TSTP Solution File: ITP027^2 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP027^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n011.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon May 20 22:22:12 EDT 2024

% Result   : Theorem 259.52s 217.30s
% Output   : Refutation 261.17s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  325
% Syntax   : Number of formulae    :  597 (  42 unt;  55 typ;   0 def)
%            Number of atoms       : 3531 ( 401 equ;   0 cnn)
%            Maximal formula atoms :   30 (   6 avg)
%            Number of connectives : 20351 (  59   ~;   7   |;  54   &;18892   @)
%                                         (   0 <=>;1339  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (  13 avg)
%            Number of types       :    8 (   7 usr)
%            Number of type conns  :  672 ( 672   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   56 (  53 usr;   9 con; 0-11 aty)
%            Number of variables   : 2694 ( 118   ^2452   !;  26   ?;2694   :)
%                                         (  98  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(product_unit_type,type,
    product_unit: $tType ).

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

thf(e_type,type,
    e: $tType ).

thf(d_type,type,
    d: $tType ).

thf(c_type,type,
    c: $tType ).

thf(b_type,type,
    b: $tType ).

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

thf(plus_type,type,
    plus: 
      !>[TA: $tType] : $o ).

thf(ord_type,type,
    ord: 
      !>[TA: $tType] : $o ).

thf(group_add_type,type,
    group_add: 
      !>[TA: $tType] : $o ).

thf(semigroup_add_type,type,
    semigroup_add: 
      !>[TA: $tType] : $o ).

thf(comm_monoid_add_type,type,
    comm_monoid_add: 
      !>[TA: $tType] : $o ).

thf(ab_semigroup_add_type,type,
    ab_semigroup_add: 
      !>[TA: $tType] : $o ).

thf(cancel_semigroup_add_type,type,
    cancel_semigroup_add: 
      !>[TA: $tType] : $o ).

thf(ordere779506340up_add_type,type,
    ordere779506340up_add: 
      !>[TA: $tType] : $o ).

thf(ordere236663937imp_le_type,type,
    ordere236663937imp_le: 
      !>[TA: $tType] : $o ).

thf(carrier_type,type,
    carrier: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ TA ) > ( set @ TB ) ) ).

thf(j_rad_type,type,
    j_rad: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) ) ).

thf(ring_type,type,
    ring: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).

thf(tp_type,type,
    tp: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > TB > TB ) ).

thf(un_type,type,
    un: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB ) ).

thf(sr_type,type,
    sr: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) ) ).

thf(subring_type,type,
    subring: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TB ) ) ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TA ) ) ) > $o ) ).

thf(unit_type,type,
    unit: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > $o ) ).

thf(aGroup_type,type,
    aGroup: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > $o ) ).

thf(mop_type,type,
    mop: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > TB > TB ) ).

thf(pop_type,type,
    pop: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > TB > TB > TB ) ).

thf(aset_sum_type,type,
    aset_sum: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TB ) ) ).

thf(carr_dsumag_type,type,
    carr_dsumag: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( set @ ( TC > TB ) ) ) ).

thf(carr_prodag_type,type,
    carr_prodag: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( set @ ( TC > TB ) ) ) ).

thf(local_ring_type,type,
    local_ring: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).

thf(maximal_ideal_type,type,
    maximal_ideal: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(mul_closed_set_type,type,
    mul_closed_set: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(nscal_type,type,
    nscal: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > TB > nat > TB ) ).

thf(primary_ideal_type,type,
    primary_ideal: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(prod_mOp_type,type,
    prod_mOp: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( TC > TB ) > TC > TB ) ).

thf(prod_pOp_type,type,
    prod_pOp: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > ( TC > TB ) > ( TC > TB ) > TC > TB ) ).

thf(prod_zero_type,type,
    prod_zero: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) ) > TC > TB ) ).

thf(rInvim_type,type,
    rInvim: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( TD > TB ) > ( set @ TB ) > ( set @ TD ) ) ).

thf(sr2_type,type,
    sr2: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(zeroring_type,type,
    zeroring: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).

thf(sum_mult_type,type,
    sum_mult: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TB ) ) ).

thf(module_type,type,
    module: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TA ) ) ) > $o ) ).

thf(sprod_type,type,
    sprod: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TB @ TA ) ) ) > TB > TC > TC ) ).

thf(mHom_type,type,
    mHom: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType] : ( ( carrier_ext @ TF @ ( aGroup_ext @ TF @ ( ring_ext @ TF @ TE ) ) ) > ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TF @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TF @ TA ) ) ) > ( set @ ( TD > TB ) ) ) ).

thf(algebr1000837334_prodM_type,type,
    algebr1000837334_prodM: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType] : ( ( carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TD ) ) ) > ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TA ) ) ) ) > ( carrier_ext @ ( TC > TB ) @ ( aGroup_ext @ ( TC > TB ) @ ( module_ext @ ( TC > TB ) @ TE @ product_unit ) ) ) ) ).

thf(algebr142713364_sprod_type,type,
    algebr142713364_sprod: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType] : ( ( carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TD ) ) ) > ( set @ TC ) > ( TC > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TA ) ) ) ) > TE > ( TC > TB ) > TC > TB ) ).

thf(pi_type,type,
    pi: 
      !>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( TB > ( set @ TA ) ) > ( set @ ( TB > TA ) ) ) ).

thf(plus_plus_type,type,
    plus_plus: 
      !>[TA: $tType] : ( TA > TA > TA ) ).

thf(ord_less_eq_type,type,
    ord_less_eq: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

thf(collect_type,type,
    collect: 
      !>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).

thf(member_type,type,
    member: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).

thf(i_type,type,
    i: set @ c ).

thf(m_type,type,
    m: c > ( carrier_ext @ d @ ( aGroup_ext @ d @ ( module_ext @ d @ a @ e ) ) ) ).

thf(r_type,type,
    r: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) ).

thf(201,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: nat,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ C ) @ ( nscal @ TB @ TA @ A @ B @ D ) )
          = ( nscal @ TB @ TA @ A @ B @ ( plus_plus @ nat @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_Ring_OnsDistr) ).

thf(892,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: nat,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ C ) @ ( nscal @ TB @ TA @ A @ B @ D ) )
          = ( nscal @ TB @ TA @ A @ B @ ( plus_plus @ nat @ C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(180,axiom,
    ! [A: a,B: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( member @ a @ ( nscal @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_nsClose) ).

thf(809,plain,
    ! [A: a,B: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( member @ a @ ( nscal @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(243,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( tp @ TA @ TB @ A @ B @ C )
            = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_Ring_Oring__inv1__3) ).

thf(1024,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( tp @ TA @ TB @ A @ B @ C )
            = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(36,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
      ( ( subring @ TA @ TB @ TC @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_Subring__subset) ).

thf(365,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
      ( ( subring @ TA @ TB @ TC @ A @ B )
     => ( ord_less_eq @ ( set @ TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(190,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ C @ D ) ) )
              = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ D ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ C ) ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_ring__distrib4) ).

thf(846,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ C @ D ) ) )
              = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ D ) ) ) @ ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ C ) ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(174,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ ( TA > TA > TA ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A )
        @ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
          @ ^ [B: TA] :
              ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
              @ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_Ring_Opop__closed) ).

thf(777,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ ( TA > TA > TA ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A )
        @ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
          @ ^ [B: TA] :
              ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
              @ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(270,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TC ) ),C: TA] :
      ( ( module @ TA @ TB @ TD @ TC @ A @ B )
     => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ) @ A ) )
       => ( ( sprod @ TA @ TB @ TD @ A @ ( un @ TB @ TC @ B ) @ C )
          = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_Module_Osprod__one) ).

thf(1110,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TC ) ),C: TA] :
      ( ( module @ TA @ TB @ TD @ TC @ A @ B )
     => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TD ) ) @ A ) )
       => ( ( sprod @ TA @ TB @ TD @ A @ ( un @ TB @ TC @ B ) @ C )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(31,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ A @ D )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_group__cancel_Oadd1) ).

thf(352,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ A @ D )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(205,axiom,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
     => ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_Pi__I_H) ).

thf(909,plain,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
     => ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(123,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_ring__distrib1) ).

thf(628,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(261,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
            = ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_Ring_Oring__inv1__2) ).

thf(1083,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
            = ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(224,axiom,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > TB,D: TA > ( set @ TB )] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( ( B @ E )
            = ( C @ E ) ) )
     => ( ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ D ) )
        = ( member @ ( TA > TB ) @ C @ ( pi @ TA @ TB @ A @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_Pi__cong) ).

thf(962,plain,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > TB,D: TA > ( set @ TB )] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( ( B @ E )
            = ( C @ E ) ) )
     => ( ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ D ) )
        = ( member @ ( TA > TB ) @ C @ ( pi @ TA @ TB @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(55,axiom,
    ord @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_3) ).

thf(408,plain,
    ord @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(140,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( A = B )
         => ( ( nscal @ a @ b @ r @ A @ C )
            = ( nscal @ a @ b @ r @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_nsEqElm) ).

thf(683,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( A = B )
         => ( ( nscal @ a @ b @ r @ A @ C )
            = ( nscal @ a @ b @ r @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(3,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( C = D ) )
         => ( ( plus_plus @ TA @ A @ C )
            = ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_add__mono__thms__linordered__semiring_I4_J) ).

thf(276,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( C = D ) )
         => ( ( plus_plus @ TA @ A @ C )
            = ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(257,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( member @ ( TA > TA )
      @ ^ [B: TA] : B
      @ ( pi @ TA @ TA @ A
        @ ^ [B: TA] : A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_funcset__id) ).

thf(1073,plain,
    ! [TA: $tType,A: set @ TA] :
      ( member @ ( TA > TA )
      @ ^ [B: TA] : B
      @ ( pi @ TA @ TA @ A
        @ ^ [B: TA] : A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(68,axiom,
    ! [TA: $tType,A: set @ TA,B: TA > $o] :
      ( ord_less_eq @ ( set @ TA )
      @ ( collect @ TA
        @ ^ [C: TA] :
            ( ( member @ TA @ C @ A )
            & ( B @ C ) ) )
      @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_conditional__subset) ).

thf(449,plain,
    ! [TA: $tType,A: set @ TA,B: TA > $o] :
      ( ord_less_eq @ ( set @ TA )
      @ ( collect @ TA
        @ ^ [C: TA] :
            ( ( member @ TA @ C @ A )
            & ( B @ C ) ) )
      @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(43,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ( zeroring @ TA @ TB @ A )
       => ~ ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_Ring_Ozeroring__no__maximal) ).

thf(379,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ( zeroring @ TA @ TB @ A )
       => ~ ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(66,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ C ) @ r ) )
     => ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) )
        = ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_prodM__carr) ).

thf(440,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ C ) @ r ) )
     => ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) )
        = ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(118,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( tp @ TB @ TA @ A @ ( nscal @ TB @ TA @ A @ C @ D ) @ B )
            = ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ C @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_Ring_OnsMulDistrR) ).

thf(614,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( tp @ TB @ TA @ A @ ( nscal @ TB @ TA @ A @ C @ D ) @ B )
            = ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ C @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(195,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ B )
           => ( ( member @ TA @ E @ C )
             => ( member @ TA @ ( tp @ TA @ TB @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_Ring_Otimes__mem__sum__mult) ).

thf(861,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ B )
           => ( ( member @ TA @ E @ C )
             => ( member @ TA @ ( tp @ TA @ TB @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(259,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
           => ( ( tp @ TA @ TC @ B @ C @ D )
              = ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_Ring_OSubring__tOp__ring__tOp) ).

thf(1078,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
           => ( ( tp @ TA @ TC @ B @ C @ D )
              = ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(45,axiom,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB )] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ B @ C ) @ ( pi @ TA @ TB @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_Pi__anti__mono) ).

thf(385,plain,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB )] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ B @ C ) @ ( pi @ TA @ TB @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(95,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [F: TA] : C ) )
     => ( ( member @ ( TA > TB ) @ D
          @ ( pi @ TA @ TB @ B
            @ ^ [F: TA] : C ) )
       => ( ( A = D )
         => ( ( member @ TA @ E @ B )
           => ( ( A @ E )
              = ( D @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_funcTr) ).

thf(533,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [F: TA] : C ) )
     => ( ( member @ ( TA > TB ) @ D
          @ ( pi @ TA @ TB @ B
            @ ^ [F: TA] : C ) )
       => ( ( A = D )
         => ( ( member @ TA @ E @ B )
           => ( ( A @ E )
              = ( D @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(86,axiom,
    ! [A: set @ a,B: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_sum__mult__subR) ).

