Xs: list_nat,Ys: list_Ground_gterm_a] :
      ( ( member6622040750283622891term_a @ ( produc4697074808646628614term_a @ X2 @ Y3 ) @ ( set_Pr3203312087732499071term_a @ ( zip_na4248810256523208166term_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ~ ( member5261315023172567896term_a @ Y3 @ ( set_Ground_gterm_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_988_in__set__zipE,axiom,
    ! [X2: ground_gterm_a,Y3: nat,Xs: list_Ground_gterm_a,Ys: list_nat] :
      ( ( member2899317942255798213_a_nat @ ( produc6032213142089859112_a_nat @ X2 @ Y3 ) @ ( set_Pr8703961316559450201_a_nat @ ( zip_Gr5583948589966438664_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_989_in__set__zipE,axiom,
    ! [X2: ground_gmctxt_a,Y3: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y3 ) @ ( set_Pr5641390345007089586ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Ys ) ) )
     => ~ ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
         => ~ ( member3306386364193653941ctxt_a @ Y3 @ ( set_Ground_gmctxt_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_990_in__set__zipE,axiom,
    ! [X2: ground_gmctxt_a,Y3: ground_gterm_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a] :
      ( ( member1041919251622114333term_a @ ( produc7893155741350732600term_a @ X2 @ Y3 ) @ ( set_Pr6159641702654731441term_a @ ( zip_Gr4568060137147143192term_a @ Xs @ Ys ) ) )
     => ~ ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
         => ~ ( member5261315023172567896term_a @ Y3 @ ( set_Ground_gterm_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_991_in__set__zipE,axiom,
    ! [X2: nat,Y3: list_Ground_gterm_a,Xs: list_nat,Ys: list_l4678906206682822203term_a] :
      ( ( member825974164662248433term_a @ ( produc1055361199944718092term_a @ X2 @ Y3 ) @ ( set_Pr4194642937221624453term_a @ ( zip_na8174785387252291564term_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ~ ( member5011579089780872286term_a @ Y3 @ ( set_li7546925467519454922term_a @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_992_in__set__zipE,axiom,
    ! [X2: list_Ground_gterm_a,Y3: nat,Xs: list_l4678906206682822203term_a,Ys: list_nat] :
      ( ( member4055828667005674059_a_nat @ ( produc6328109218701688238_a_nat @ X2 @ Y3 ) @ ( set_Pr7424497439565050079_a_nat @ ( zip_li4224161369154485902_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5011579089780872286term_a @ X2 @ ( set_li7546925467519454922term_a @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_993_in__set__zipE,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gmctxt_a,Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a] :
      ( ( member3925204351666028307ctxt_a @ ( produc6025636961954769526ctxt_a @ X2 @ Y3 ) @ ( set_Pr9042926802698645415ctxt_a @ ( zip_Gr2700541357751180118ctxt_a @ Xs @ Ys ) ) )
     => ~ ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
         => ~ ( member3306386364193653941ctxt_a @ Y3 @ ( set_Ground_gmctxt_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_994_set__zip__leftD,axiom,
    ! [X2: ground_gmctxt_a,Y3: list_Ground_gterm_a,Xs: list_Ground_gmctxt_a,Ys: list_l4678906206682822203term_a] :
      ( ( member8767046467196472611term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) @ ( set_Pr8846649574831596471term_a @ ( zip_Gr7514910105460098334term_a @ Xs @ Ys ) ) )
     => ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_995_set__zip__leftD,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gterm_a,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) ) )
     => ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_996_set__zip__leftD,axiom,
    ! [X2: nat > nat,Y3: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat_nat @ X2 @ ( set_nat_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_997_set__zip__rightD,axiom,
    ! [X2: ground_gmctxt_a,Y3: list_Ground_gterm_a,Xs: list_Ground_gmctxt_a,Ys: list_l4678906206682822203term_a] :
      ( ( member8767046467196472611term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) @ ( set_Pr8846649574831596471term_a @ ( zip_Gr7514910105460098334term_a @ Xs @ Ys ) ) )
     => ( member5011579089780872286term_a @ Y3 @ ( set_li7546925467519454922term_a @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_998_set__zip__rightD,axiom,
    ! [X2: ground_gterm_a,Y3: ground_gterm_a,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) ) )
     => ( member5261315023172567896term_a @ Y3 @ ( set_Ground_gterm_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_999_set__zip__rightD,axiom,
    ! [X2: nat > nat,Y3: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1000_zip__fst,axiom,
    ! [P2: produc1525979690750518759term_a,As: list_Ground_gterm_a,Bs: list_Ground_gterm_a] :
      ( ( member4351891425000572176term_a @ P2 @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ As @ Bs ) ) )
     => ( member5261315023172567896term_a @ ( produc5097907526672912195term_a @ P2 ) @ ( set_Ground_gterm_a2 @ As ) ) ) ).

% zip_fst
thf(fact_1001_zip__snd,axiom,
    ! [P2: produc1525979690750518759term_a,As: list_Ground_gterm_a,Bs: list_Ground_gterm_a] :
      ( ( member4351891425000572176term_a @ P2 @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ As @ Bs ) ) )
     => ( member5261315023172567896term_a @ ( produc5323275678763266181term_a @ P2 ) @ ( set_Ground_gterm_a2 @ Bs ) ) ) ).