thf(500,plain,
    ! [A: set @ a,B: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ord_less_eq @ ( set @ a ) @ ( sum_mult @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(182,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
           => ( ( pop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C @ D )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_Ring_OSubring__pOp__ring__pOp) ).

thf(814,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
           => ( ( pop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C @ D )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(145,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
        | ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
     => ( ~ ( member @ TA @ C @ A )
       => ( ( member @ TA @ C @ B )
         => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_sub__which2) ).

thf(698,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
        | ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
     => ( ~ ( member @ TA @ C @ A )
       => ( ( member @ TA @ C @ B )
         => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(217,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( ( member @ TA @ D @ B )
           => ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_Ring_OSr__tOp__closed) ).

thf(945,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( ( member @ TA @ D @ B )
           => ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(27,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less_eq @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_add__le__cancel__right) ).

thf(340,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less_eq @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(211,axiom,
    ! [A: set @ a] :
      ( ( mul_closed_set @ a @ b @ r @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_mul__closed__set__sub) ).

thf(930,plain,
    ! [A: set @ a] :
      ( ( mul_closed_set @ a @ b @ r @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(236,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_aGroup_OpOp__assocTr41) ).

thf(1001,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(149,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),B: TA > TD,C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),D: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) )] :
      ( ( ring @ TC @ TF @ A )
     => ( ( member @ ( TA > TD ) @ B @ ( mHom @ TC @ TF @ TA @ TE @ TD @ TB @ A @ C @ D ) )
       => ( member @ ( TA > TD ) @ B
          @ ( pi @ TA @ TD @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ C )
            @ ^ [E: TA] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_Ring_OmHom__func) ).

thf(709,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),B: TA > TD,C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),D: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) )] :
      ( ( ring @ TC @ TF @ A )
     => ( ( member @ ( TA > TD ) @ B @ ( mHom @ TC @ TF @ TA @ TE @ TD @ TB @ A @ C @ D ) )
       => ( member @ ( TA > TD ) @ B
          @ ( pi @ TA @ TD @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ C )
            @ ^ [E: TA] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(59,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) )
                = D )
             => ( ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ B ) )
                = ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_aGroup_Oag__diff__minus) ).

thf(417,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) )
                = D )
             => ( ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ B ) )
                = ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(227,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( aGroup @ TA @ ( module_ext @ TA @ TB @ TC ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_Module_Omodule__is__ag) ).

thf(973,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( aGroup @ TA @ ( module_ext @ TA @ TB @ TC ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(104,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_ring__l__one) ).

thf(565,plain,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(130,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( B != C )
           => ( ( mop @ TA @ TB @ A @ B )
             != ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_aGroup_Oag__inv__inj) ).

thf(651,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( B != C )
           => ( ( mop @ TA @ TB @ A @ B )
             != ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(47,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ( zeroring @ TA @ TB @ A )
       => ( ( j_rad @ TA @ TB @ A )
          = ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_Ring_Ozeroring__J__rad__empty) ).

thf(389,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ( zeroring @ TA @ TB @ A )
       => ( ( j_rad @ TA @ TB @ A )
          = ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(155,axiom,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_mem__subring__mem__ring) ).

thf(724,plain,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(106,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
              = ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_aGroup_Oaassoc) ).

thf(570,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
              = ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(153,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( member @ TA @ ( tp @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_Ring_Oring__tOp__closed) ).

thf(719,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( member @ TA @ ( tp @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(41,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ~ ( zeroring @ TA @ TB @ A )
       => ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_Ring_Oid__maximal__Exist) ).

thf(372,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ~ ( zeroring @ TA @ TB @ A )
       => ? [B: set @ TA] : ( maximal_ideal @ TA @ TB @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(72,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] :
              ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
              @ ^ [D: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_dsum__pOp__func) ).

thf(460,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] :
              ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
              @ ^ [D: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(111,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ? [C: TA] :
          ( ( member @ TA @ C @ A )
          & ~ ( member @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_not__subseteq) ).

thf(587,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ? [C: TA] :
          ( ( member @ TA @ C @ A )
          & ~ ( member @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(120,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_aGroup_OpOp__assocTr42) ).

thf(619,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D ) @ E )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(208,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TB > TC > TA,B: set @ TB,C: set @ TC,D: set @ TA,E: TB,F: TC] :
      ( ( member @ ( TB > TC > TA ) @ A
        @ ( pi @ TB @ ( TC > TA ) @ B
          @ ^ [G: TB] :
              ( pi @ TC @ TA @ C
              @ ^ [H: TC] : D ) ) )
     => ( ( member @ TB @ E @ B )
       => ( ( member @ TC @ F @ C )
         => ( member @ TA @ ( A @ E @ F ) @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_bivar__fun__mem) ).

thf(917,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TB > TC > TA,B: set @ TB,C: set @ TC,D: set @ TA,E: TB,F: TC] :
      ( ( member @ ( TB > TC > TA ) @ A
        @ ( pi @ TB @ ( TC > TA ) @ B
          @ ^ [G: TB] :
              ( pi @ TC @ TA @ C
              @ ^ [H: TC] : D ) ) )
     => ( ( member @ TB @ E @ B )
       => ( ( member @ TC @ F @ C )
         => ( member @ TA @ ( A @ E @ F ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(166,axiom,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ring @ a @ TA @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_subring__Ring) ).

thf(751,plain,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ring @ a @ TA @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(183,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
      ( ( aGroup @ TA @ TB @ A )
     => ( member @ ( TA > TA > TA ) @ ( pop @ TA @ TB @ A )
        @ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
          @ ^ [B: TA] :
              ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
              @ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_aGroup_Opop__closed) ).

thf(817,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
      ( ( aGroup @ TA @ TB @ A )
     => ( member @ ( TA > TA > TA ) @ ( pop @ TA @ TB @ A )
        @ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
          @ ^ [B: TA] :
              ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
              @ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(115,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
           => ( ( sprod @ TA @ TC @ TB @ A @ ( tp @ TC @ TD @ B @ C @ D ) @ E )
              = ( sprod @ TA @ TC @ TB @ A @ C @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_Module_Osprod__assoc) ).

thf(606,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
           => ( ( sprod @ TA @ TC @ TB @ A @ ( tp @ TC @ TD @ B @ C @ D ) @ E )
              = ( sprod @ TA @ TC @ TB @ A @ C @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(7,axiom,
    cancel_semigroup_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add) ).

thf(289,plain,
    cancel_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(19,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
          = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_add_Oleft__commute) ).

thf(316,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
          = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(267,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB] :
      ( ( module @ TA @ TD @ TE @ TF @ A @ B )
     => ( ( module @ TB @ TD @ TC @ TF @ C @ B )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
           => ( ( E @ ( mop @ TA @ ( module_ext @ TA @ TD @ TE ) @ A @ D ) )
              = ( mop @ TB @ ( module_ext @ TB @ TD @ TC ) @ C @ ( E @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_Module_OmHom__inv) ).

thf(1100,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB] :
      ( ( module @ TA @ TD @ TE @ TF @ A @ B )
     => ( ( module @ TB @ TD @ TC @ TF @ C @ B )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
           => ( ( E @ ( mop @ TA @ ( module_ext @ TA @ TD @ TE ) @ A @ D ) )
              = ( mop @ TB @ ( module_ext @ TB @ TD @ TC ) @ C @ ( E @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(60,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
            = ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_ring__tOp__assoc) ).

thf(421,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
            = ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(198,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( B = C )
             => ( ( tp @ TA @ TB @ A @ B @ D )
                = ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_Ring_OrEQMulR) ).

thf(870,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( B = C )
             => ( ( tp @ TA @ TB @ A @ B @ D )
                = ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(5,axiom,
    ordere236663937imp_le @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(284,plain,
    ordere236663937imp_le @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(172,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) )
                = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ B @ E ) ) @ ( tp @ TA @ TB @ A @ C @ D ) ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_Ring_Oring__distrib3) ).

thf(770,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) )
                = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ B @ E ) ) @ ( tp @ TA @ TB @ A @ C @ D ) ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(77,axiom,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
         => ( ( tp @ a @ TA @ A @ B @ C )
            = ( tp @ a @ b @ r @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_Subring__tOp__ring__tOp) ).

thf(475,plain,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
         => ( ( tp @ a @ TA @ A @ B @ C )
            = ( tp @ a @ b @ r @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(83,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) ) @ ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ D ) ) )
              = ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_aGroup_OpOp__assoc__cancel) ).

thf(492,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ C ) ) @ ( pop @ TA @ TB @ A @ C @ ( mop @ TA @ TB @ A @ D ) ) )
              = ( pop @ TA @ TB @ A @ B @ ( mop @ TA @ TB @ A @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(168,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A != B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ? [C: TA] :
            ( ( member @ TA @ C @ A )
            & ~ ( member @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_sets__not__eq) ).

thf(756,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A != B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ? [C: TA] :
            ( ( member @ TA @ C @ A )
            & ~ ( member @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less_eq @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_add__le__cancel__left) ).

thf(321,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less_eq @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(129,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( A = B )
           => ( ( tp @ a @ b @ r @ A @ C )
              = ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_rEQMulR) ).

thf(646,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( A = B )
           => ( ( tp @ a @ b @ r @ A @ C )
              = ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(90,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
          = ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_Ring_Oring__times__minusr) ).

thf(514,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
          = ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(250,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
            = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_Ring_Oring__inv1__1) ).

thf(1051,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
            = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(204,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ C ) @ ( nscal @ a @ b @ r @ B @ C ) )
          = ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_nsDistrL) ).

thf(905,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ C ) @ ( nscal @ a @ b @ r @ B @ C ) )
          = ( nscal @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(114,axiom,
    ! [A: a] :
      ( ~ ( zeroring @ a @ b @ r )
     => ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ~ ( unit @ a @ b @ r @ A )
         => ? [B: set @ a] :
              ( ( maximal_ideal @ a @ b @ r @ B )
              & ( member @ a @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_nonunit__contained__maxid) ).

thf(602,plain,
    ! [A: a] :
      ( ~ ( zeroring @ a @ b @ r )
     => ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ~ ( unit @ a @ b @ r @ A )
         => ? [B: set @ a] :
              ( ( maximal_ideal @ a @ b @ r @ B )
              & ( member @ a @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(4,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_nat__eq__le) ).

thf(280,plain,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(143,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_dsum__zero__func) ).

thf(692,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(160,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_prod__zero__func) ).

thf(734,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( TA > TB ) @ ( prod_zero @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(197,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ B )
          = ( tp @ a @ b @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_ring__tOp__commute) ).

thf(866,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ B )
          = ( tp @ a @ b @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(49,axiom,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_2) ).

thf(396,plain,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(235,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ C @ B ) @ ( tp @ TA @ TB @ A @ D @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_Ring_Oring__distrib2) ).

thf(998,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ C @ B ) @ ( tp @ TA @ TB @ A @ D @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(232,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ~ ( member @ TA @ C @ A )
       => ( ( member @ TA @ C @ B )
         => ( A != B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_proper__subset) ).

thf(988,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ~ ( member @ TA @ C @ A )
       => ( ( member @ TA @ C @ B )
         => ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(88,axiom,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > ( set @ TB ),C: TA > ( set @ TB )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( ord_less_eq @ ( set @ TB ) @ ( B @ D ) @ ( C @ D ) ) )
     => ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ A @ B ) @ ( pi @ TA @ TB @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_Pi__mono) ).

thf(507,plain,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > ( set @ TB ),C: TA > ( set @ TB )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( ord_less_eq @ ( set @ TB ) @ ( B @ D ) @ ( C @ D ) ) )
     => ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( pi @ TA @ TB @ A @ B ) @ ( pi @ TA @ TB @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(18,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ D @ A )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_group__cancel_Oadd2) ).

thf(312,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ D @ A )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(10,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ring @ TA @ TB @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_Ring_ORing) ).

thf(293,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(222,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ D @ B )
                = ( pop @ TA @ TB @ A @ D @ C ) )
             => ( B = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_aGroup_OpOp__cancel__l) ).

thf(956,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ D @ B )
                = ( pop @ TA @ TB @ A @ D @ C ) )
             => ( B = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(15,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(307,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(9,axiom,
    ordere779506340up_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).

thf(292,plain,
    ordere779506340up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(134,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType] :
      ( ( rInvim @ TA @ TB @ TD @ TC )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TA > TD,D: set @ TD] :
            ( collect @ TA
            @ ^ [E: TA] :
                ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
                & ( member @ TD @ ( C @ E ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_rInvim__def) ).

thf(661,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType] :
      ( ( rInvim @ TA @ TB @ TD @ TC )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TA > TD,D: set @ TD] :
            ( collect @ TA
            @ ^ [E: TA] :
                ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
                & ( member @ TD @ ( C @ E ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(99,axiom,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: set @ TB] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( member @ TB @ ( B @ D ) @ C ) )
     => ( member @ ( TA > TB ) @ B
        @ ( pi @ TA @ TB @ A
          @ ^ [D: TA] : C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_funcsetI) ).

thf(546,plain,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: set @ TB] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( member @ TB @ ( B @ D ) @ C ) )
     => ( member @ ( TA > TB ) @ B
        @ ( pi @ TA @ TB @ A
          @ ^ [D: TA] : C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(169,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ D @ E ) ) )
                = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ E ) ) ) @ ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_Ring_Oring__distrib4) ).

thf(761,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ D @ E ) ) )
                = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ E ) ) ) @ ( tp @ TA @ TB @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(44,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_eqsets__sub) ).

thf(381,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( A = B )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(255,axiom,
    ! [A: set @ a,B: set @ a,C: a,D: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( sum_mult @ a @ b @ r @ A @ B ) )
         => ( ( member @ a @ D @ ( sum_mult @ a @ b @ r @ A @ B ) )
           => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_sum__mult__pOp__closed) ).

thf(1067,plain,
    ! [A: set @ a,B: set @ a,C: a,D: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( sum_mult @ a @ b @ r @ A @ B ) )
         => ( ( member @ a @ D @ ( sum_mult @ a @ b @ r @ A @ B ) )
           => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(178,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( mul_closed_set @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( ( member @ TA @ D @ B )
           => ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_Ring_Omul__closed__set__tOp__closed) ).

thf(798,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( mul_closed_set @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( ( member @ TA @ D @ B )
           => ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(252,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ B @ C )
                = D )
             => ( C
                = ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_aGroup_Oag__eq__sol1) ).

thf(1057,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ B @ C )
                = D )
             => ( C
                = ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(25,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_add__left__cancel) ).

thf(334,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(249,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( prod_zero @ TA @ TB @ TC @ A @ B )
          = ( prod_zero @ TA @ TB @ TC @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_prodag__sameTr4) ).

thf(1042,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( prod_zero @ TA @ TB @ TC @ A @ B )
          = ( prod_zero @ TA @ TB @ TC @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(30,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_le__imp__add__int) ).

thf(349,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(48,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_add__mono__thms__linordered__semiring_I3_J) ).

thf(392,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(12,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_add_Oright__cancel) ).

thf(297,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(63,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
              = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) )
            & ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
              = ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_Ring_Oring__inv1) ).

thf(429,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
              = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B ) @ C ) )
            & ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) )
              = ( tp @ TA @ TB @ A @ B @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(164,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( pop @ TA @ TB @ A @ B @ C )
            = ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_aGroup_Oag__pOp__commute) ).

thf(746,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( pop @ TA @ TB @ A @ B @ C )
            = ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(264,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_prod__pOp__mem) ).

thf(1091,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(75,axiom,
    ! [A: set @ a,B: a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_Sr__mOp__closed) ).

thf(467,plain,
    ! [A: set @ a,B: a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(80,axiom,
    ! [A: a,B: set @ a,C: a,D: set @ a] :
      ( ( member @ a @ A @ B )
     => ( ( member @ a @ C @ D )
       => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( ord_less_eq @ ( set @ a ) @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ C ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_set__sum__mem) ).

thf(484,plain,
    ! [A: a,B: set @ a,C: a,D: set @ a] :
      ( ( member @ a @ A @ B )
     => ( ( member @ a @ C @ D )
       => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( ord_less_eq @ ( set @ a ) @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ C ) @ ( aset_sum @ a @ ( ring_ext @ a @ b ) @ r @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(34,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ord_less_eq @ ( set @ TA ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_Ring_Osum__mult__subR) ).

thf(361,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ord_less_eq @ ( set @ TA ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(122,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( mop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) )
            = ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_aGroup_Oag__p__inv) ).

thf(625,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( mop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) )
            = ( pop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) @ ( mop @ TA @ TB @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(192,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_Ring_Oring__distrib1) ).

thf(852,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(244,axiom,
    ! [A: set @ a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_Sr__one) ).

thf(1027,plain,
    ! [A: set @ a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(22,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_add_Oleft__cancel) ).

thf(324,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(239,axiom,
    ( ( zeroring @ a @ b @ r )
   => ( ( j_rad @ a @ b @ r )
      = ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_zeroring__J__rad__empty) ).

thf(1012,plain,
    ( ( zeroring @ a @ b @ r )
   => ( ( j_rad @ a @ b @ r )
      = ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ A @ C )
           => ( ord_less_eq @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_le__convert) ).

thf(285,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ A @ C )
           => ( ord_less_eq @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(110,axiom,
    ! [A: set @ a] :
      ( ~ ( zeroring @ a @ b @ r )
     => ( ( maximal_ideal @ a @ b @ r @ A )
       => ( ! [B: a] :
              ( ( member @ a @ B @ A )
             => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( un @ a @ b @ r ) ) ) )
         => ( local_ring @ a @ b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_localring__unit) ).

thf(583,plain,
    ! [A: set @ a] :
      ( ~ ( zeroring @ a @ b @ r )
     => ( ( maximal_ideal @ a @ b @ r @ A )
       => ( ! [B: a] :
              ( ( member @ a @ B @ A )
             => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ ( un @ a @ b @ r ) ) ) )
         => ( local_ring @ a @ b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(186,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB,E: TA] :
      ( ! [F: TA] :
          ( ( member @ TA @ F @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ F ) @ r ) )
     => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
         => ( ( member @ TA @ E @ A )
           => ( ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D @ E )
              = ( sprod @ TB @ a @ TC @ ( B @ E ) @ C @ ( D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_prodM__sprod__val) ).

thf(830,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB,E: TA] :
      ( ! [F: TA] :
          ( ( member @ TA @ F @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ F ) @ r ) )
     => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
         => ( ( member @ TA @ E @ A )
           => ( ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D @ E )
              = ( sprod @ TB @ a @ TC @ ( B @ E ) @ C @ ( D @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(263,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
           => ( ( member @ TA @ E @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
             => ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_Ring_Osum__mult__pOp__closed) ).

thf(1089,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: set @ TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
           => ( ( member @ TA @ E @ ( sum_mult @ TA @ TB @ A @ B @ C ) )
             => ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D @ E ) @ ( sum_mult @ TA @ TB @ A @ B @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(199,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ! [E: TA] :
                ( ( member @ TA @ E @ A )
               => ( ( C @ E )
                  = ( D @ E ) ) )
           => ( C = D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_carr__prodag__mem__eq) ).

thf(874,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ! [E: TA] :
                ( ( member @ TA @ E @ A )
               => ( ( C @ E )
                  = ( D @ E ) ) )
           => ( C = D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(219,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA > TB] :
      ( ! [F: TA] :
          ( ( member @ TA @ F @ A )
         => ( aGroup @ TB @ TC @ ( B @ F ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
           => ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ E )
              = ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_prod__pOp__assoc) ).

thf(948,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA > TB] :
      ( ! [F: TA] :
          ( ( member @ TA @ F @ A )
         => ( aGroup @ TB @ TC @ ( B @ F ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
           => ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ E )
              = ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(254,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
        = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_ring__times__minusl) ).

thf(1064,plain,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
        = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(53,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
         => ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_add__left__imp__eq) ).

thf(403,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
         => ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(32,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TC @ A )
     => ( ( subring @ TA @ TC @ TB @ A @ B )
       => ( ring @ TA @ TB @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_Ring_Osubring__Ring) ).

thf(356,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TC @ A )
     => ( ( subring @ TA @ TC @ TB @ A @ B )
       => ( ring @ TA @ TB @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(113,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( carr_prodag @ TA @ TB @ TC @ A @ B )
          = ( carr_prodag @ TA @ TB @ TC @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_prodag__sameTr1) ).

thf(593,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( carr_prodag @ TA @ TB @ TC @ A @ B )
          = ( carr_prodag @ TA @ TB @ TC @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(167,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_Ring_Org__distrib) ).

thf(753,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(96,axiom,
    ! [A: set @ a] :
      ( ( primary_ideal @ a @ b @ r @ A )
     => ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_primary__ideal__proper1) ).

thf(537,plain,
    ! [A: set @ a] :
      ( ( primary_ideal @ a @ b @ r @ A )
     => ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(258,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_Ring_Oring__l__one) ).

thf(1075,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(112,axiom,
    ! [A: set @ a,B: set @ a,C: a,D: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ A )
         => ( ( member @ a @ D @ B )
           => ( member @ a @ ( tp @ a @ b @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_times__mem__sum__mult) ).

thf(590,plain,
    ! [A: set @ a,B: set @ a,C: a,D: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ A )
         => ( ( member @ a @ D @ B )
           => ( member @ a @ ( tp @ a @ b @ r @ C @ D ) @ ( sum_mult @ a @ b @ r @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(226,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
           => ( ( sprod @ TA @ TC @ TB @ A @ ( pop @ TC @ ( ring_ext @ TC @ TD ) @ B @ C @ D ) @ E )
              = ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_Module_Osprod__l__distr) ).

thf(970,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TC,E: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
           => ( ( sprod @ TA @ TC @ TB @ A @ ( pop @ TC @ ( ring_ext @ TC @ TD ) @ B @ C @ D ) @ E )
              = ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) @ ( sprod @ TA @ TC @ TB @ A @ D @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(148,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ? [C: TA] :
          ( ( member @ TA @ C @ A )
          & ~ ( member @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_not__sub) ).

thf(706,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ~ ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ? [C: TA] :
          ( ( member @ TA @ C @ A )
          & ~ ( member @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(121,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_Ring_Opop__aassoc) ).

thf(622,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C ) @ D )
              = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(146,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) )
            = ( sprod @ TA @ TC @ TB @ A @ C @ ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_Module_Osc__minus__am) ).

thf(701,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) )
            = ( sprod @ TA @ TC @ TB @ A @ C @ ( mop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(131,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( ( member @ TA @ D @ B )
           => ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_Ring_OSr__pOp__closed) ).

thf(654,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( ( member @ TA @ D @ B )
           => ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ D ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(71,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_prod__mOp__func) ).

thf(457,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(231,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( B = C )
           => ( ( nscal @ TB @ TA @ A @ B @ D )
              = ( nscal @ TB @ TA @ A @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_Ring_OnsEqElm) ).

thf(984,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( B = C )
           => ( ( nscal @ TB @ TA @ A @ B @ D )
              = ( nscal @ TB @ TA @ A @ C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(154,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
     => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
         => ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_prodM__sprod__mem) ).

thf(721,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: a,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
     => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) )
         => ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ a @ b @ TA @ TB @ TC @ r @ A @ B @ C @ D ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ a @ TC ) @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(175,axiom,
    ( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
      @ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
        @ ^ [A: c > d] :
            ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
            @ ^ [B: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ! [B: c > d] :
            ( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
           => ! [C: c > d] :
                ( ( member @ ( c > d ) @ C @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
               => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B ) @ C )
                  = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ C ) ) ) ) ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ! [B: c > d] :
            ( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
           => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B )
              = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ A ) ) ) )
    & ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
      @ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
        @ ^ [A: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A ) @ A )
          = ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
    & ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).

thf(779,plain,
    ( ( member @ ( ( c > d ) > ( c > d ) > c > d ) @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
      @ ( pi @ ( c > d ) @ ( ( c > d ) > c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
        @ ^ [A: c > d] :
            ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
            @ ^ [B: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ! [B: c > d] :
            ( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
           => ! [C: c > d] :
                ( ( member @ ( c > d ) @ C @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
               => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B ) @ C )
                  = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ C ) ) ) ) ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ! [B: c > d] :
            ( ( member @ ( c > d ) @ B @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
           => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ B )
              = ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ B @ A ) ) ) )
    & ( member @ ( ( c > d ) > c > d ) @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
      @ ( pi @ ( c > d ) @ ( c > d ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m )
        @ ^ [A: c > d] : ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_mOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A ) @ A )
          = ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) ) )
    & ( member @ ( c > d ) @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
    & ! [A: c > d] :
        ( ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
       => ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(40,axiom,
    ord @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_1) ).

thf(371,plain,
    ord @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(87,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) )
            = ( tp @ a @ b @ r @ B @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_rMulLC) ).

thf(503,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) )
            = ( tp @ a @ b @ r @ B @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(28,axiom,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_subset__self) ).

thf(343,plain,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(17,axiom,
    semigroup_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add) ).