% zip_snd
thf(fact_1002_P__as__bs__extend,axiom,
    ! [As: list_nat,Bs: list_nat,Cs2: list_nat,Ds3: list_nat,P: nat > nat > $o] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_size_list_nat @ Cs2 )
          = ( size_size_list_nat @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_nat @ As @ I ) @ ( nth_nat @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ds3 ) )
               => ( P @ ( nth_nat @ Cs2 @ I ) @ ( nth_nat @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_nat @ ( append_nat @ As @ Cs2 ) @ I2 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1003_P__as__bs__extend,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_nat,Cs2: list_Ground_gmctxt_a,Ds3: list_nat,P: ground_gmctxt_a > nat > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
          = ( size_size_list_nat @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_Ground_gmctxt_a @ As @ I ) @ ( nth_nat @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ds3 ) )
               => ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I ) @ ( nth_nat @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ As @ Cs2 ) @ I2 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1004_P__as__bs__extend,axiom,
    ! [As: list_Ground_gterm_a,Bs: list_nat,Cs2: list_Ground_gterm_a,Ds3: list_nat,P: ground_gterm_a > nat > $o] :
      ( ( ( size_s1893836384466666913term_a @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_s1893836384466666913term_a @ Cs2 )
          = ( size_size_list_nat @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_Ground_gterm_a @ As @ I ) @ ( nth_nat @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ds3 ) )
               => ( P @ ( nth_Ground_gterm_a @ Cs2 @ I ) @ ( nth_nat @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ As @ Cs2 ) @ I2 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1005_P__as__bs__extend,axiom,
    ! [As: list_nat,Bs: list_Ground_gmctxt_a,Cs2: list_nat,Ds3: list_Ground_gmctxt_a,P: nat > ground_gmctxt_a > $o] :
      ( ( ( size_size_list_nat @ As )
        = ( size_s962548663205356226ctxt_a @ Bs ) )
     => ( ( ( size_size_list_nat @ Cs2 )
          = ( size_s962548663205356226ctxt_a @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Bs ) )
             => ( P @ ( nth_nat @ As @ I ) @ ( nth_Ground_gmctxt_a @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds3 ) )
               => ( P @ ( nth_nat @ Cs2 @ I ) @ ( nth_Ground_gmctxt_a @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_nat @ ( append_nat @ As @ Cs2 ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1006_P__as__bs__extend,axiom,
    ! [As: list_nat,Bs: list_Ground_gterm_a,Cs2: list_nat,Ds3: list_Ground_gterm_a,P: nat > ground_gterm_a > $o] :
      ( ( ( size_size_list_nat @ As )
        = ( size_s1893836384466666913term_a @ Bs ) )
     => ( ( ( size_size_list_nat @ Cs2 )
          = ( size_s1893836384466666913term_a @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Bs ) )
             => ( P @ ( nth_nat @ As @ I ) @ ( nth_Ground_gterm_a @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ds3 ) )
               => ( P @ ( nth_nat @ Cs2 @ I ) @ ( nth_Ground_gterm_a @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( append7798372865478268874term_a @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_nat @ ( append_nat @ As @ Cs2 ) @ I2 ) @ ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1007_P__as__bs__extend,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_Ground_gmctxt_a,Cs2: list_Ground_gmctxt_a,Ds3: list_Ground_gmctxt_a,P: ground_gmctxt_a > ground_gmctxt_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_s962548663205356226ctxt_a @ Bs ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
          = ( size_s962548663205356226ctxt_a @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Bs ) )
             => ( P @ ( nth_Ground_gmctxt_a @ As @ I ) @ ( nth_Ground_gmctxt_a @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds3 ) )
               => ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I ) @ ( nth_Ground_gmctxt_a @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ As @ Cs2 ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1008_P__as__bs__extend,axiom,
    ! [As: list_Ground_gmctxt_a,Bs: list_Ground_gterm_a,Cs2: list_Ground_gmctxt_a,Ds3: list_Ground_gterm_a,P: ground_gmctxt_a > ground_gterm_a > $o] :
      ( ( ( size_s962548663205356226ctxt_a @ As )
        = ( size_s1893836384466666913term_a @ Bs ) )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs2 )
          = ( size_s1893836384466666913term_a @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Bs ) )
             => ( P @ ( nth_Ground_gmctxt_a @ As @ I ) @ ( nth_Ground_gterm_a @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ds3 ) )
               => ( P @ ( nth_Ground_gmctxt_a @ Cs2 @ I ) @ ( nth_Ground_gterm_a @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( append7798372865478268874term_a @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ As @ Cs2 ) @ I2 ) @ ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1009_P__as__bs__extend,axiom,
    ! [As: list_Ground_gterm_a,Bs: list_Ground_gmctxt_a,Cs2: list_Ground_gterm_a,Ds3: list_Ground_gmctxt_a,P: ground_gterm_a > ground_gmctxt_a > $o] :
      ( ( ( size_s1893836384466666913term_a @ As )
        = ( size_s962548663205356226ctxt_a @ Bs ) )
     => ( ( ( size_s1893836384466666913term_a @ Cs2 )
          = ( size_s962548663205356226ctxt_a @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Bs ) )
             => ( P @ ( nth_Ground_gterm_a @ As @ I ) @ ( nth_Ground_gmctxt_a @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds3 ) )
               => ( P @ ( nth_Ground_gterm_a @ Cs2 @ I ) @ ( nth_Ground_gmctxt_a @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ As @ Cs2 ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1010_P__as__bs__extend,axiom,
    ! [As: list_Ground_gterm_a,Bs: list_Ground_gterm_a,Cs2: list_Ground_gterm_a,Ds3: list_Ground_gterm_a,P: ground_gterm_a > ground_gterm_a > $o] :
      ( ( ( size_s1893836384466666913term_a @ As )
        = ( size_s1893836384466666913term_a @ Bs ) )
     => ( ( ( size_s1893836384466666913term_a @ Cs2 )
          = ( size_s1893836384466666913term_a @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Bs ) )
             => ( P @ ( nth_Ground_gterm_a @ As @ I ) @ ( nth_Ground_gterm_a @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s1893836384466666913term_a @ Ds3 ) )
               => ( P @ ( nth_Ground_gterm_a @ Cs2 @ I ) @ ( nth_Ground_gterm_a @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( append7798372865478268874term_a @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ As @ Cs2 ) @ I2 ) @ ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1011_P__as__bs__extend,axiom,
    ! [As: list_l4678906206682822203term_a,Bs: list_nat,Cs2: list_l4678906206682822203term_a,Ds3: list_nat,P: list_Ground_gterm_a > nat > $o] :
      ( ( ( size_s5321119603959729319term_a @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ( ( size_s5321119603959729319term_a @ Cs2 )
          = ( size_size_list_nat @ Ds3 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs ) )
             => ( P @ ( nth_li6172171122907299772term_a @ As @ I ) @ ( nth_nat @ Bs @ I ) ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ds3 ) )
               => ( P @ ( nth_li6172171122907299772term_a @ Cs2 @ I ) @ ( nth_nat @ Ds3 @ I ) ) )
           => ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( append_nat @ Bs @ Ds3 ) ) )
               => ( P @ ( nth_li6172171122907299772term_a @ ( append488786809011849168term_a @ As @ Cs2 ) @ I2 ) @ ( nth_nat @ ( append_nat @ Bs @ Ds3 ) @ I2 ) ) ) ) ) ) ) ).