thf(311,plain,
    semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(251,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ),D: TA > TB,E: TA,F: TA] :
      ( ( module @ TA @ TC @ TE @ TF @ A @ B )
     => ( ( module @ TB @ TC @ TD @ TF @ C @ B )
       => ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TF @ TA @ TE @ TB @ TD @ B @ A @ C ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
           => ( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
             => ( ( D @ ( pop @ TA @ ( module_ext @ TA @ TC @ TE ) @ A @ E @ F ) )
                = ( pop @ TB @ ( module_ext @ TB @ TC @ TD ) @ C @ ( D @ E ) @ ( D @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_Module_OmHom__add) ).

thf(1054,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ),D: TA > TB,E: TA,F: TA] :
      ( ( module @ TA @ TC @ TE @ TF @ A @ B )
     => ( ( module @ TB @ TC @ TD @ TF @ C @ B )
       => ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TF @ TA @ TE @ TB @ TD @ B @ A @ C ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
           => ( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TE ) ) @ A ) )
             => ( ( D @ ( pop @ TA @ ( module_ext @ TA @ TC @ TE ) @ A @ E @ F ) )
                = ( pop @ TB @ ( module_ext @ TB @ TC @ TD ) @ C @ ( D @ E ) @ ( D @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(220,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA] :
          ( ( member @ TA @ A
            @ ( collect @ TA
              @ ^ [C: TA] : ( ord_less_eq @ TA @ C @ B ) ) )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_Nset__le) ).

thf(953,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA] :
          ( ( member @ TA @ A
            @ ( collect @ TA
              @ ^ [C: TA] : ( ord_less_eq @ TA @ C @ B ) ) )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(262,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) )
              = ( tp @ TA @ TB @ A @ C @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_Ring_OrMulLC) ).

thf(1086,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) )
              = ( tp @ TA @ TB @ A @ C @ ( tp @ TA @ TB @ A @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(214,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: nat] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( member @ TA @ ( nscal @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_Ring_OnsClose) ).

thf(938,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: nat] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( member @ TA @ ( nscal @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(50,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).

thf(398,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(247,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( tp @ a @ b @ r @ A @ ( un @ a @ b @ r ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_ring__r__one) ).

thf(1035,plain,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( tp @ a @ b @ r @ A @ ( un @ a @ b @ r ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(173,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [F: TA] : C ) )
     => ( ( member @ ( TA > TB ) @ D
          @ ( pi @ TA @ TB @ B
            @ ^ [F: TA] : C ) )
       => ( ( A = D )
         => ( ( member @ TA @ E @ B )
           => ( ( A @ E )
              = ( D @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_eq__funcs) ).

thf(773,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB,E: TA] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [F: TA] : C ) )
     => ( ( member @ ( TA > TB ) @ D
          @ ( pi @ TA @ TB @ B
            @ ^ [F: TA] : C ) )
       => ( ( A = D )
         => ( ( member @ TA @ E @ B )
           => ( ( A @ E )
              = ( D @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(176,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: set @ TC,C: set @ TA,D: set @ TB,E: TC] :
      ( ( member @ ( TC > TA > TB ) @ A
        @ ( pi @ TC @ ( TA > TB ) @ B
          @ ^ [F: TC] :
              ( pi @ TA @ TB @ C
              @ ^ [G: TA] : D ) ) )
     => ( ( member @ TC @ E @ B )
       => ( member @ ( TA > TB ) @ ( A @ E )
          @ ( pi @ TA @ TB @ C
            @ ^ [F: TA] : D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_bivar__fun) ).

thf(794,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: set @ TC,C: set @ TA,D: set @ TB,E: TC] :
      ( ( member @ ( TC > TA > TB ) @ A
        @ ( pi @ TC @ ( TA > TB ) @ B
          @ ^ [F: TC] :
              ( pi @ TA @ TB @ C
              @ ^ [G: TA] : D ) ) )
     => ( ( member @ TC @ E @ B )
       => ( member @ ( TA > TB ) @ ( A @ E )
          @ ( pi @ TA @ TB @ C
            @ ^ [F: TA] : D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(57,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( aGroup @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_Ring_Oring__is__ag) ).

thf(412,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( aGroup @ TA @ ( ring_ext @ TA @ TB ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(61,axiom,
    member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_un__closed) ).

thf(425,plain,
    member @ a @ ( un @ a @ b @ r ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(268,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
          = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_pop__commute) ).

thf(1103,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B )
          = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(107,axiom,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
     => ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_Pi__I) ).

thf(573,plain,
    ! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB,C: TA > ( set @ TB )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( member @ TB @ ( B @ D ) @ ( C @ D ) ) )
     => ( member @ ( TA > TB ) @ B @ ( pi @ TA @ TB @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(142,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) ),D: TC,E: TA > TB] :
      ( ( ring @ TC @ TE @ A )
     => ( ! [F: TA] :
            ( ( member @ TA @ F @ B )
           => ( module @ TB @ TC @ TD @ TE @ ( C @ F ) @ A ) )
       => ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) )
           => ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ TC @ TE @ TA @ TB @ TD @ A @ B @ C @ D @ E ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_Ring_OprodM__sprod__mem) ).

thf(689,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) ),D: TC,E: TA > TB] :
      ( ( ring @ TC @ TE @ A )
     => ( ! [F: TA] :
            ( ( member @ TA @ F @ B )
           => ( module @ TB @ TC @ TD @ TE @ ( C @ F ) @ A ) )
       => ( ( member @ TC @ D @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) )
           => ( member @ ( TA > TB ) @ ( algebr142713364_sprod @ TC @ TE @ TA @ TB @ TD @ A @ B @ C @ D @ E ) @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(159,axiom,
    ( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
    @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
      @ ^ [A: a] :
          ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
          @ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_pop__closed) ).

thf(733,plain,
    ( member @ ( a > a > a ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r )
    @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
      @ ^ [A: a] :
          ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
          @ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(184,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TC ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TD ) ) ),D: TA > TB,E: TA > TB] :
      ( ( ring @ TE @ TC @ A )
     => ( ! [F: TA] :
            ( ( member @ TA @ F @ B )
           => ( module @ TB @ TE @ TD @ TC @ ( C @ F ) @ A ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
           => ( ! [F: TA] :
                  ( ( member @ TA @ F @ B )
                 => ( ( D @ F )
                    = ( E @ F ) ) )
             => ( D = E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_Ring_OprodM__mem__eq) ).

thf(819,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TE @ ( aGroup_ext @ TE @ ( ring_ext @ TE @ TC ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TE @ TD ) ) ),D: TA > TB,E: TA > TB] :
      ( ( ring @ TE @ TC @ A )
     => ( ! [F: TA] :
            ( ( member @ TA @ F @ B )
           => ( module @ TB @ TE @ TD @ TC @ ( C @ F ) @ A ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TE @ product_unit ) ) @ ( algebr1000837334_prodM @ TE @ TC @ TA @ TB @ TD @ A @ B @ C ) ) )
           => ( ! [F: TA] :
                  ( ( member @ TA @ F @ B )
                 => ( ( D @ F )
                    = ( E @ F ) ) )
             => ( D = E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(163,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_dsum__iOp__func) ).

thf(743,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] : ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(150,axiom,
    ! [A: set @ a] :
      ( ( maximal_ideal @ a @ b @ r @ A )
     => ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_maximal__ideal__proper) ).

thf(711,plain,
    ! [A: set @ a] :
      ( ( maximal_ideal @ a @ b @ r @ A )
     => ~ ( member @ a @ ( un @ a @ b @ r ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(103,axiom,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( un @ a @ TA @ A )
        = ( un @ a @ b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_Subring__one__ring__one) ).

thf(562,plain,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) )] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( un @ a @ TA @ A )
        = ( un @ a @ b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(228,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
              = ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_Ring_Otp__assoc) ).

thf(975,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
              = ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(127,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ ( TA > TA ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A )
        @ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
          @ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_Ring_Omop__closed) ).

thf(641,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ ( TA > TA ) @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A )
        @ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
          @ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(67,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
            = ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_tp__assoc) ).

thf(445,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C )
            = ( tp @ a @ b @ r @ A @ ( tp @ a @ b @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(269,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( member @ a @ ( tp @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_ring__tOp__closed) ).

thf(1107,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( member @ a @ ( tp @ a @ b @ r @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(74,axiom,
    ring @ a @ b @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_Ring__axioms) ).

thf(466,plain,
    ring @ a @ b @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(209,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( prod_mOp @ TA @ TB @ TC @ A @ B )
          = ( prod_mOp @ TA @ TB @ TC @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_prodag__sameTr3) ).

thf(919,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( prod_mOp @ TA @ TB @ TC @ A @ B )
          = ( prod_mOp @ TA @ TB @ TC @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(147,axiom,
    ( ~ ( zeroring @ a @ b @ r )
   => ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_id__maximal__Exist) ).

thf(704,plain,
    ( ~ ( zeroring @ a @ b @ r )
   => ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(218,axiom,
    ( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
    @ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
      @ ^ [A: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_mop__closed) ).

thf(947,plain,
    ( member @ ( a > a ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r )
    @ ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
      @ ^ [A: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(151,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB,F: TD] :
      ( ( module @ TA @ TD @ TE @ TF @ A @ B )
     => ( ( module @ TB @ TD @ TC @ TF @ C @ B )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
           => ( ( member @ TD @ F @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ) @ B ) )
             => ( ( E @ ( sprod @ TA @ TD @ TE @ A @ F @ D ) )
                = ( sprod @ TB @ TD @ TC @ C @ F @ ( E @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_Module_OmHom__lin) ).

thf(713,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ),D: TA,E: TA > TB,F: TD] :
      ( ( module @ TA @ TD @ TE @ TF @ A @ B )
     => ( ( module @ TB @ TD @ TC @ TF @ C @ B )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( mHom @ TD @ TF @ TA @ TE @ TB @ TC @ B @ A @ C ) )
           => ( ( member @ TD @ F @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TF ) ) @ B ) )
             => ( ( E @ ( sprod @ TA @ TD @ TE @ A @ F @ D ) )
                = ( sprod @ TB @ TD @ TC @ C @ F @ ( E @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(85,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: set @ TA,D: TA,E: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ C )
       => ( ( member @ TA @ D @ E )
         => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( ord_less_eq @ ( set @ TA ) @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ D ) @ ( aset_sum @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_Ring_Oset__sum__mem) ).

thf(498,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: set @ TA,D: TA,E: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ C )
       => ( ( member @ TA @ D @ E )
         => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( ord_less_eq @ ( set @ TA ) @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ D ) @ ( aset_sum @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(170,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( ( mop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C )
            = ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_Ring_OSubring__minus__ring__minus) ).

thf(764,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( ( mop @ TA @ ( ring_ext @ TA @ TC ) @ B @ C )
            = ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(54,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ring @ TA @ TB @ ( sr @ TA @ TB @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_Ring_OSr__ring) ).

thf(406,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ring @ TA @ TB @ ( sr @ TA @ TB @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(69,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_Ring_Org__l__unit) ).

thf(451,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( tp @ TA @ TB @ A @ ( un @ TA @ TB @ A ) @ B )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(161,axiom,
    ! [A: set @ a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ring @ a @ b @ ( sr @ a @ b @ r @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_Sr__ring) ).

thf(737,plain,
    ! [A: set @ a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ring @ a @ b @ ( sr @ a @ b @ r @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(64,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_rg__l__unit) ).

thf(434,plain,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( tp @ a @ b @ r @ ( un @ a @ b @ r ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(14,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ C @ A )
           => ( ord_less_eq @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_ge__convert) ).

thf(303,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ C @ A )
           => ( ord_less_eq @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(196,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ D @ E ) )
                = ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_Ring_Oring__tOp__rel) ).

thf(863,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA,E: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
             => ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ ( tp @ TA @ TB @ A @ D @ E ) )
                = ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ D ) @ ( tp @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(46,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( mul_closed_set @ TA @ TB @ A @ B )
       => ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_Ring_Omul__closed__set__sub) ).

thf(387,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( mul_closed_set @ TA @ TB @ A @ B )
       => ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(116,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( mul_closed_set @ TA @ TB )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
            ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
            & ! [C: TA] :
                ( ( member @ TA @ C @ B )
               => ! [D: TA] :
                    ( ( member @ TA @ D @ B )
                   => ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_mul__closed__set__def) ).

thf(609,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( mul_closed_set @ TA @ TB )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
            ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
            & ! [C: TA] :
                ( ( member @ TA @ C @ B )
               => ! [D: TA] :
                    ( ( member @ TA @ D @ B )
                   => ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(156,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( j_rad @ TA @ TB @ A ) )
       => ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_Ring_OJ__rad__mem) ).

thf(726,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( j_rad @ TA @ TB @ A ) )
       => ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(92,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) )
              = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ A @ D ) ) @ ( tp @ a @ b @ r @ B @ C ) ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_ring__distrib3) ).

thf(520,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ C @ D ) )
              = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ A @ D ) ) @ ( tp @ a @ b @ r @ B @ C ) ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(207,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TA > TC,B: set @ TA,C: TA > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ TB ) ),D: TA] :
      ( ( member @ ( TA > TC ) @ A @ ( carr_prodag @ TA @ TC @ TB @ B @ C ) )
     => ( ( member @ TA @ D @ B )
       => ( member @ TC @ ( A @ D ) @ ( carrier @ TC @ ( aGroup_ext @ TC @ TB ) @ ( C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_prodag__comp__i) ).

thf(915,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: TA > TC,B: set @ TA,C: TA > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ TB ) ),D: TA] :
      ( ( member @ ( TA > TC ) @ A @ ( carr_prodag @ TA @ TC @ TB @ B @ C ) )
     => ( ( member @ TA @ D @ B )
       => ( member @ TC @ ( A @ D ) @ ( carrier @ TC @ ( aGroup_ext @ TC @ TB ) @ ( C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(136,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
        | ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
     => ( ( member @ TA @ C @ A )
       => ( ~ ( member @ TA @ C @ B )
         => ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_sub__which1) ).

thf(668,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
      ( ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
        | ( ord_less_eq @ ( set @ TA ) @ B @ A ) )
     => ( ( member @ TA @ C @ A )
       => ( ~ ( member @ TA @ C @ B )
         => ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(24,axiom,
    ! [TA: $tType] :
      ( ( plus @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( plus_plus @ TA @ C @ A )
            = ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_add__bothl) ).

thf(330,plain,
    ! [TA: $tType] :
      ( ( plus @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( plus_plus @ TA @ C @ A )
            = ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(29,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_add__mono__thms__linordered__semiring_I2_J) ).

thf(345,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(194,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
         => ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_dsum__pOp__mem) ).

thf(858,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
         => ( member @ ( TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(37,axiom,
    ord @ product_unit,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Ounit___Orderings_Oord_4) ).

thf(367,plain,
    ord @ product_unit,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(101,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TF ) ),D: TA > TB,E: TA > TB] :
      ( ( module @ TA @ TC @ TD @ TE @ A @ B )
     => ( ( module @ TB @ TC @ TF @ TE @ C @ B )
       => ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
         => ( ( member @ ( TA > TB ) @ E @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
           => ( ! [F: TA] :
                  ( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ) @ A ) )
                 => ( ( D @ F )
                    = ( E @ F ) ) )
             => ( D = E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_Module_OmHom__eq) ).

thf(552,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),C: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TF ) ),D: TA > TB,E: TA > TB] :
      ( ( module @ TA @ TC @ TD @ TE @ A @ B )
     => ( ( module @ TB @ TC @ TF @ TE @ C @ B )
       => ( ( member @ ( TA > TB ) @ D @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
         => ( ( member @ ( TA > TB ) @ E @ ( mHom @ TC @ TE @ TA @ TD @ TB @ TF @ B @ A @ C ) )
           => ( ! [F: TA] :
                  ( ( member @ TA @ F @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TD ) ) @ A ) )
                 => ( ( D @ F )
                    = ( E @ F ) ) )
             => ( D = E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(245,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ B )
          = ( tp @ a @ b @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_tp__commute) ).

thf(1029,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ B )
          = ( tp @ a @ b @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(202,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( B = E )
               => ( ( C = D )
                 => ( ( pop @ TA @ TB @ A @ B @ C )
                    = ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_aGroup_OgEQAddcross) ).

thf(895,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( B = E )
               => ( ( C = D )
                 => ( ( pop @ TA @ TB @ A @ B @ C )
                    = ( pop @ TA @ TB @ A @ D @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(216,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_Ring_OSr__one) ).

thf(943,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(242,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ ( TA > TA > TA ) @ ( tp @ TA @ TB @ A )
        @ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
          @ ^ [B: TA] :
              ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
              @ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_Ring_Otp__closed) ).

thf(1022,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ ( TA > TA > TA ) @ ( tp @ TA @ TB @ A )
        @ ( pi @ TA @ ( TA > TA ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
          @ ^ [B: TA] :
              ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A )
              @ ^ [C: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(212,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( tp @ TA @ TB @ A @ B @ C )
            = ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_Ring_Otp__commute) ).

thf(932,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( tp @ TA @ TB @ A @ B @ C )
            = ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(253,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( tp @ TA @ TB @ A @ B @ C )
            = ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_Ring_Oring__tOp__commute) ).

thf(1061,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( tp @ TA @ TB @ A @ B @ C )
            = ( tp @ TA @ TB @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(206,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( tp @ TA @ TB @ A @ B @ ( un @ TA @ TB @ A ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_Ring_Oring__r__one) ).

thf(912,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( tp @ TA @ TB @ A @ B @ ( un @ TA @ TB @ A ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(223,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C )
            = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_Ring_Opop__commute) ).

thf(959,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B @ C )
            = ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(144,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] :
              ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
              @ ^ [D: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_prod__pOp__func) ).

thf(695,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( member @ ( ( TA > TB ) > ( TA > TB ) > TA > TB ) @ ( prod_pOp @ TA @ TB @ TC @ A @ B )
        @ ( pi @ ( TA > TB ) @ ( ( TA > TB ) > TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
          @ ^ [C: TA > TB] :
              ( pi @ ( TA > TB ) @ ( TA > TB ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B )
              @ ^ [D: TA > TB] : ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(200,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( prod_pOp @ TA @ TB @ TC @ A @ B )
          = ( prod_pOp @ TA @ TB @ TC @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_prodag__sameTr2) ).

thf(883,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ A )
           => ( ( B @ D )
              = ( C @ D ) ) )
       => ( ( prod_pOp @ TA @ TB @ TC @ A @ B )
          = ( prod_pOp @ TA @ TB @ TC @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(240,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
          = ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_ring__inv1__2) ).

thf(1015,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
          = ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(271,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( tp @ TB @ TA @ A @ B @ ( nscal @ TB @ TA @ A @ C @ D ) )
            = ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ B @ C ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_Ring_OnsMulDistrL) ).

thf(1113,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( tp @ TB @ TA @ A @ B @ ( nscal @ TB @ TA @ A @ C @ D ) )
            = ( nscal @ TB @ TA @ A @ ( tp @ TB @ TA @ A @ B @ C ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(62,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_aGroup_OpOp__assocTr43) ).

thf(426,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(139,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ B @ D )
                = ( pop @ TA @ TB @ A @ C @ D ) )
             => ( B = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_aGroup_OpOp__cancel__r) ).

thf(680,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ B @ D )
                = ( pop @ TA @ TB @ A @ C @ D ) )
             => ( B = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(42,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( plus_plus @ nat @ A @ C )
        = ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_add__both) ).

thf(374,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( plus_plus @ nat @ A @ C )
        = ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(81,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_dsumag__sub__prodag) ).

thf(487,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) )] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( aGroup @ TB @ TC @ ( B @ C ) ) )
     => ( ord_less_eq @ ( set @ ( TA > TB ) ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(181,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( member @ TA @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_Ring_OSr__mOp__closed) ).

thf(812,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA,C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( sr2 @ TA @ TB @ A @ B )
       => ( ( member @ TA @ C @ B )
         => ( member @ TA @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(119,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
      ( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
     => ( ~ ( member @ TB @ ( A @ D ) @ ( C @ D ) )
       => ~ ( member @ TA @ D @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_PiE) ).

thf(617,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
      ( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
     => ( ~ ( member @ TB @ ( A @ D ) @ ( C @ D ) )
       => ~ ( member @ TA @ D @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(82,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( member @ nat @ A
        @ ( collect @ nat
          @ ^ [C: nat] : ( ord_less_eq @ nat @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_mem__of__Nset) ).

thf(490,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( member @ nat @ A
        @ ( collect @ nat
          @ ^ [C: nat] : ( ord_less_eq @ nat @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(138,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ) ),D: TD,E: TA > TB,F: TA] :
      ( ( ring @ TD @ TE @ A )
     => ( ! [G: TA] :
            ( ( member @ TA @ G @ B )
           => ( module @ TB @ TD @ TC @ TE @ ( C @ G ) @ A ) )
       => ( ( member @ TD @ D @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TD @ TC ) @ B @ C ) )
           => ( ( member @ TA @ F @ B )
             => ( ( algebr142713364_sprod @ TD @ TE @ TA @ TB @ TC @ A @ B @ C @ D @ E @ F )
                = ( sprod @ TB @ TD @ TC @ ( C @ F ) @ D @ ( E @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_Ring_OprodM__sprod__val) ).

thf(675,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TC ) ) ),D: TD,E: TA > TB,F: TA] :
      ( ( ring @ TD @ TE @ A )
     => ( ! [G: TA] :
            ( ( member @ TA @ G @ B )
           => ( module @ TB @ TD @ TC @ TE @ ( C @ G ) @ A ) )
       => ( ( member @ TD @ D @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TE ) ) @ A ) )
         => ( ( member @ ( TA > TB ) @ E @ ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TD @ TC ) @ B @ C ) )
           => ( ( member @ TA @ F @ B )
             => ( ( algebr142713364_sprod @ TD @ TE @ TA @ TB @ TC @ A @ B @ C @ D @ E @ F )
                = ( sprod @ TB @ TD @ TC @ ( C @ F ) @ D @ ( E @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(260,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: TA > TC,B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ),C: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) )] :
      ( ( member @ ( TA > TC ) @ A @ ( mHom @ a @ b @ TA @ TD @ TC @ TB @ r @ B @ C ) )
     => ( member @ ( TA > TC ) @ A
        @ ( pi @ TA @ TC @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ) @ B )
          @ ^ [D: TA] : ( carrier @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_mHom__func) ).

thf(1081,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: TA > TC,B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ),C: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) )] :
      ( ( member @ ( TA > TC ) @ A @ ( mHom @ a @ b @ TA @ TD @ TC @ TB @ r @ B @ C ) )
     => ( member @ ( TA > TC ) @ A
        @ ( pi @ TA @ TC @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ a @ TD ) ) @ B )
          @ ^ [D: TA] : ( carrier @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ a @ TB ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(165,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TE ) ),C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),D: TD > TA,E: TD] :
      ( ( module @ TD @ TB @ TF @ TE @ A @ B )
     => ( ( module @ TA @ TB @ TC @ TE @ C @ B )
       => ( ( member @ ( TD > TA ) @ D @ ( mHom @ TB @ TE @ TD @ TF @ TA @ TC @ B @ A @ C ) )
         => ( ( member @ TD @ E @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ) @ A ) )
           => ( member @ TA @ ( D @ E ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_Module_OmHom__mem) ).

thf(749,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TE ) ),C: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),D: TD > TA,E: TD] :
      ( ( module @ TD @ TB @ TF @ TE @ A @ B )
     => ( ( module @ TA @ TB @ TC @ TE @ C @ B )
       => ( ( member @ ( TD > TA ) @ D @ ( mHom @ TB @ TE @ TD @ TF @ TA @ TC @ B @ A @ C ) )
         => ( ( member @ TD @ E @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TB @ TF ) ) @ A ) )
           => ( member @ TA @ ( D @ E ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(102,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: set @ TA,C: set @ TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( aset_sum @ TA @ TB @ A @ B @ C ) )
           => ? [E: TA] :
                ( ( member @ TA @ E @ B )
                & ? [F: TA] :
                    ( ( member @ TA @ F @ C )
                    & ( D
                      = ( pop @ TA @ TB @ A @ E @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_aGroup_Omem__set__sum) ).

thf(557,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: set @ TA,C: set @ TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( aset_sum @ TA @ TB @ A @ B @ C ) )
           => ? [E: TA] :
                ( ( member @ TA @ E @ B )
                & ? [F: TA] :
                    ( ( member @ TA @ F @ C )
                    & ( D
                      = ( pop @ TA @ TB @ A @ E @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(193,axiom,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
         => ( ( pop @ a @ ( ring_ext @ a @ TA ) @ A @ B @ C )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_Subring__pOp__ring__pOp) ).

thf(855,plain,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a,C: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
         => ( ( pop @ a @ ( ring_ext @ a @ TA ) @ A @ B @ C )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(256,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ D ) @ ( pop @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_aGroup_Oag__add4__rel) ).