% P_as_bs_extend
thf(fact_1012_list__update__append1,axiom,
    ! [I4: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( ord_less_nat @ I4 @ ( size_s962548663205356226ctxt_a @ Xs ) )
     => ( ( list_u7950812632479372336ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ I4 @ X2 )
        = ( append5238750768297430211ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1013_list__update__append1,axiom,
    ! [I4: nat,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s1893836384466666913term_a @ Xs ) )
     => ( ( list_u6582344683876525341term_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ I4 @ X2 )
        = ( append7798372865478268874term_a @ ( list_u6582344683876525341term_a @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1014_list__update__append1,axiom,
    ! [I4: nat,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,X2: list_Ground_gterm_a] :
      ( ( ord_less_nat @ I4 @ ( size_s5321119603959729319term_a @ Xs ) )
     => ( ( list_u412135983231242147term_a @ ( append488786809011849168term_a @ Xs @ Ys ) @ I4 @ X2 )
        = ( append488786809011849168term_a @ ( list_u412135983231242147term_a @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1015_list__update__append1,axiom,
    ! [I4: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I4 @ X2 )
        = ( append_nat @ ( list_update_nat @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1016_list__update__append1,axiom,
    ! [I4: nat,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a] :
      ( ( ord_less_nat @ I4 @ ( size_s281034491502894169term_a @ Xs ) )
     => ( ( list_u8102571067576484949term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) @ I4 @ X2 )
        = ( append7909858224561987714term_a @ ( list_u8102571067576484949term_a @ Xs @ I4 @ X2 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1017_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1018_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,X2: ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member1226896217267473912_a_nat @ ( produc2030840239341956231_a_nat @ X2 @ Y4 ) @ ( set_Pr2336437649969777252_a_nat @ ( zip_Gr434538988109263783_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1019_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_nat,X2: ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member2899317942255798213_a_nat @ ( produc6032213142089859112_a_nat @ X2 @ Y4 ) @ ( set_Pr8703961316559450201_a_nat @ ( zip_Gr5583948589966438664_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1020_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gmctxt_a,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y4: ground_gmctxt_a] :
              ~ ( member2885886537299685624ctxt_a @ ( produc3104537575318011527ctxt_a @ X2 @ Y4 ) @ ( set_Pr3995427970001988964ctxt_a @ ( zip_na1508236324085319079ctxt_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1021_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gterm_a,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y4: ground_gterm_a] :
              ~ ( member6622040750283622891term_a @ ( produc4697074808646628614term_a @ X2 @ Y4 ) @ ( set_Pr3203312087732499071term_a @ ( zip_na4248810256523208166term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1022_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,X2: nat > nat] :
      ( ( ( size_s8208510060688613859at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat_nat @ X2 @ ( set_nat_nat2 @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y4 ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1023_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
       => ~ ! [Y4: ground_gmctxt_a] :
              ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X2 @ Y4 ) @ ( set_Pr5641390345007089586ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1024_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a,X2: ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ( member3306386364193653941ctxt_a @ X2 @ ( set_Ground_gmctxt_a2 @ Xs ) )
       => ~ ! [Y4: ground_gterm_a] :
              ~ ( member1041919251622114333term_a @ ( produc7893155741350732600term_a @ X2 @ Y4 ) @ ( set_Pr6159641702654731441term_a @ ( zip_Gr4568060137147143192term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1025_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a,X2: ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
       => ~ ! [Y4: ground_gmctxt_a] :
              ~ ( member3925204351666028307ctxt_a @ ( produc6025636961954769526ctxt_a @ X2 @ Y4 ) @ ( set_Pr9042926802698645415ctxt_a @ ( zip_Gr2700541357751180118ctxt_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1026_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ( member5261315023172567896term_a @ X2 @ ( set_Ground_gterm_a2 @ Xs ) )
       => ~ ! [Y4: ground_gterm_a] :
              ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X2 @ Y4 ) @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1027_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1028_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_nat,Y3: nat] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X: ground_gmctxt_a] :
              ~ ( member1226896217267473912_a_nat @ ( produc2030840239341956231_a_nat @ X @ Y3 ) @ ( set_Pr2336437649969777252_a_nat @ ( zip_Gr434538988109263783_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1029_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_nat,Y3: nat] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X: ground_gterm_a] :
              ~ ( member2899317942255798213_a_nat @ ( produc6032213142089859112_a_nat @ X @ Y3 ) @ ( set_Pr8703961316559450201_a_nat @ ( zip_Gr5583948589966438664_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1030_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gmctxt_a,Y3: ground_gmctxt_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( member3306386364193653941ctxt_a @ Y3 @ ( set_Ground_gmctxt_a2 @ Ys ) )
       => ~ ! [X: nat] :
              ~ ( member2885886537299685624ctxt_a @ ( produc3104537575318011527ctxt_a @ X @ Y3 ) @ ( set_Pr3995427970001988964ctxt_a @ ( zip_na1508236324085319079ctxt_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1031_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_Ground_gterm_a,Y3: ground_gterm_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ( member5261315023172567896term_a @ Y3 @ ( set_Ground_gterm_a2 @ Ys ) )
       => ~ ! [X: nat] :
              ~ ( member6622040750283622891term_a @ ( produc4697074808646628614term_a @ X @ Y3 ) @ ( set_Pr3203312087732499071term_a @ ( zip_na4248810256523208166term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1032_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,Y3: nat] :
      ( ( ( size_s8208510060688613859at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X: nat > nat] :
              ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1033_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,Y3: ground_gmctxt_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( member3306386364193653941ctxt_a @ Y3 @ ( set_Ground_gmctxt_a2 @ Ys ) )
       => ~ ! [X: ground_gmctxt_a] :
              ~ ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ X @ Y3 ) @ ( set_Pr5641390345007089586ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1034_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a,Y3: ground_gterm_a] :
      ( ( ( size_s962548663205356226ctxt_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ( member5261315023172567896term_a @ Y3 @ ( set_Ground_gterm_a2 @ Ys ) )
       => ~ ! [X: ground_gmctxt_a] :
              ~ ( member1041919251622114333term_a @ ( produc7893155741350732600term_a @ X @ Y3 ) @ ( set_Pr6159641702654731441term_a @ ( zip_Gr4568060137147143192term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1035_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a,Y3: ground_gmctxt_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s962548663205356226ctxt_a @ Ys ) )
     => ( ( member3306386364193653941ctxt_a @ Y3 @ ( set_Ground_gmctxt_a2 @ Ys ) )
       => ~ ! [X: ground_gterm_a] :
              ~ ( member3925204351666028307ctxt_a @ ( produc6025636961954769526ctxt_a @ X @ Y3 ) @ ( set_Pr9042926802698645415ctxt_a @ ( zip_Gr2700541357751180118ctxt_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1036_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,Y3: ground_gterm_a] :
      ( ( ( size_s1893836384466666913term_a @ Xs )
        = ( size_s1893836384466666913term_a @ Ys ) )
     => ( ( member5261315023172567896term_a @ Y3 @ ( set_Ground_gterm_a2 @ Ys ) )
       => ~ ! [X: ground_gterm_a] :
              ~ ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ X @ Y3 ) @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1037_nth__append,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N )
          = ( nth_Ground_gmctxt_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( nth_Ground_gmctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N )
          = ( nth_Ground_gmctxt_a @ Ys @ ( minus_minus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1038_nth__append,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
       => ( ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ N )
          = ( nth_Ground_gterm_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
       => ( ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ N )
          = ( nth_Ground_gterm_a @ Ys @ ( minus_minus_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1039_nth__append,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
       => ( ( nth_li6172171122907299772term_a @ ( append488786809011849168term_a @ Xs @ Ys ) @ N )
          = ( nth_li6172171122907299772term_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
       => ( ( nth_li6172171122907299772term_a @ ( append488786809011849168term_a @ Xs @ Ys ) @ N )
          = ( nth_li6172171122907299772term_a @ Ys @ ( minus_minus_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1040_nth__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1041_nth__append,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
       => ( ( nth_Pr5943965302187034734term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) @ N )
          = ( nth_Pr5943965302187034734term_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
       => ( ( nth_Pr5943965302187034734term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) @ N )
          = ( nth_Pr5943965302187034734term_a @ Ys @ ( minus_minus_nat @ N @ ( size_s281034491502894169term_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1042_list__update__append,axiom,
    ! [N: nat,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a,X2: ground_gmctxt_a] :
      ( ( ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( list_u7950812632479372336ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N @ X2 )
          = ( append5238750768297430211ctxt_a @ ( list_u7950812632479372336ctxt_a @ Xs @ N @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) )
       => ( ( list_u7950812632479372336ctxt_a @ ( append5238750768297430211ctxt_a @ Xs @ Ys ) @ N @ X2 )
          = ( append5238750768297430211ctxt_a @ Xs @ ( list_u7950812632479372336ctxt_a @ Ys @ ( minus_minus_nat @ N @ ( size_s962548663205356226ctxt_a @ Xs ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_1043_list__update__append,axiom,
    ! [N: nat,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a,X2: ground_gterm_a] :
      ( ( ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
       => ( ( list_u6582344683876525341term_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ N @ X2 )
          = ( append7798372865478268874term_a @ ( list_u6582344683876525341term_a @ Xs @ N @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) )
       => ( ( list_u6582344683876525341term_a @ ( append7798372865478268874term_a @ Xs @ Ys ) @ N @ X2 )
          = ( append7798372865478268874term_a @ Xs @ ( list_u6582344683876525341term_a @ Ys @ ( minus_minus_nat @ N @ ( size_s1893836384466666913term_a @ Xs ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_1044_list__update__append,axiom,
    ! [N: nat,Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a,X2: list_Ground_gterm_a] :
      ( ( ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
       => ( ( list_u412135983231242147term_a @ ( append488786809011849168term_a @ Xs @ Ys ) @ N @ X2 )
          = ( append488786809011849168term_a @ ( list_u412135983231242147term_a @ Xs @ N @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) )
       => ( ( list_u412135983231242147term_a @ ( append488786809011849168term_a @ Xs @ Ys ) @ N @ X2 )
          = ( append488786809011849168term_a @ Xs @ ( list_u412135983231242147term_a @ Ys @ ( minus_minus_nat @ N @ ( size_s5321119603959729319term_a @ Xs ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_1045_list__update__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
          = ( append_nat @ ( list_update_nat @ Xs @ N @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X2 )
          = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_1046_list__update__append,axiom,
    ! [N: nat,Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a] :
      ( ( ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
       => ( ( list_u8102571067576484949term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) @ N @ X2 )
          = ( append7909858224561987714term_a @ ( list_u8102571067576484949term_a @ Xs @ N @ X2 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s281034491502894169term_a @ Xs ) )
       => ( ( list_u8102571067576484949term_a @ ( append7909858224561987714term_a @ Xs @ Ys ) @ N @ X2 )
          = ( append7909858224561987714term_a @ Xs @ ( list_u8102571067576484949term_a @ Ys @ ( minus_minus_nat @ N @ ( size_s281034491502894169term_a @ Xs ) ) @ X2 ) ) ) ) ) ).

% list_update_append
thf(fact_1047_in__set__zip,axiom,
    ! [P2: product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_nat @ Xs @ N2 )
              = ( product_fst_nat_nat @ P2 ) )
            & ( ( nth_nat @ Ys @ N2 )
              = ( product_snd_nat_nat @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1048_in__set__zip,axiom,
    ! [P2: produc8497099580089875791_a_nat,Xs: list_Ground_gmctxt_a,Ys: list_nat] :
      ( ( member1226896217267473912_a_nat @ P2 @ ( set_Pr2336437649969777252_a_nat @ ( zip_Gr434538988109263783_a_nat @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Ground_gmctxt_a @ Xs @ N2 )
              = ( produc7558069788972888179_a_nat @ P2 ) )
            & ( ( nth_nat @ Ys @ N2 )
              = ( produc2479047237538536629_a_nat @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1049_in__set__zip,axiom,
    ! [P2: produc3386756434963026606_a_nat,Xs: list_Ground_gterm_a,Ys: list_nat] :
      ( ( member2899317942255798213_a_nat @ P2 @ ( set_Pr8703961316559450201_a_nat @ ( zip_Gr5583948589966438664_a_nat @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Ground_gterm_a @ Xs @ N2 )
              = ( produc8007197682390958396_a_nat @ P2 ) )
            & ( ( nth_nat @ Ys @ N2 )
              = ( produc5543064479622145146_a_nat @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1050_in__set__zip,axiom,
    ! [P2: produc932717863267311695ctxt_a,Xs: list_nat,Ys: list_Ground_gmctxt_a] :
      ( ( member2885886537299685624ctxt_a @ P2 @ ( set_Pr3995427970001988964ctxt_a @ ( zip_na1508236324085319079ctxt_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_nat @ Xs @ N2 )
              = ( produc8631767124948943475ctxt_a @ P2 ) )
            & ( ( nth_Ground_gmctxt_a @ Ys @ N2 )
              = ( produc3552744573514591925ctxt_a @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1051_in__set__zip,axiom,
    ! [P2: produc7109479242990851284term_a,Xs: list_nat,Ys: list_Ground_gterm_a] :
      ( ( member6622040750283622891term_a @ P2 @ ( set_Pr3203312087732499071term_a @ ( zip_na4248810256523208166term_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_nat @ Xs @ N2 )
              = ( produc6672059348947727898term_a @ P2 ) )
            & ( ( nth_Ground_gterm_a @ Ys @ N2 )
              = ( produc4207926146178914648term_a @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1052_in__set__zip,axiom,
    ! [P2: produc8175205341597284509ctxt_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( member461722496487281478ctxt_a @ P2 @ ( set_Pr5641390345007089586ctxt_a @ ( zip_Gr6926631713258005493ctxt_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Ground_gmctxt_a @ Xs @ N2 )
              = ( produc844137399493384385ctxt_a @ P2 ) )
            & ( ( nth_Ground_gmctxt_a @ Ys @ N2 )
              = ( produc3308216634374912259ctxt_a @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1053_in__set__zip,axiom,
    ! [P2: produc4612632926506977542term_a,Xs: list_Ground_gmctxt_a,Ys: list_Ground_gterm_a] :
      ( ( member1041919251622114333term_a @ P2 @ ( set_Pr6159641702654731441term_a @ ( zip_Gr4568060137147143192term_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Ground_gmctxt_a @ Xs @ N2 )
              = ( produc3363849087924776012term_a @ P2 ) )
            & ( ( nth_Ground_gterm_a @ Ys @ N2 )
              = ( produc2223882984850283402term_a @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1054_in__set__zip,axiom,
    ! [P2: produc7495918026550891516ctxt_a,Xs: list_Ground_gterm_a,Ys: list_Ground_gmctxt_a] :
      ( ( member3925204351666028307ctxt_a @ P2 @ ( set_Pr9042926802698645415ctxt_a @ ( zip_Gr2700541357751180118ctxt_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Ground_gterm_a @ Xs @ N2 )
              = ( produc1496330308528812938ctxt_a @ P2 ) )
            & ( ( nth_Ground_gmctxt_a @ Ys @ N2 )
              = ( produc356364205454320328ctxt_a @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s962548663205356226ctxt_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1055_in__set__zip,axiom,
    ! [P2: produc1525979690750518759term_a,Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( member4351891425000572176term_a @ P2 @ ( set_Pr5405221470436175740term_a @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_Ground_gterm_a @ Xs @ N2 )
              = ( produc5097907526672912195term_a @ P2 ) )
            & ( ( nth_Ground_gterm_a @ Ys @ N2 )
              = ( produc5323275678763266181term_a @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_s1893836384466666913term_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1056_in__set__zip,axiom,
    ! [P2: produc1258604012075459764_a_nat,Xs: list_l4678906206682822203term_a,Ys: list_nat] :
      ( ( member4055828667005674059_a_nat @ P2 @ ( set_Pr7424497439565050079_a_nat @ ( zip_li4224161369154485902_a_nat @ Xs @ Ys ) ) )
      = ( ? [N2: nat] :
            ( ( ( nth_li6172171122907299772term_a @ Xs @ N2 )
              = ( produc1636621001250972866_a_nat @ P2 ) )
            & ( ( nth_nat @ Ys @ N2 )
              = ( produc4911619286566473984_a_nat @ P2 ) )
            & ( ord_less_nat @ N2 @ ( size_s5321119603959729319term_a @ Xs ) )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1057_less__sup__gmctxt__args__funas__gmctxt,axiom,
    ! [C: ground_gmctxt_a,D: ground_gmctxt_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le1940079914304802430ctxt_a @ C @ D )
     => ( ( ord_le8666007276011122963_a_nat @ ( ground4323624445104842080ctxt_a @ C ) @ F5 )
       => ( ! [X: ground_gmctxt_a] :
              ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ ( ground7778985915038999110args_a @ C @ D ) ) )
             => ( ord_le8666007276011122963_a_nat @ ( ground4323624445104842080ctxt_a @ X ) @ F5 ) )
         => ( ord_le8666007276011122963_a_nat @ ( ground4323624445104842080ctxt_a @ D ) @ F5 ) ) ) ) ).

% less_sup_gmctxt_args_funas_gmctxt
thf(fact_1058_partition2__sound__nP,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( produc7615485059283090437st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_nat @ ( produc1382935764643595205st_nat @ ( produc7615485059283090437st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc5865812112468994567st_nat @ ( produc7615485059283090437st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1059_partition2__sound__nP,axiom,
    ! [P: ground_gmctxt_a > nat > $o,As: list_Ground_gmctxt_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc4208855429218208275st_nat @ ( produc5378828341999431813st_nat @ ( missin7537391439621851856_a_nat @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_Ground_gmctxt_a @ ( produc4208855429218208275st_nat @ ( produc5378828341999431813st_nat @ ( missin7537391439621851856_a_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc317929709919118421st_nat @ ( produc5378828341999431813st_nat @ ( missin7537391439621851856_a_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1060_partition2__sound__nP,axiom,
    ! [P: ground_gterm_a > nat > $o,As: list_Ground_gterm_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc2323902495782985426st_nat @ ( produc2378332124388746191st_nat @ ( missin723996838057557919_a_nat @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_Ground_gterm_a @ ( produc2323902495782985426st_nat @ ( produc2378332124388746191st_nat @ ( missin723996838057557919_a_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc1106367634080142352st_nat @ ( produc2378332124388746191st_nat @ ( missin723996838057557919_a_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1061_partition2__sound__nP,axiom,
    ! [P: nat > ground_gterm_a > $o,As: list_nat,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc3239241062991153424term_a @ ( produc6410121591457334235term_a @ ( missin8612230541469103229term_a @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_nat @ ( produc3239241062991153424term_a @ ( produc6410121591457334235term_a @ ( missin8612230541469103229term_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc2021706201288310350term_a @ ( produc6410121591457334235term_a @ ( missin8612230541469103229term_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1062_partition2__sound__nP,axiom,
    ! [P: nat > ground_gmctxt_a > $o,As: list_nat,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc8940406564501060371ctxt_a @ ( produc1946485147793191045ctxt_a @ ( missin8611088775597907152ctxt_a @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_nat @ ( produc8940406564501060371ctxt_a @ ( produc1946485147793191045ctxt_a @ ( missin8611088775597907152ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc5049480845201970517ctxt_a @ ( produc1946485147793191045ctxt_a @ ( missin8611088775597907152ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1063_partition2__sound__nP,axiom,
    ! [P: ground_gmctxt_a > ground_gterm_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc8197280458369426242term_a @ ( produc2844248634981145407term_a @ ( missin7282809207517305007term_a @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_Ground_gmctxt_a @ ( produc8197280458369426242term_a @ ( produc2844248634981145407term_a @ ( missin7282809207517305007term_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc7064337315879363712term_a @ ( produc2844248634981145407term_a @ ( missin7282809207517305007term_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1064_partition2__sound__nP,axiom,
    ! [P: ground_gmctxt_a > ground_gmctxt_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc7598568373860330849ctxt_a @ ( produc6377299910269472005ctxt_a @ ( missin6883611333491606814ctxt_a @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_Ground_gmctxt_a @ ( produc7598568373860330849ctxt_a @ ( produc6377299910269472005ctxt_a @ ( missin6883611333491606814ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc369865107177110435ctxt_a @ ( produc6377299910269472005ctxt_a @ ( missin6883611333491606814ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1065_partition2__sound__nP,axiom,
    ! [P: ground_gterm_a > ground_gterm_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc139599297396156611term_a @ ( produc786530669614394501term_a @ ( missin8685998045867055648term_a @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_Ground_gterm_a @ ( produc139599297396156611term_a @ ( produc786530669614394501term_a @ ( missin8685998045867055648term_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc7009130510273001477term_a @ ( produc786530669614394501term_a @ ( missin8685998045867055648term_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1066_partition2__sound__nP,axiom,
    ! [P: ground_gterm_a > ground_gmctxt_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc4776777761837811744ctxt_a @ ( produc5139866272286260587ctxt_a @ ( missin5415290428121341933ctxt_a @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_Ground_gterm_a @ ( produc4776777761837811744ctxt_a @ ( produc5139866272286260587ctxt_a @ ( missin5415290428121341933ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc3643834619347749214ctxt_a @ ( produc5139866272286260587ctxt_a @ ( missin5415290428121341933ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1067_partition2__sound__nP,axiom,
    ! [P: list_Ground_gterm_a > nat > $o,As: list_l4678906206682822203term_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ ( produc4097921998028667608st_nat @ ( produc7879605611266285659st_nat @ ( missin8017854170469291429_a_nat @ P @ As @ Bs ) ) ) ) )
     => ~ ( P @ ( nth_li6172171122907299772term_a @ ( produc4097921998028667608st_nat @ ( produc7879605611266285659st_nat @ ( missin8017854170469291429_a_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc4169894101998860566st_nat @ ( produc7879605611266285659st_nat @ ( missin8017854170469291429_a_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_nP
thf(fact_1068_partition2__sound__P,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( produc7458156121412276931st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_nat @ ( produc1382935764643595205st_nat @ ( produc7458156121412276931st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc5865812112468994567st_nat @ ( produc7458156121412276931st_nat @ ( missin5516618432537284226at_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1069_partition2__sound__P,axiom,
    ! [P: ground_gmctxt_a > nat > $o,As: list_Ground_gmctxt_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc4208855429218208275st_nat @ ( produc5577118523729256259st_nat @ ( missin7537391439621851856_a_nat @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_Ground_gmctxt_a @ ( produc4208855429218208275st_nat @ ( produc5577118523729256259st_nat @ ( missin7537391439621851856_a_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc317929709919118421st_nat @ ( produc5577118523729256259st_nat @ ( missin7537391439621851856_a_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1070_partition2__sound__P,axiom,
    ! [P: ground_gterm_a > nat > $o,As: list_Ground_gterm_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc2323902495782985426st_nat @ ( produc5901039043000570253st_nat @ ( missin723996838057557919_a_nat @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_Ground_gterm_a @ ( produc2323902495782985426st_nat @ ( produc5901039043000570253st_nat @ ( missin723996838057557919_a_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc1106367634080142352st_nat @ ( produc5901039043000570253st_nat @ ( missin723996838057557919_a_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1071_partition2__sound__P,axiom,
    ! [P: nat > ground_gterm_a > $o,As: list_nat,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc3239241062991153424term_a @ ( produc709456473214382489term_a @ ( missin8612230541469103229term_a @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_nat @ ( produc3239241062991153424term_a @ ( produc709456473214382489term_a @ ( missin8612230541469103229term_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc2021706201288310350term_a @ ( produc709456473214382489term_a @ ( missin8612230541469103229term_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1072_partition2__sound__P,axiom,
    ! [P: nat > ground_gmctxt_a > $o,As: list_nat,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc8940406564501060371ctxt_a @ ( produc2144775329523015491ctxt_a @ ( missin8611088775597907152ctxt_a @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_nat @ ( produc8940406564501060371ctxt_a @ ( produc2144775329523015491ctxt_a @ ( missin8611088775597907152ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc5049480845201970517ctxt_a @ ( produc2144775329523015491ctxt_a @ ( missin8611088775597907152ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1073_partition2__sound__P,axiom,
    ! [P: ground_gmctxt_a > ground_gterm_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc8197280458369426242term_a @ ( produc2289174639566004477term_a @ ( missin7282809207517305007term_a @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_Ground_gmctxt_a @ ( produc8197280458369426242term_a @ ( produc2289174639566004477term_a @ ( missin7282809207517305007term_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc7064337315879363712term_a @ ( produc2289174639566004477term_a @ ( missin7282809207517305007term_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1074_partition2__sound__P,axiom,
    ! [P: ground_gmctxt_a > ground_gmctxt_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc7598568373860330849ctxt_a @ ( produc7323287651441988547ctxt_a @ ( missin6883611333491606814ctxt_a @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_Ground_gmctxt_a @ ( produc7598568373860330849ctxt_a @ ( produc7323287651441988547ctxt_a @ ( missin6883611333491606814ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc369865107177110435ctxt_a @ ( produc7323287651441988547ctxt_a @ ( missin6883611333491606814ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1075_partition2__sound__P,axiom,
    ! [P: ground_gterm_a > ground_gterm_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc139599297396156611term_a @ ( produc770064522850120003term_a @ ( missin8685998045867055648term_a @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_Ground_gterm_a @ ( produc139599297396156611term_a @ ( produc770064522850120003term_a @ ( missin8685998045867055648term_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc7009130510273001477term_a @ ( produc770064522850120003term_a @ ( missin8685998045867055648term_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1076_partition2__sound__P,axiom,
    ! [P: ground_gterm_a > ground_gmctxt_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc4776777761837811744ctxt_a @ ( produc4584792276871119657ctxt_a @ ( missin5415290428121341933ctxt_a @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_Ground_gterm_a @ ( produc4776777761837811744ctxt_a @ ( produc4584792276871119657ctxt_a @ ( missin5415290428121341933ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc3643834619347749214ctxt_a @ ( produc4584792276871119657ctxt_a @ ( missin5415290428121341933ctxt_a @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1077_partition2__sound__P,axiom,
    ! [P: list_Ground_gterm_a > nat > $o,As: list_l4678906206682822203term_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ ( produc4097921998028667608st_nat @ ( produc774801194911817753st_nat @ ( missin8017854170469291429_a_nat @ P @ As @ Bs ) ) ) ) )
     => ( P @ ( nth_li6172171122907299772term_a @ ( produc4097921998028667608st_nat @ ( produc774801194911817753st_nat @ ( missin8017854170469291429_a_nat @ P @ As @ Bs ) ) ) @ I2 ) @ ( nth_nat @ ( produc4169894101998860566st_nat @ ( produc774801194911817753st_nat @ ( missin8017854170469291429_a_nat @ P @ As @ Bs ) ) ) @ I2 ) ) ) ).