thf(1070,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA,E: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
             => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ ( pop @ TA @ TB @ A @ D @ E ) )
                = ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ D ) @ ( pop @ TA @ TB @ A @ C @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(58,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( pop @ TA @ TB @ A @ B @ C )
            = ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_aGroup_Oag__add__commute) ).

thf(414,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( pop @ TA @ TB @ A @ B @ C )
            = ( pop @ TA @ TB @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(23,axiom,
    ! [TA: $tType] :
      ( ( semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_add_Oassoc) ).

thf(327,plain,
    ! [TA: $tType] :
      ( ( semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(162,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ B )
          = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_ring__inv1__3) ).

thf(739,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ B )
          = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(56,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_add__right__cancel) ).

thf(409,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(16,axiom,
    plus @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oplus) ).

thf(310,plain,
    plus @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(73,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ D ) @ ( nscal @ TB @ TA @ A @ C @ D ) )
            = ( nscal @ TB @ TA @ A @ ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ B @ C ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_Ring_OnsDistrL) ).

thf(463,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ),B: TB,C: TB,D: nat] :
      ( ( ring @ TB @ TA @ A )
     => ( ( member @ TB @ B @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
       => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) @ A ) )
         => ( ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ ( nscal @ TB @ TA @ A @ B @ D ) @ ( nscal @ TB @ TA @ A @ C @ D ) )
            = ( nscal @ TB @ TA @ A @ ( pop @ TB @ ( ring_ext @ TB @ TA ) @ A @ B @ C ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(128,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
        = ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_ring__times__minusr) ).

thf(643,plain,
    ! [A: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A )
        = ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(248,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [E: TA] : C ) )
     => ( ( A = D )
       => ( member @ ( TA > TB ) @ D
          @ ( pi @ TA @ TB @ B
            @ ^ [E: TA] : C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_eq__fun) ).

thf(1038,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: TA > TB] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [E: TA] : C ) )
     => ( ( A = D )
       => ( member @ ( TA > TB ) @ D
          @ ( pi @ TA @ TB @ B
            @ ^ [E: TA] : C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(39,axiom,
    ab_semigroup_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add) ).

thf(370,plain,
    ab_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(76,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA] :
      ( ! [F: TA] :
          ( ( member @ TA @ F @ A )
         => ( aGroup @ TB @ TC @ ( B @ F ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ( member @ TA @ E @ A )
           => ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D @ E )
              = ( pop @ TB @ TC @ ( B @ E ) @ ( C @ E ) @ ( D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_prod__pOp__mem__i) ).

thf(470,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB,E: TA] :
      ( ! [F: TA] :
          ( ( member @ TA @ F @ A )
         => ( aGroup @ TB @ TC @ ( B @ F ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ( member @ TA @ E @ A )
           => ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D @ E )
              = ( pop @ TB @ TC @ ( B @ E ) @ ( C @ E ) @ ( D @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(108,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D )
            = ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_prod__pOp__commute) ).

thf(576,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
         => ( ( prod_pOp @ TA @ TB @ TC @ A @ B @ C @ D )
            = ( prod_pOp @ TA @ TB @ TC @ A @ B @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(132,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
              = ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_aGroup_Oag__pOp__assoc) ).

thf(656,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( pop @ TA @ TB @ A @ ( pop @ TA @ TB @ A @ B @ C ) @ D )
              = ( pop @ TA @ TB @ A @ B @ ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(238,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( B = C )
             => ( ( pop @ TA @ TB @ A @ B @ D )
                = ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_aGroup_Oag__pOp__add__r) ).

thf(1008,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( B = C )
             => ( ( pop @ TA @ TB @ A @ B @ D )
                = ( pop @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(117,axiom,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: set @ TB,C: set @ TA,D: TB] :
      ( ( member @ ( TB > TA ) @ A
        @ ( pi @ TB @ TA @ B
          @ ^ [E: TB] : C ) )
     => ( ( member @ TB @ D @ B )
       => ( member @ TA @ ( A @ D ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_Algebra1_Ofuncset__mem) ).

thf(612,plain,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: set @ TB,C: set @ TA,D: TB] :
      ( ( member @ ( TB > TA ) @ A
        @ ( pi @ TB @ TA @ B
          @ ^ [E: TB] : C ) )
     => ( ( member @ TB @ D @ B )
       => ( member @ TA @ ( A @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(265,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ~ ( zeroring @ TA @ TB @ A )
       => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ~ ( unit @ TA @ TB @ A @ B )
           => ? [C: set @ TA] :
                ( ( maximal_ideal @ TA @ TB @ A @ C )
                & ( member @ TA @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_Ring_Ononunit__contained__maxid) ).

thf(1094,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ~ ( zeroring @ TA @ TB @ A )
       => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ~ ( unit @ TA @ TB @ A @ B )
           => ? [C: set @ TA] :
                ( ( maximal_ideal @ TA @ TB @ A @ C )
                & ( member @ TA @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(229,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ B @ A ) @ ( tp @ a @ b @ r @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_ring__distrib2) ).

thf(978,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ B @ A ) @ ( tp @ a @ b @ r @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(35,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_self__le) ).

thf(363,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(97,axiom,
    ! [A: a,B: nat,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ B ) @ ( nscal @ a @ b @ r @ A @ C ) )
        = ( nscal @ a @ b @ r @ A @ ( plus_plus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_nsDistr) ).

thf(539,plain,
    ! [A: a,B: nat,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( nscal @ a @ b @ r @ A @ B ) @ ( nscal @ a @ b @ r @ A @ C ) )
        = ( nscal @ a @ b @ r @ A @ ( plus_plus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(79,axiom,
    ( ( zeroring @ a @ b @ r )
   => ~ ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_zeroring__no__maximal) ).

thf(482,plain,
    ( ( zeroring @ a @ b @ r )
   => ~ ? [A: set @ a] : ( maximal_ideal @ a @ b @ r @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(137,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( B = C )
             => ( ( pop @ TA @ TB @ A @ D @ B )
                = ( pop @ TA @ TB @ A @ D @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_aGroup_Oag__pOp__add__l) ).

thf(671,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( B = C )
             => ( ( pop @ TA @ TB @ A @ D @ B )
                = ( pop @ TA @ TB @ A @ D @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(141,axiom,
    ( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
    @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
      @ ^ [A: a] :
          ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
          @ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_tp__closed) ).

thf(688,plain,
    ( member @ ( a > a > a ) @ ( tp @ a @ b @ r )
    @ ( pi @ a @ ( a > a ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
      @ ^ [A: a] :
          ( pi @ a @ a @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
          @ ^ [B: a] : ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(84,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).

thf(495,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(93,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_rg__distrib) ).

thf(524,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( tp @ a @ b @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(109,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
      ( ( aGroup @ TA @ TB @ A )
     => ( member @ ( TA > TA ) @ ( mop @ TA @ TB @ A )
        @ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
          @ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_aGroup_Omop__closed) ).

thf(581,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB )] :
      ( ( aGroup @ TA @ TB @ A )
     => ( member @ ( TA > TA ) @ ( mop @ TA @ TB @ A )
        @ ( pi @ TA @ TA @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A )
          @ ^ [B: TA] : ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(38,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( module @ TC @ TA @ TD @ TB @ A @ B )
     => ( ring @ TA @ TB @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_Module_Osc__Ring) ).

thf(368,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( module @ TC @ TA @ TD @ TB @ A @ B )
     => ( ring @ TA @ TB @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(189,axiom,
    ! [A: nat] :
      ( member @ nat @ A
      @ ( collect @ nat
        @ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_n__in__Nsetn) ).

thf(844,plain,
    ! [A: nat] :
      ( member @ nat @ A
      @ ( collect @ nat
        @ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(91,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( sr2 @ TA @ TB )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
            ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
            & ( member @ TA @ ( un @ TA @ TB @ A ) @ B )
            & ! [C: TA] :
                ( ( member @ TA @ C @ B )
               => ! [D: TA] :
                    ( ( member @ TA @ D @ B )
                   => ( ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ B )
                      & ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_sr__def) ).

thf(517,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( sr2 @ TA @ TB )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
            ( ( ord_less_eq @ ( set @ TA ) @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
            & ( member @ TA @ ( un @ TA @ TB @ A ) @ B )
            & ! [C: TA] :
                ( ( member @ TA @ C @ B )
               => ! [D: TA] :
                    ( ( member @ TA @ D @ B )
                   => ( ( member @ TA @ ( pop @ TA @ ( ring_ext @ TA @ TB ) @ A @ C @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ D ) ) @ B )
                      & ( member @ TA @ ( tp @ TA @ TB @ A @ C @ D ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(20,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_add__mono__thms__linordered__semiring_I1_J) ).

thf(319,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(124,axiom,
    ! [A: a] :
      ( ~ ( zeroring @ a @ b @ r )
     => ( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
       => ! [B: a] :
            ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_J__rad__unit) ).

thf(632,plain,
    ! [A: a] :
      ( ~ ( zeroring @ a @ b @ r )
     => ( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
       => ! [B: a] :
            ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( unit @ a @ b @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( un @ a @ b @ r ) @ ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(52,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_add__left__mono) ).

thf(401,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(158,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( maximal_ideal @ TA @ TB @ A @ B )
       => ~ ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_Ring_Omaximal__ideal__proper) ).

thf(731,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: set @ TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( maximal_ideal @ TA @ TB @ A @ B )
       => ~ ( member @ TA @ ( un @ TA @ TB @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(234,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( mop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_aGroup_Oag__inv__inv) ).

thf(995,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( mop @ TA @ TB @ A @ ( mop @ TA @ TB @ A @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(105,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( member @ TA @ ( mop @ TA @ TB @ A @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_aGroup_Oag__mOp__closed) ).

thf(568,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( member @ TA @ ( mop @ TA @ TB @ A @ B ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(8,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_add__mono) ).

thf(290,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(89,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_pop__aassoc) ).

thf(510,plain,
    ! [A: a,B: a,C: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( pop @ a @ ( ring_ext @ a @ b ) @ r @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ B ) @ C )
            = ( pop @ a @ ( ring_ext @ a @ b ) @ r @ A @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(78,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
          = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_ring__inv1__1) ).

thf(478,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
          = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(1,conjecture,
    ! [A: c > d] :
      ( ~ ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
      | ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).

thf(2,negated_conjecture,
    ~ ! [A: c > d] :
        ( ~ ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
        | ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
          = A ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(272,plain,
    ~ ! [A: c > d] :
        ( ~ ( member @ ( c > d ) @ A @ ( carr_prodag @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
        | ( ( prod_pOp @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m @ A @ ( prod_zero @ c @ d @ ( module_ext @ d @ a @ e ) @ i @ m ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(203,axiom,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
            = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) )
          & ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
            = ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_ring__inv1) ).

thf(899,plain,
    ! [A: a,B: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
            = ( tp @ a @ b @ r @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ A ) @ B ) )
          & ( ( mop @ a @ ( ring_ext @ a @ b ) @ r @ ( tp @ a @ b @ r @ A @ B ) )
            = ( tp @ a @ b @ r @ A @ ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(100,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_prod__mOp__mem) ).

thf(549,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(152,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TD,D: TA] :
      ( ( module @ TA @ TD @ TB @ TC @ A @ B )
     => ( ( member @ TD @ C @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( module_ext @ TA @ TD @ TB ) @ A @ ( sprod @ TA @ TD @ TB @ A @ C @ D ) )
            = ( sprod @ TA @ TD @ TB @ A @ ( mop @ TD @ ( ring_ext @ TD @ TC ) @ B @ C ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_Module_Osc__minus__am1) ).

thf(716,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ),C: TD,D: TA] :
      ( ( module @ TA @ TD @ TB @ TC @ A @ B )
     => ( ( member @ TD @ C @ ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TB ) ) @ A ) )
         => ( ( mop @ TA @ ( module_ext @ TA @ TD @ TB ) @ A @ ( sprod @ TA @ TD @ TB @ A @ C @ D ) )
            = ( sprod @ TA @ TD @ TB @ A @ ( mop @ TD @ ( ring_ext @ TD @ TC ) @ B @ C ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(179,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
         => ( ! [E: TA] :
                ( ( member @ TA @ E @ A )
               => ( ( C @ E )
                  = ( D @ E ) ) )
           => ( C = D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_prodM__mem__eq) ).

thf(800,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ a @ TC ) ) ),C: TA > TB,D: TA > TB] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( module @ TB @ a @ TC @ b @ ( B @ E ) @ r ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
       => ( ( member @ ( TA > TB ) @ D @ ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ a @ product_unit ) ) @ ( algebr1000837334_prodM @ a @ b @ TA @ TB @ TC @ r @ A @ B ) ) )
         => ( ! [E: TA] :
                ( ( member @ TA @ E @ A )
               => ( ( C @ E )
                  = ( D @ E ) ) )
           => ( C = D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(13,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__cong) ).