% partition2_sound_P
thf(fact_1078_filter2__sound,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_nat @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) @ I2 ) @ ( nth_nat @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1079_filter2__sound,axiom,
    ! [P: ground_gmctxt_a > nat > $o,As: list_Ground_gmctxt_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc4208855429218208275st_nat @ ( missin6138776448348237362_a_nat @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_Ground_gmctxt_a @ ( produc4208855429218208275st_nat @ ( missin6138776448348237362_a_nat @ P @ As @ Bs ) ) @ I2 ) @ ( nth_nat @ ( produc317929709919118421st_nat @ ( missin6138776448348237362_a_nat @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1080_filter2__sound,axiom,
    ! [P: ground_gterm_a > nat > $o,As: list_Ground_gterm_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc2323902495782985426st_nat @ ( missin8695283833933308861_a_nat @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_Ground_gterm_a @ ( produc2323902495782985426st_nat @ ( missin8695283833933308861_a_nat @ P @ As @ Bs ) ) @ I2 ) @ ( nth_nat @ ( produc1106367634080142352st_nat @ ( missin8695283833933308861_a_nat @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1081_filter2__sound,axiom,
    ! [P: nat > ground_gterm_a > $o,As: list_nat,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc3239241062991153424term_a @ ( missin7360145500490078363term_a @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_nat @ ( produc3239241062991153424term_a @ ( missin7360145500490078363term_a @ P @ As @ Bs ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc2021706201288310350term_a @ ( missin7360145500490078363term_a @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1082_filter2__sound,axiom,
    ! [P: nat > ground_gmctxt_a > $o,As: list_nat,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ ( produc8940406564501060371ctxt_a @ ( missin7212473784324292658ctxt_a @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_nat @ ( produc8940406564501060371ctxt_a @ ( missin7212473784324292658ctxt_a @ P @ As @ Bs ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc5049480845201970517ctxt_a @ ( missin7212473784324292658ctxt_a @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1083_filter2__sound,axiom,
    ! [P: ground_gmctxt_a > ground_gterm_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc8197280458369426242term_a @ ( missin2184941372115355853term_a @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_Ground_gmctxt_a @ ( produc8197280458369426242term_a @ ( missin2184941372115355853term_a @ P @ As @ Bs ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc7064337315879363712term_a @ ( missin2184941372115355853term_a @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1084_filter2__sound,axiom,
    ! [P: ground_gmctxt_a > ground_gmctxt_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s962548663205356226ctxt_a @ ( produc7598568373860330849ctxt_a @ ( missin3651417044635524224ctxt_a @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_Ground_gmctxt_a @ ( produc7598568373860330849ctxt_a @ ( missin3651417044635524224ctxt_a @ P @ As @ Bs ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc369865107177110435ctxt_a @ ( missin3651417044635524224ctxt_a @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1085_filter2__sound,axiom,
    ! [P: ground_gterm_a > ground_gterm_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gterm_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc139599297396156611term_a @ ( missin2697948438808920706term_a @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_Ground_gterm_a @ ( produc139599297396156611term_a @ ( missin2697948438808920706term_a @ P @ As @ Bs ) ) @ I2 ) @ ( nth_Ground_gterm_a @ ( produc7009130510273001477term_a @ ( missin2697948438808920706term_a @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1086_filter2__sound,axiom,
    ! [P: ground_gterm_a > ground_gmctxt_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gmctxt_a,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1893836384466666913term_a @ ( produc4776777761837811744ctxt_a @ ( missin317422592719392779ctxt_a @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_Ground_gterm_a @ ( produc4776777761837811744ctxt_a @ ( missin317422592719392779ctxt_a @ P @ As @ Bs ) ) @ I2 ) @ ( nth_Ground_gmctxt_a @ ( produc3643834619347749214ctxt_a @ ( missin317422592719392779ctxt_a @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1087_filter2__sound,axiom,
    ! [P: list_Ground_gterm_a > nat > $o,As: list_l4678906206682822203term_a,Bs: list_nat,I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s5321119603959729319term_a @ ( produc4097921998028667608st_nat @ ( missin490755688877953731_a_nat @ P @ As @ Bs ) ) ) )
     => ( P @ ( nth_li6172171122907299772term_a @ ( produc4097921998028667608st_nat @ ( missin490755688877953731_a_nat @ P @ As @ Bs ) ) @ I2 ) @ ( nth_nat @ ( produc4169894101998860566st_nat @ ( missin490755688877953731_a_nat @ P @ As @ Bs ) ) @ I2 ) ) ) ).

% filter2_sound
thf(fact_1088_funas__gterm__gterm__of__gmctxt,axiom,
    ! [C: ground_gmctxt_a] :
      ( ( ( ground_num_gholes_a @ C )
        = zero_zero_nat )
     => ( ( ground_funas_gterm_a @ ( ground2297065866753446215ctxt_a @ C ) )
        = ( ground4323624445104842080ctxt_a @ C ) ) ) ).

% funas_gterm_gterm_of_gmctxt
thf(fact_1089_set__list__diff,axiom,
    ! [Xs: list_Ground_gmctxt_a,Ys: list_Ground_gmctxt_a] :
      ( ( set_Ground_gmctxt_a2 @ ( missin6797617701560608538ctxt_a @ Xs @ Ys ) )
      = ( minus_4616606335744337325ctxt_a @ ( set_Ground_gmctxt_a2 @ Xs ) @ ( set_Ground_gmctxt_a2 @ Ys ) ) ) ).

% set_list_diff
thf(fact_1090_set__list__diff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( missin818507234016924876ff_nat @ Xs @ Ys ) )
      = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_list_diff
thf(fact_1091_set__list__diff,axiom,
    ! [Xs: list_l4678906206682822203term_a,Ys: list_l4678906206682822203term_a] :
      ( ( set_li7546925467519454922term_a @ ( missin6335751071628959545term_a @ Xs @ Ys ) )
      = ( minus_3383382438752481404term_a @ ( set_li7546925467519454922term_a @ Xs ) @ ( set_li7546925467519454922term_a @ Ys ) ) ) ).

% set_list_diff
thf(fact_1092_set__list__diff,axiom,
    ! [Xs: list_Ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( set_Ground_gterm_a2 @ ( missin8668443827398277043term_a @ Xs @ Ys ) )
      = ( minus_80403276131473398term_a @ ( set_Ground_gterm_a2 @ Xs ) @ ( set_Ground_gterm_a2 @ Ys ) ) ) ).

% set_list_diff
thf(fact_1093_set__list__diff,axiom,
    ! [Xs: list_P2433084232546440685term_a,Ys: list_P2433084232546440685term_a] :
      ( ( set_Pr5405221470436175740term_a @ ( missin8456351267525757931term_a @ Xs @ Ys ) )
      = ( minus_551215046067487790term_a @ ( set_Pr5405221470436175740term_a @ Xs ) @ ( set_Pr5405221470436175740term_a @ Ys ) ) ) ).

% set_list_diff
thf(fact_1094_filter2_Osimps_I1_J,axiom,
    ! [P: nat > nat > $o,Uu: list_nat] :
      ( ( missin7656342814955748836at_nat @ P @ nil_nat @ Uu )
      = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).

% filter2.simps(1)
thf(fact_1095__092_060T_062_092_060_094sub_062G__funas__gterm__conv,axiom,
    ! [S: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( member5261315023172567896term_a @ S @ ( ground_T_G_a @ F5 ) )
      = ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ S ) @ F5 ) ) ).

% \<T>\<^sub>G_funas_gterm_conv
thf(fact_1096__092_060T_062_092_060_094sub_062G__complete,axiom,
    ! [S: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ S ) @ F5 )
     => ( member5261315023172567896term_a @ S @ ( ground_T_G_a @ F5 ) ) ) ).

% \<T>\<^sub>G_complete
thf(fact_1097__092_060T_062_092_060_094sub_062G__sound,axiom,
    ! [S: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat] :
      ( ( member5261315023172567896term_a @ S @ ( ground_T_G_a @ F5 ) )
     => ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ S ) @ F5 ) ) ).

% \<T>\<^sub>G_sound
thf(fact_1098_list__diff_Osimps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( missin818507234016924876ff_nat @ nil_nat @ Ys )
      = nil_nat ) ).

% list_diff.simps(1)
thf(fact_1099_gmctxt__cl__refl,axiom,
    ! [T: ground_gterm_a,F5: set_Pr4934435412358123699_a_nat,R: set_Pr8282532413760598727term_a] :
      ( ( ord_le8666007276011122963_a_nat @ ( ground_funas_gterm_a @ T ) @ F5 )
     => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ T @ T ) @ ( ground_gmctxt_cl_a @ F5 @ R ) ) ) ).

% gmctxt_cl_refl
thf(fact_1100_filter2__length,axiom,
    ! [P: nat > nat > $o,As: list_nat,Bs: list_nat] :
      ( ( size_size_list_nat @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) )
      = ( size_size_list_nat @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1101_filter2__length,axiom,
    ! [P: ground_gmctxt_a > nat > $o,As: list_Ground_gmctxt_a,Bs: list_nat] :
      ( ( size_s962548663205356226ctxt_a @ ( produc4208855429218208275st_nat @ ( missin6138776448348237362_a_nat @ P @ As @ Bs ) ) )
      = ( size_size_list_nat @ ( produc317929709919118421st_nat @ ( missin6138776448348237362_a_nat @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1102_filter2__length,axiom,
    ! [P: ground_gterm_a > nat > $o,As: list_Ground_gterm_a,Bs: list_nat] :
      ( ( size_s1893836384466666913term_a @ ( produc2323902495782985426st_nat @ ( missin8695283833933308861_a_nat @ P @ As @ Bs ) ) )
      = ( size_size_list_nat @ ( produc1106367634080142352st_nat @ ( missin8695283833933308861_a_nat @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1103_filter2__length,axiom,
    ! [P: nat > ground_gmctxt_a > $o,As: list_nat,Bs: list_Ground_gmctxt_a] :
      ( ( size_size_list_nat @ ( produc8940406564501060371ctxt_a @ ( missin7212473784324292658ctxt_a @ P @ As @ Bs ) ) )
      = ( size_s962548663205356226ctxt_a @ ( produc5049480845201970517ctxt_a @ ( missin7212473784324292658ctxt_a @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1104_filter2__length,axiom,
    ! [P: nat > ground_gterm_a > $o,As: list_nat,Bs: list_Ground_gterm_a] :
      ( ( size_size_list_nat @ ( produc3239241062991153424term_a @ ( missin7360145500490078363term_a @ P @ As @ Bs ) ) )
      = ( size_s1893836384466666913term_a @ ( produc2021706201288310350term_a @ ( missin7360145500490078363term_a @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1105_filter2__length,axiom,
    ! [P: ground_gmctxt_a > ground_gmctxt_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gmctxt_a] :
      ( ( size_s962548663205356226ctxt_a @ ( produc7598568373860330849ctxt_a @ ( missin3651417044635524224ctxt_a @ P @ As @ Bs ) ) )
      = ( size_s962548663205356226ctxt_a @ ( produc369865107177110435ctxt_a @ ( missin3651417044635524224ctxt_a @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1106_filter2__length,axiom,
    ! [P: ground_gmctxt_a > ground_gterm_a > $o,As: list_Ground_gmctxt_a,Bs: list_Ground_gterm_a] :
      ( ( size_s962548663205356226ctxt_a @ ( produc8197280458369426242term_a @ ( missin2184941372115355853term_a @ P @ As @ Bs ) ) )
      = ( size_s1893836384466666913term_a @ ( produc7064337315879363712term_a @ ( missin2184941372115355853term_a @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1107_filter2__length,axiom,
    ! [P: ground_gterm_a > ground_gmctxt_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gmctxt_a] :
      ( ( size_s1893836384466666913term_a @ ( produc4776777761837811744ctxt_a @ ( missin317422592719392779ctxt_a @ P @ As @ Bs ) ) )
      = ( size_s962548663205356226ctxt_a @ ( produc3643834619347749214ctxt_a @ ( missin317422592719392779ctxt_a @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1108_filter2__length,axiom,
    ! [P: ground_gterm_a > ground_gterm_a > $o,As: list_Ground_gterm_a,Bs: list_Ground_gterm_a] :
      ( ( size_s1893836384466666913term_a @ ( produc139599297396156611term_a @ ( missin2697948438808920706term_a @ P @ As @ Bs ) ) )
      = ( size_s1893836384466666913term_a @ ( produc7009130510273001477term_a @ ( missin2697948438808920706term_a @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1109_filter2__length,axiom,
    ! [P: list_Ground_gterm_a > nat > $o,As: list_l4678906206682822203term_a,Bs: list_nat] :
      ( ( size_s5321119603959729319term_a @ ( produc4097921998028667608st_nat @ ( missin490755688877953731_a_nat @ P @ As @ Bs ) ) )
      = ( size_size_list_nat @ ( produc4169894101998860566st_nat @ ( missin490755688877953731_a_nat @ P @ As @ Bs ) ) ) ) ).