thf(300,plain,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(225,axiom,
    ! [A: set @ a,B: a,C: a] :
      ( ( mul_closed_set @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( ( member @ a @ C @ A )
         => ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_mul__closed__set__tOp__closed) ).

thf(967,plain,
    ! [A: set @ a,B: a,C: a] :
      ( ( mul_closed_set @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( ( member @ a @ C @ A )
         => ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(266,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA,E: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
           => ( ( sprod @ TA @ TC @ TB @ A @ C @ ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D @ E ) )
              = ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_Module_Osprod__r__distr) ).

thf(1097,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ),B: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ),C: TC,D: TA,E: TA] :
      ( ( module @ TA @ TC @ TB @ TD @ A @ B )
     => ( ( member @ TC @ C @ ( carrier @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TD ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
         => ( ( member @ TA @ E @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TC @ TB ) ) @ A ) )
           => ( ( sprod @ TA @ TC @ TB @ A @ C @ ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ D @ E ) )
              = ( pop @ TA @ ( module_ext @ TA @ TC @ TB ) @ A @ ( sprod @ TA @ TC @ TB @ A @ C @ D ) @ ( sprod @ TA @ TC @ TB @ A @ C @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(187,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ C @ B )
                = D )
             => ( C
                = ( pop @ TA @ TB @ A @ D @ ( mop @ TA @ TB @ A @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_aGroup_Oag__eq__sol2) ).

thf(835,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA,D: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
           => ( ( ( pop @ TA @ TB @ A @ C @ B )
                = D )
             => ( C
                = ( pop @ TA @ TB @ A @ D @ ( mop @ TA @ TB @ A @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(11,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
         => ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_add__right__imp__eq) ).

thf(294,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
         => ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(33,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_add_Ocommute) ).

thf(358,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(65,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( unit @ TA @ TB )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
            ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
            & ? [C: TA] :
                ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
                & ( ( tp @ TA @ TB @ A @ B @ C )
                  = ( un @ TA @ TB @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_Unit__def) ).

thf(437,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( unit @ TA @ TB )
      = ( ^ [A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
            ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
            & ? [C: TA] :
                ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
                & ( ( tp @ TA @ TB @ A @ B @ C )
                  = ( un @ TA @ TB @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(171,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
       => ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_dsum__iOp__mem) ).

thf(767,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( aGroup @ TB @ TC @ ( B @ D ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) )
       => ( member @ ( TA > TB ) @ ( prod_mOp @ TA @ TB @ TC @ A @ B @ C ) @ ( carr_dsumag @ TA @ TB @ TC @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(135,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ B @ C ) @ A )
          = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_nsMulDistrR) ).

thf(664,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ ( nscal @ a @ b @ r @ B @ C ) @ A )
          = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(185,axiom,
    ! [A: c] :
      ( ( member @ c @ A @ i )
     => ( module @ d @ a @ e @ b @ ( m @ A ) @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(828,plain,
    ! [A: c] :
      ( ( member @ c @ A @ i )
     => ( module @ d @ a @ e @ b @ ( m @ A ) @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(125,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ C @ D ) )
              = ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_ring__tOp__rel) ).

thf(635,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( member @ a @ C @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
         => ( ( member @ a @ D @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
           => ( ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ ( tp @ a @ b @ r @ C @ D ) )
              = ( tp @ a @ b @ r @ ( tp @ a @ b @ r @ A @ C ) @ ( tp @ a @ b @ r @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(221,axiom,
    aGroup @ a @ ( ring_ext @ a @ b ) @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_ring__is__ag) ).

thf(955,plain,
    aGroup @ a @ ( ring_ext @ a @ b ) @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(177,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
      ( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
     => ( ( member @ TA @ D @ B )
       => ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_Pi__mem) ).

thf(796,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB ),D: TA] :
      ( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
     => ( ( member @ TA @ D @ B )
       => ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(210,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: TB,D: TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
         => ( member @ TA @ ( sprod @ TA @ TB @ TC @ A @ C @ D ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_Module_Osprod__closed) ).

thf(928,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: TB,D: TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( member @ TB @ C @ ( carrier @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ) @ B ) )
       => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
         => ( member @ TA @ ( sprod @ TA @ TB @ TC @ A @ C @ D ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(246,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_Ring_Omem__subring__mem__ring) ).

thf(1033,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ),C: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) ) @ B ) )
         => ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(51,axiom,
    comm_monoid_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).

thf(400,plain,
    comm_monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(188,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ TA @ D @ A )
         => ( ( prod_mOp @ TA @ TB @ TC @ A @ B @ C @ D )
            = ( mop @ TB @ TC @ ( B @ D ) @ ( C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_prod__mOp__mem__i) ).

thf(839,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: set @ TA,B: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TC ) ),C: TA > TB,D: TA] :
      ( ! [E: TA] :
          ( ( member @ TA @ E @ A )
         => ( aGroup @ TB @ TC @ ( B @ E ) ) )
     => ( ( member @ ( TA > TB ) @ C @ ( carr_prodag @ TA @ TB @ TC @ A @ B ) )
       => ( ( member @ TA @ D @ A )
         => ( ( prod_mOp @ TA @ TB @ TC @ A @ B @ C @ D )
            = ( mop @ TB @ TC @ ( B @ D ) @ ( C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(157,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
              = ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_Ring_Oring__tOp__assoc) ).

thf(728,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA,C: TA,D: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
         => ( ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
           => ( ( tp @ TA @ TB @ A @ ( tp @ TA @ TB @ A @ B @ C ) @ D )
              = ( tp @ TA @ TB @ A @ B @ ( tp @ TA @ TB @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(241,axiom,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( ( mop @ a @ ( ring_ext @ a @ TA ) @ A @ B )
          = ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_Subring__minus__ring__minus) ).

thf(1019,plain,
    ! [TA: $tType,A: carrier_ext @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ),B: a] :
      ( ( subring @ a @ b @ TA @ r @ A )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ TA ) ) @ A ) )
       => ( ( mop @ a @ ( ring_ext @ a @ TA ) @ A @ B )
          = ( mop @ a @ ( ring_ext @ a @ b ) @ r @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(94,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) )] :
      ( ( ring @ TC @ TE @ A )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ B )
           => ( module @ TB @ TC @ TD @ TE @ ( C @ D ) @ A ) )
       => ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TC @ product_unit ) ) @ ( algebr1000837334_prodM @ TC @ TE @ TA @ TB @ TD @ A @ B @ C ) )
          = ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_Ring_OprodM__carr) ).

thf(528,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TE ) ),B: set @ TA,C: TA > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TC @ TD ) ) )] :
      ( ( ring @ TC @ TE @ A )
     => ( ! [D: TA] :
            ( ( member @ TA @ D @ B )
           => ( module @ TB @ TC @ TD @ TE @ ( C @ D ) @ A ) )
       => ( ( carrier @ ( TA > TB ) @ ( aGroup_ext @ ( TA > TB ) @ ( module_ext @ ( TA > TB ) @ TC @ product_unit ) ) @ ( algebr1000837334_prodM @ TC @ TE @ TA @ TB @ TD @ A @ B @ C ) )
          = ( carr_prodag @ TA @ TB @ ( module_ext @ TB @ TC @ TD ) @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(233,axiom,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).

thf(992,plain,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(191,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ TA @ ( un @ TA @ TB @ A ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_Ring_Oun__closed) ).

thf(850,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( member @ TA @ ( un @ TA @ TB @ A ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(230,axiom,
    ! [A: a] :
      ( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
     => ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_J__rad__mem) ).

thf(982,plain,
    ! [A: a] :
      ( ( member @ a @ A @ ( j_rad @ a @ b @ r ) )
     => ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(237,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ ( nscal @ a @ b @ r @ B @ C ) )
          = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_nsMulDistrL) ).

thf(1004,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member @ a @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
     => ( ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
       => ( ( tp @ a @ b @ r @ A @ ( nscal @ a @ b @ r @ B @ C ) )
          = ( nscal @ a @ b @ r @ ( tp @ a @ b @ r @ A @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(26,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( un @ TA @ TC @ B )
          = ( un @ TA @ TB @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_Ring_OSubring__one__ring__one) ).

thf(337,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TC ) )] :
      ( ( ring @ TA @ TB @ A )
     => ( ( subring @ TA @ TB @ TC @ A @ B )
       => ( ( un @ TA @ TC @ B )
          = ( un @ TA @ TB @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(215,axiom,
    ! [A: set @ a,B: a,C: a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( ( member @ a @ C @ A )
         => ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_Sr__tOp__closed) ).

thf(940,plain,
    ! [A: set @ a,B: a,C: a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( ( member @ a @ C @ A )
         => ( member @ a @ ( tp @ a @ b @ r @ B @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(213,axiom,
    ! [A: set @ a,B: a,C: a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( ( member @ a @ C @ A )
         => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_Sr__pOp__closed) ).

thf(935,plain,
    ! [A: set @ a,B: a,C: a] :
      ( ( sr2 @ a @ b @ r @ A )
     => ( ( member @ a @ B @ A )
       => ( ( member @ a @ C @ A )
         => ( member @ a @ ( pop @ a @ ( ring_ext @ a @ b ) @ r @ B @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(133,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( member @ TA @ ( pop @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_aGroup_Oag__pOp__closed) ).

thf(659,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ TB ),B: TA,C: TA] :
      ( ( aGroup @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
       => ( ( member @ TA @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) )
         => ( member @ TA @ ( pop @ TA @ TB @ A @ B @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ TB ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(98,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
          = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_Ring_Oring__times__minusl) ).

thf(543,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),B: TA] :
      ( ( ring @ TA @ TB @ A )
     => ( ( member @ TA @ B @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ) @ A ) )
       => ( ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ B )
          = ( tp @ TA @ TB @ A @ ( mop @ TA @ ( ring_ext @ TA @ TB ) @ A @ ( un @ TA @ TB @ A ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(126,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: set @ TB] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [E: TA] : C ) )
     => ( ( ord_less_eq @ ( set @ TB ) @ C @ D )
       => ( member @ ( TA > TB ) @ A
          @ ( pi @ TA @ TB @ B
            @ ^ [E: TA] : D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_extend__fun) ).

thf(639,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: set @ TB,D: set @ TB] :
      ( ( member @ ( TA > TB ) @ A
        @ ( pi @ TA @ TB @ B
          @ ^ [E: TA] : C ) )
     => ( ( ord_less_eq @ ( set @ TB ) @ C @ D )
       => ( member @ ( TA > TB ) @ A
          @ ( pi @ TA @ TB @ B
            @ ^ [E: TA] : D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(70,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB )] :
      ( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
      = ( ! [D: TA] :
            ( ( member @ TA @ D @ B )
           => ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_Pi__iff) ).