% filter2_length
thf(fact_1110_filter2_Oelims,axiom,
    ! [X2: nat > nat > $o,Xa: list_nat,Xb: list_nat,Y3: produc1828647624359046049st_nat] :
      ( ( ( missin7656342814955748836at_nat @ X2 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3
           != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) )
       => ( ( ? [V2: nat,Va: list_nat] :
                ( Xa
                = ( cons_nat @ V2 @ Va ) )
           => ( ( Xb = nil_nat )
             => ( Y3
               != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ) )
         => ~ ! [A3: nat,As2: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A3 @ As2 ) )
               => ! [B3: nat,Bs2: list_nat] :
                    ( ( Xb
                      = ( cons_nat @ B3 @ Bs2 ) )
                   => ~ ( ( ( X2 @ A3 @ B3 )
                         => ( Y3
                            = ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ ( produc1382935764643595205st_nat @ ( missin7656342814955748836at_nat @ X2 @ As2 @ Bs2 ) ) ) @ ( cons_nat @ B3 @ ( produc5865812112468994567st_nat @ ( missin7656342814955748836at_nat @ X2 @ As2 @ Bs2 ) ) ) ) ) )
                        & ( ~ ( X2 @ A3 @ B3 )
                         => ( Y3
                            = ( missin7656342814955748836at_nat @ X2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ) ).

% filter2.elims
thf(fact_1111_comp__gmctxt_OGMFun,axiom,
    ! [F: a,G: a,Cs: list_Ground_gmctxt_a,Ds: list_Ground_gmctxt_a] :
      ( ( F = G )
     => ( ( ( size_s962548663205356226ctxt_a @ Cs )
          = ( size_s962548663205356226ctxt_a @ Ds ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds ) )
             => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( nth_Ground_gmctxt_a @ Cs @ I ) @ ( nth_Ground_gmctxt_a @ Ds @ I ) ) @ ground_comp_gmctxt_a ) )
         => ( member461722496487281478ctxt_a @ ( produc6347872285926497493ctxt_a @ ( ground_GMFun_a @ F @ Cs ) @ ( ground_GMFun_a @ G @ Ds ) ) @ ground_comp_gmctxt_a ) ) ) ) ).

% comp_gmctxt.GMFun
thf(fact_1112_concat__eq__append__conv,axiom,
    ! [Xss: list_list_nat,Ys: list_nat,Zs2: list_nat] :
      ( ( ( concat_nat @ Xss )
        = ( append_nat @ Ys @ Zs2 ) )
      = ( ( ( Xss = nil_list_nat )
         => ( ( Ys = nil_nat )
            & ( Zs2 = nil_nat ) ) )
        & ( ( Xss != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs2: list_nat,Xs4: list_nat,Xss2: list_list_nat] :
              ( ( Xss
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs4 ) @ Xss2 ) ) )
              & ( Ys
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
              & ( Zs2
                = ( append_nat @ Xs4 @ ( concat_nat @ Xss2 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1113_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_1114_nth__Cons__0,axiom,
    ! [X2: ground_gterm_a,Xs: list_Ground_gterm_a] :
      ( ( nth_Ground_gterm_a @ ( cons_Ground_gterm_a @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1115_nth__Cons__0,axiom,
    ! [X2: ground_gmctxt_a,Xs: list_Ground_gmctxt_a] :
      ( ( nth_Ground_gmctxt_a @ ( cons_Ground_gmctxt_a @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1116_nth__Cons__0,axiom,
    ! [X2: list_Ground_gterm_a,Xs: list_l4678906206682822203term_a] :
      ( ( nth_li6172171122907299772term_a @ ( cons_l7239566772969696491term_a @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1117_nth__Cons__0,axiom,
    ! [X2: produc1525979690750518759term_a,Xs: list_P2433084232546440685term_a] :
      ( ( nth_Pr5943965302187034734term_a @ ( cons_P5554873272834883997term_a @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1118_nth__Cons__0,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1119_append1__eq__conv,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_1120_zip__Cons__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1121_zip__Cons__Cons,axiom,
    ! [X2: ground_gmctxt_a,Xs: list_Ground_gmctxt_a,Y3: list_Ground_gterm_a,Ys: list_l4678906206682822203term_a] :
      ( ( zip_Gr7514910105460098334term_a @ ( cons_Ground_gmctxt_a @ X2 @ Xs ) @ ( cons_l7239566772969696491term_a @ Y3 @ Ys ) )
      = ( cons_P6376868564677574230term_a @ ( produc4720362732934906942term_a @ X2 @ Y3 ) @ ( zip_Gr7514910105460098334term_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1122_zip__Cons__Cons,axiom,
    ! [X2: ground_gterm_a,Xs: list_Ground_gterm_a,Y3: ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( zip_Gr212557436252222583term_a @ ( cons_Ground_gterm_a @ X2 @ Xs ) @ ( cons_Ground_gterm_a @ Y3 @ Ys ) )
      = ( cons_P5554873272834883997term_a @ ( produc7419039783125785943term_a @ X2 @ Y3 ) @ ( zip_Gr212557436252222583term_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1123_zip__Cons__Cons,axiom,
    ! [X2: nat > nat,Xs: list_nat_nat,Y3: nat,Ys: list_nat] :
      ( ( zip_nat_nat_nat @ ( cons_nat_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
      = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1124_nth__append__length,axiom,
    ! [Xs: list_Ground_gterm_a,X2: ground_gterm_a,Ys: list_Ground_gterm_a] :
      ( ( nth_Ground_gterm_a @ ( append7798372865478268874term_a @ Xs @ ( cons_Ground_gterm_a @ X2 @ Ys ) ) @ ( size_s1893836384466666913term_a @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_1125_nth__append__length,axiom,
    ! [Xs: list_l4678906206682822203term_a,X2: list_Ground_gterm_a,Ys: list_l4678906206682822203term_a] :
      ( ( nth_li6172171122907299772term_a @ ( append488786809011849168term_a @ Xs @ ( cons_l7239566772969696491term_a @ X2 @ Ys ) ) @ ( size_s5321119603959729319term_a @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_1126_nth__append__length,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_1127_nth__append__length,axiom,
    ! [Xs: list_P2433084232546440685term_a,X2: produc1525979690750518759term_a,Ys: list_P2433084232546440685term_a] :
      ( ( nth_Pr5943965302187034734term_a @ ( append7909858224561987714term_a @ Xs @ ( cons_P5554873272834883997term_a @ X2 @ Ys ) ) @ ( size_s281034491502894169term_a @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_1128_length__upt,axiom,
    ! [I4: nat,J3: nat] :
      ( ( size_size_list_nat @ ( upt @ I4 @ J3 ) )
      = ( minus_minus_nat @ J3 @ I4 ) ) ).

% length_upt
thf(fact_1129_nth__upt,axiom,
    ! [I4: nat,K: nat,J3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I4 @ K ) @ J3 )
     => ( ( nth_nat @ ( upt @ I4 @ J3 ) @ K )
        = ( plus_plus_nat @ I4 @ K ) ) ) ).

% nth_upt
thf(fact_1130_upt__conv__Nil,axiom,
    ! [J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ J3 @ I4 )
     => ( ( upt @ I4 @ J3 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1131_upt__eq__Nil__conv,axiom,
    ! [I4: nat,J3: nat] :
      ( ( ( upt @ I4 @ J3 )
        = nil_nat )
      = ( ( J3 = zero_zero_nat )
        | ( ord_less_eq_nat @ J3 @ I4 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1132_upt__add__eq__append,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( upt @ I4 @ ( plus_plus_nat @ J3 @ K ) )
        = ( append_nat @ ( upt @ I4 @ J3 ) @ ( upt @ J3 @ ( plus_plus_nat @ J3 @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_1133_upt__0,axiom,
    ! [I4: nat] :
      ( ( upt @ I4 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1134_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1135_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_1136_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_1137_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1138_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1139_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1140_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_1141_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1142_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1143_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1144_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1145_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1146_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) @ I4 )
        = ( minus_minus_nat @ ( suc @ J3 ) @ ( plus_plus_nat @ K @ I4 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1147_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J3: nat,I4: nat] :
      ( ( ord_less_eq_nat @ K @ J3 )
     => ( ( minus_minus_nat @ I4 @ ( suc @ ( minus_minus_nat @ J3 @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I4 @ K ) @ ( suc @ J3 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1148_Nat_OlessE,axiom,
    ! [I4: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ K )
     => ( ( K
         != ( suc @ I4 ) )
       => ~ ! [J: nat] :
              ( ( ord_less_nat @ I4 @ J )
             => ( K
               != ( suc @ J ) ) ) ) ) ).

% Nat.lessE
thf(fact_1149_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1150_Suc__lessE,axiom,
    ! [I4: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I4 ) @ K )
     => ~ ! [J: nat] :
            ( ( ord_less_nat @ I4 @ J )
           => ( K
             != ( suc @ J ) ) ) ) ).

% Suc_lessE
thf(fact_1151_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1152_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1153_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1154_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1155_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1156_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1157_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_1158_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1159_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1160_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1161_less__trans__Suc,axiom,
    ! [I4: nat,J3: nat,K: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ( ord_less_nat @ J3 @ K )
       => ( ord_less_nat @ ( suc @ I4 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1162_less__Suc__induct,axiom,
    ! [I4: nat,J3: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
       => ( ! [I: nat,J: nat,K4: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( ( ord_less_nat @ J @ K4 )
               => ( ( P @ I @ J )
                 => ( ( P @ J @ K4 )
                   => ( P @ I @ K4 ) ) ) ) )
         => ( P @ I4 @ J3 ) ) ) ) ).

% less_Suc_induct
thf(fact_1163_strict__inc__induct,axiom,
    ! [I4: nat,J3: nat,P: nat > $o] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ! [I: nat] :
            ( ( J3
              = ( suc @ I ) )
           => ( P @ I ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ J3 )
             => ( ( P @ ( suc @ I ) )
               => ( P @ I ) ) )
         => ( P @ I4 ) ) ) ) ).

% strict_inc_induct
thf(fact_1164_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1165_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1166_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1167_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J4: nat] :
            ( ( M
              = ( suc @ J4 ) )
            & ( ord_less_nat @ J4 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1168_ex__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% ex_Suc_conv
thf(fact_1169_all__Suc__conv,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% all_Suc_conv
thf(fact_1170_all__less__two,axiom,
    ! [P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ ( suc @ zero_zero_nat ) ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ( P @ ( suc @ zero_zero_nat ) ) ) ) ).

% all_less_two
thf(fact_1171_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1172_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1173_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K4: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1174_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K5: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M3 @ K5 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1175_less__add__Suc2,axiom,
    ! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ M @ I4 ) ) ) ).

% less_add_Suc2
thf(fact_1176_less__add__Suc1,axiom,
    ! [I4: nat,M: nat] : ( ord_less_nat @ I4 @ ( suc @ ( plus_plus_nat @ I4 @ M ) ) ) ).

% less_add_Suc1
thf(fact_1177_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).

% less_natE
thf(fact_1178_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1179_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1180_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I4: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I4 ) ) ) ) ).

% zero_induct_lemma
thf(fact_1181_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1182_Suc__inject,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y3 ) )
     => ( X2 = Y3 ) ) ).

% Suc_inject
thf(fact_1183_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1184_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1185_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1186_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1187_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1188_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1189_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1190_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1191_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X: nat] : ( P @ X @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X: nat,Y4: nat] :
              ( ( P @ X @ Y4 )
             => ( P @ ( suc @ X ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1192_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1193_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1194_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1195_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1196_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1197_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1198_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1199_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1200_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1201_dec__induct,axiom,
    ! [I4: nat,J3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( P @ I4 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I4 @ N3 )
             => ( ( ord_less_nat @ N3 @ J3 )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J3 ) ) ) ) ).

% dec_induct
thf(fact_1202_inc__induct,axiom,
    ! [I4: nat,J3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( P @ J3 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I4 @ N3 )
             => ( ( ord_less_nat @ N3 @ J3 )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I4 ) ) ) ) ).

% inc_induct
thf(fact_1203_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1204_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1205_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1206_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1207_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1208_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R5: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X: nat] : ( R5 @ X @ X )
       => ( ! [X: nat,Y4: nat,Z2: nat] :
              ( ( R5 @ X @ Y4 )
             => ( ( R5 @ Y4 @ Z2 )
               => ( R5 @ X @ Z2 ) ) )
         => ( ! [N3: nat] : ( R5 @ N3 @ ( suc @ N3 ) )
           => ( R5 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1209_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1210_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1211_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1212_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1213_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1214_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M4: nat] :
          ( M6
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1215_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1216_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1217_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1218_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_1219_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K4: nat] :
            ( ( ord_less_nat @ K4 @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K4 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K4 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1220_diff__Suc__less,axiom,
    ! [N: nat,I4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1221_upt__conv__Cons,axiom,
    ! [I4: nat,J3: nat] :
      ( ( ord_less_nat @ I4 @ J3 )
     => ( ( upt @ I4 @ J3 )
        = ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) ) ) ).

% upt_conv_Cons
thf(fact_1222_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J4: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J4 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J4 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1223_upt__Suc__append,axiom,
    ! [I4: nat,J3: nat] :
      ( ( ord_less_eq_nat @ I4 @ J3 )
     => ( ( upt @ I4 @ ( suc @ J3 ) )
        = ( append_nat @ ( upt @ I4 @ J3 ) @ ( cons_nat @ J3 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1224_upt__Suc,axiom,
    ! [I4: nat,J3: nat] :
      ( ( ( ord_less_eq_nat @ I4 @ J3 )
       => ( ( upt @ I4 @ ( suc @ J3 ) )
          = ( append_nat @ ( upt @ I4 @ J3 ) @ ( cons_nat @ J3 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I4 @ J3 )
       => ( ( upt @ I4 @ ( suc @ J3 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1225_inf__concat__simple_Ocases,axiom,
    ! [X2: produc8199716216217303280at_nat] :
      ( ! [F2: nat > nat] :
          ( X2
         != ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
     => ~ ! [F2: nat > nat,N3: nat] :
            ( X2
           != ( produc72220940542539688at_nat @ F2 @ ( suc @ N3 ) ) ) ) ).

% inf_concat_simple.cases

% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [C6: ground_gmctxt_a,Ss4: list_Ground_gterm_a] :
      ( ( ground_eq_gfill_a @ s @ ( produc4720362732934906942term_a @ C6 @ Ss4 ) )
     => ! [Ts4: list_Ground_gterm_a] :
          ( ( ground_eq_gfill_a @ t @ ( produc4720362732934906942term_a @ C6 @ Ts4 ) )
         => ( ( p @ C6 )
           => ! [Ds4: list_Ground_gmctxt_a] :
                ( ! [X: ground_gmctxt_a] :
                    ( ( member3306386364193653941ctxt_a @ X @ ( set_Ground_gmctxt_a2 @ Ds4 ) )
                   => ( q @ X ) )
               => ( ( ( ground_num_gholes_a @ C6 )
                    = ( size_s962548663205356226ctxt_a @ Ds4 ) )
                 => ( ( ( size_s962548663205356226ctxt_a @ Ds4 )
                      = ( size_s1893836384466666913term_a @ Ss4 ) )
                   => ( ( ( size_s1893836384466666913term_a @ Ss4 )
                        = ( size_s1893836384466666913term_a @ Ts4 ) )
                     => ! [Us4: list_l4678906206682822203term_a] :
                          ( ( ( size_s1893836384466666913term_a @ Ts4 )
                            = ( size_s5321119603959729319term_a @ Us4 ) )
                         => ! [Vs3: list_l4678906206682822203term_a] :
                              ( ( ( size_s5321119603959729319term_a @ Us4 )
                                = ( size_s5321119603959729319term_a @ Vs3 ) )
                             => ( ! [I: nat] :
                                    ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds4 ) )
                                   => ( ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ss4 @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds4 @ I ) @ ( nth_li6172171122907299772term_a @ Us4 @ I ) ) )
                                      & ( ground_eq_gfill_a @ ( nth_Ground_gterm_a @ Ts4 @ I ) @ ( produc4720362732934906942term_a @ ( nth_Ground_gmctxt_a @ Ds4 @ I ) @ ( nth_li6172171122907299772term_a @ Vs3 @ I ) ) ) ) )
                               => ( ! [I: nat] :
                                      ( ( ord_less_nat @ I @ ( size_s962548663205356226ctxt_a @ Ds4 ) )
                                     => ! [J: nat] :
                                          ( ( ord_less_nat @ J @ ( size_s1893836384466666913term_a @ ( nth_li6172171122907299772term_a @ Vs3 @ I ) ) )
                                         => ( member4351891425000572176term_a @ ( produc7419039783125785943term_a @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Us4 @ I ) @ J ) @ ( nth_Ground_gterm_a @ ( nth_li6172171122907299772term_a @ Vs3 @ I ) @ J ) ) @ r ) ) )
                                 => thesis ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------