thf(454,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: set @ TA,C: TA > ( set @ TB )] :
      ( ( member @ ( TA > TB ) @ A @ ( pi @ TA @ TB @ B @ C ) )
      = ( ! [D: TA] :
            ( ( member @ TA @ D @ B )
           => ( member @ TB @ ( A @ D ) @ ( C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(1117,plain,
    $false,
    inference(cvc4,[status(thm)],[892,809,1024,365,846,777,1110,352,909,628,1083,962,408,683,276,1073,449,379,440,614,861,1078,385,533,500,814,698,945,340,930,1001,709,417,973,565,651,389,724,570,719,372,460,587,619,917,751,817,606,289,316,1100,421,870,284,770,475,492,756,321,646,514,1051,905,602,280,692,734,866,396,998,988,507,312,293,956,307,292,661,546,761,381,1067,798,1057,334,1042,349,392,297,429,746,1091,467,484,361,625,852,1027,324,1012,285,583,830,1089,874,948,1064,403,356,593,753,537,1075,590,970,706,622,701,654,457,984,721,779,371,503,343,311,1054,953,1086,938,398,1035,773,794,412,425,1103,573,689,733,819,743,711,562,975,641,445,1107,466,919,704,947,713,498,764,406,451,737,434,303,863,387,609,726,520,915,668,330,345,858,367,552,1029,895,943,1022,932,1061,912,959,695,883,1015,1113,426,680,374,487,812,617,490,675,1081,749,557,855,1070,414,327,739,409,310,463,643,1038,370,470,576,656,1008,612,1094,978,363,539,482,671,688,495,524,581,368,844,517,319,632,401,731,995,568,290,510,478,272,899,549,716,800,300,967,1097,835,294,358,437,767,664,828,635,955,796,928,1033,400,839,728,1019,528,992,850,982,1004,337,940,935,659,543,639,454]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13  % Problem  : ITP027^2 : TPTP v8.2.0. Released v7.5.0.
% 0.08/0.16  % Command  : run_Leo-III %s %d
% 0.15/0.37  % Computer : n011.cluster.edu
% 0.15/0.37  % Model    : x86_64 x86_64
% 0.15/0.37  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37  % Memory   : 8042.1875MB
% 0.15/0.37  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37  % CPULimit : 300
% 0.15/0.37  % WCLimit  : 300
% 0.15/0.37  % DateTime : Sat May 18 17:02:38 EDT 2024
% 0.15/0.37  % CPUTime  : 
% 1.20/1.16  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 2.71/1.65  % [INFO] 	 Parsing done (486ms). 
% 2.71/1.66  % [INFO] 	 Running in sequential loop mode. 
% 4.28/2.09  % [INFO] 	 eprover registered as external prover. 
% 4.28/2.09  % [INFO] 	 cvc4 registered as external prover. 
% 4.28/2.10  % [INFO] 	 Scanning for conjecture ... 
% 4.83/2.36  % [INFO] 	 Found a conjecture (or negated_conjecture) and 270 axioms. Running axiom selection ... 
% 5.58/2.56  % [INFO] 	 Axiom selection finished. Selected 269 axioms (removed 1 axioms). 
% 7.76/3.14  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 7.76/3.19  % [INFO] 	 Type checking passed. 
% 8.17/3.19  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 259.52/217.29  % External prover 'cvc4' found a proof!
% 259.52/217.29  % [INFO] 	 Killing All external provers ... 
% 259.52/217.29  % Time passed: 216752ms (effective reasoning time: 215622ms)
% 259.52/217.29  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 259.52/217.30  % Axioms used in derivation (269): fact_124_Ring_OmHom__func, fact_246_add__mono__thms__linordered__semiring_I4_J, fact_13_ring__is__ag, fact_228_Ring_Oun__closed, fact_150_ring__inv1__2, fact_41_Sr__ring, fact_240_add_Ocommute, fact_119_Ring_OprodM__sprod__val, fact_217_ring__r__one, fact_244_group__cancel_Oadd2, fact_132_Module_Osprod__r__distr, fact_71_nsDistrL, fact_30_prod__mOp__mem, fact_198_Module_OmHom__inv, fact_80_Ring_Org__distrib, fact_191_Ring_Oring__distrib4, fact_173_Subring__subset, fact_42_mHom__func, fact_128_nsDistr, fact_145_id__maximal__Exist, fact_65_rg__distrib, fact_200_Module_Osc__minus__am, fact_12_prod__pOp__func, fact_235_Module_Osprod__one, fact_75_Ring_OSr__tOp__closed, fact_131_Module_OmHom__add, fact_64_Sr__pOp__closed, fact_26_Ring_OprodM__sprod__mem, fact_112_dsum__pOp__mem, fact_221_add__le__cancel__left, fact_227_mem__of__Nset, fact_118_Ring_Opop__closed, fact_202_Module_Osc__minus__am1, fact_122_Module_OmHom__lin, fact_212_Sr__one, fact_109_Ring_OnsEqElm, fact_2_bivar__fun__mem, fact_34_Pi__mem, fact_79_Ring_Oring__distrib1, fact_197_le__imp__add__int, fact_154_not__sub, fact_83_Ring_OnsDistrL, tcon_Nat_Onat___Orderings_Oord_1, fact_167_Pi__anti__mono, fact_59_tp__assoc, tcon_fun___Orderings_Oord, fact_209_add__le__cancel__right, fact_229_Ring_Omaximal__ideal__proper, fact_165_Nset__le, fact_176_Ring_Oring__inv1__1, fact_126_Ring_OSubring__tOp__ring__tOp, fact_172_Ring_OSr__mOp__closed, fact_186_aGroup_Oag__p__inv, fact_98_aGroup_Oaassoc, fact_31_prod__zero__func, fact_233_Ring_Oring__l__one, fact_93_aGroup_Oag__pOp__assoc, fact_169_aGroup_Oag__mOp__closed, fact_210_un__closed, fact_84_Ring_OSr__ring, fact_96_aGroup_OgEQAddcross, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, fact_54_rMulLC, fact_137_Ring_Ozeroring__J__rad__empty, fact_0_Ring__axioms, fact_153_Subring__minus__ring__minus, fact_14_prodM__mem__eq, fact_245_group__cancel_Oadd1, fact_43_rInvim__def, fact_44_dsum__pOp__func, fact_220_ring__times__minusl, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_238_add__left__imp__eq, fact_77_Ring_Oring__distrib3, fact_248_Ring_Oring__times__minusl, fact_74_mul__closed__set__tOp__closed, fact_66_ring__distrib1, fact_62_pop__commute, fact_215_rg__l__unit, tcon_Product__Type_Ounit___Orderings_Oord_4, fact_52_pop__closed, fact_63_Sr__tOp__closed, fact_178_Ring_Oid__maximal__Exist, fact_113_dsum__iOp__mem, tcon_Nat_Onat___Groups_Ocancel__semigroup__add, fact_129_add__bothl, fact_214_add__left__cancel, fact_46_Collect__mem__eq, fact_72_Subring__tOp__ring__tOp, fact_23_prodag__sameTr3, fact_87_aGroup_OpOp__assocTr43, fact_35_Pi__iff, tcon_Nat_Onat___Groups_Oplus, fact_57_ring__tOp__commute, fact_40_prodM__sprod__val, fact_133_Module_Omodule__is__ag, fact_237_add__right__imp__eq, fact_103_Ring_Otp__commute, fact_130_add__both, tcon_Set_Oset___Orderings_Oord_2, fact_146_mop__closed, fact_187_Ring_Omul__closed__set__sub, fact_182_aGroup_OpOp__assoc__cancel, fact_158_sub__which1, fact_234_Ring_Org__l__unit, fact_161_subset__self, fact_16_Ring_ORing, fact_149_ring__inv1__1, fact_199_dsumag__sub__prodag, fact_51_tp__closed, fact_4_prodM__sprod__mem, fact_32_prod__mOp__func, fact_243_add_Oassoc, fact_49_nsEqElm, fact_219_ring__times__minusr, fact_155_eqsets__sub, fact_183_aGroup_Oag__diff__minus, fact_218_Subring__one__ring__one, fact_53_rEQMulR, fact_151_ring__inv1__3, fact_82_Ring_OnsMulDistrL, fact_104_Ring_Otp__assoc, fact_69_nsMulDistrL, fact_206_Ring_Ononunit__contained__maxid, fact_144_zeroring__no__maximal, fact_204_Ring_Osum__mult__pOp__closed, fact_205_Ring_Oset__sum__mem, fact_110_Ring_OnsClose, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_136_Ring_Omul__closed__set__tOp__closed, fact_177_Ring_Oring__inv1, fact_89_aGroup_OpOp__assocTr41, fact_38_mem__subring__mem__ring, fact_10_funcset__id, fact_192_nonunit__contained__maxid, fact_135_Ring_OnsDistr, fact_115_aGroup_Opop__closed, fact_88_aGroup_OpOp__assocTr42, fact_45_mem__Collect__eq, fact_152_ring__distrib4, fact_67_ring__distrib2, fact_90_aGroup_Oag__pOp__closed, fact_226_n__in__Nsetn, fact_81_Ring_OnsMulDistrR, fact_143_mul__closed__set__sub, fact_5_prodM__carr, fact_239_add_Oleft__commute, fact_201_Ring_Osum__mult__subR, fact_7_funcTr, fact_120_dsum__iOp__func, fact_170_aGroup_Oag__inv__inv, fact_106_Ring_OrMulLC, fact_162_not__subseteq, fact_252_add__mono__thms__linordered__semiring_I2_J, fact_166_Pi__mono, fact_24_prodag__sameTr4, fact_156_ge__convert, fact_56_ring__tOp__closed, fact_255_add__left__mono, fact_163_proper__subset, fact_127_Module_Osprod__closed, fact_116_Ring_OJ__rad__mem, fact_6_eq__fun, fact_225_nat__eq__le, fact_15_J__rad__mem, fact_37_PiE, fact_18_Ring_OprodM__mem__eq, fact_95_aGroup_Oag__pOp__add__l, fact_39_subring__Ring, fact_47_Collect__cong, fact_36_Pi__I_H, fact_33_Pi__cong, fact_193_set__sum__mem, fact_76_Ring_OSr__pOp__closed, fact_142_Ring_Omem__subring__mem__ring, fact_117_Ring_Otp__closed, fact_159_sub__which2, fact_223_primary__ideal__proper1, fact_171_aGroup_Oag__inv__inj, fact_55_ring__tOp__assoc, fact_50_nsClose, fact_27_prod__pOp__commute, fact_28_prod__pOp__assoc, fact_231_Ring_OSr__one, fact_222_localring__unit, fact_101_Ring_Oring__tOp__assoc, fact_203_Ring_Otimes__mem__sum__mult, fact_160_sets__not__eq, fact_134_Module_Osprod__l__distr, fact_179_Ring_Ozeroring__no__maximal, fact_174_Ring_Oring__inv1__3, fact_11_Algebra1_Ofuncset__mem, fact_194_sum__mult__pOp__closed, fact_121_zeroring__J__rad__empty, fact_157_le__convert, fact_188_mul__closed__set__def, fact_207_aGroup_Omem__set__sum, fact_92_aGroup_OpOp__cancel__l, fact_180_Ring_OSubring__minus__ring__minus, fact_17_Ring_Oring__is__ag, fact_94_aGroup_Oag__pOp__add__r, fact_175_Ring_Oring__inv1__2, fact_251_add__mono__thms__linordered__semiring_I3_J, fact_108_Ring_Opop__aassoc, conj_1, fact_184_aGroup_Oag__eq__sol2, fact_181_Ring_Omop__closed, fact_29_prod__pOp__mem, fact_99_Ring_Oring__tOp__commute, fact_216_ring__l__one, fact_20_prodag__sameTr1, fact_139_Ring_Osubring__Ring, fact_85_aGroup_Oag__pOp__commute, fact_208_J__rad__unit, fact_1_bivar__fun, fact_21_prodag__comp__i, fact_190_prod__mOp__mem__i, fact_211_maximal__ideal__proper, tcon_HOL_Obool___Orderings_Oord_3, fact_196_times__mem__sum__mult, fact_61_pop__aassoc, fact_70_nsMulDistrR, fact_195_sum__mult__subR, fact_247_ab__semigroup__add__class_Oadd__ac_I1_J, tcon_Nat_Onat___Groups_Osemigroup__add, fact_58_ring__tOp__rel, fact_141_Module_OmHom__mem, fact_8_funcsetI, fact_232_Ring_Oring__r__one, fact_78_Ring_Oring__distrib2, fact_253_add__mono__thms__linordered__semiring_I1_J, fact_25_Ring_OprodM__carr, fact_125_Ring_OSubring__pOp__ring__pOp, fact_213_add__right__cancel, fact_105_Ring_OrEQMulR, fact_73_Subring__pOp__ring__pOp, fact_224_self__le, fact_22_prodag__sameTr2, fact_102_Ring_Oring__tOp__rel, fact_138_Module_Osc__Ring, fact_60_tp__commute, fact_249_Ring_Oring__times__minusr, fact_97_aGroup_Oag__add4__rel, tcon_Nat_Onat___Groups_Oab__semigroup__add, fact_91_aGroup_OpOp__cancel__r, fact_241_add_Oright__cancel, fact_250_sr__def, conj_0, fact_147_Sr__mOp__closed, fact_9_eq__funcs, fact_111_prod__pOp__mem__i, fact_185_aGroup_Oag__eq__sol1, fact_230_Ring_OSubring__one__ring__one, fact_236_Unit__def, fact_189_aGroup_Omop__closed, fact_164_conditional__subset, fact_19_carr__prodag__mem__eq, fact_242_add_Oleft__cancel, fact_168_extend__fun, fact_107_Ring_Opop__commute, fact_254_add__mono, fact_68_ring__distrib3, fact_123_Module_Osprod__assoc, fact_3_Pi__I, fact_86_aGroup_Oag__add__commute, fact_148_ring__inv1, fact_114_dsum__zero__func, fact_100_Ring_Oring__tOp__closed, fact_140_Module_OmHom__eq
% 259.52/217.30  % No. of inferences in proof: 542
% 259.52/217.30  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 216752 ms resp. 215622 ms w/o parsing
% 261.17/217.63  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 261.17/217.63  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------