TSTP Solution File: ITP048^2 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP048^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n018.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 22:22:23 EDT 2024

% Result   : Theorem 65.45s 42.58s
% Output   : Refutation 66.10s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  352
% Syntax   : Number of formulae    :  654 ( 156 unt;  52 typ;   0 def)
%            Number of atoms       : 1821 ( 331 equ;   6 cnn)
%            Maximal formula atoms :   13 (   3 avg)
%            Number of connectives : 7022 ( 126   ~;  20   |;  96   &;5859   @)
%                                         (   0 <=>; 921  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   8 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :  210 ( 210   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   57 (  53 usr;  19 con; 0-5 aty)
%            Number of variables   : 1936 ( 108   ^1727   !;  66   ?;1936   :)
%                                         (  35  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(capacity_type,type,
    capacity: $tType ).

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

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

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

thf(size_type,type,
    size: 
      !>[TA: $tType] : $o ).

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

thf(order_type,type,
    order: 
      !>[TA: $tType] : $o ).

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

thf(no_bot_type,type,
    no_bot: 
      !>[TA: $tType] : $o ).

thf(no_top_type,type,
    no_top: 
      !>[TA: $tType] : $o ).

thf(linorder_type,type,
    linorder: 
      !>[TA: $tType] : $o ).

thf(preorder_type,type,
    preorder: 
      !>[TA: $tType] : $o ).

thf(wellorder_type,type,
    wellorder: 
      !>[TA: $tType] : $o ).

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

thf(dense_order_type,type,
    dense_order: 
      !>[TA: $tType] : $o ).

thf(linordered_idom_type,type,
    linordered_idom: 
      !>[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(canoni770627133id_add_type,type,
    canoni770627133id_add: 
      !>[TA: $tType] : $o ).

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

thf(ordere223160158up_add_type,type,
    ordere223160158up_add: 
      !>[TA: $tType] : $o ).

thf(strict2144017051up_add_type,type,
    strict2144017051up_add: 
      !>[TA: $tType] : $o ).

thf(connected_type,type,
    connected: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > $o ) ).

thf(dist_type,type,
    dist: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > nat > $o ) ).

thf(isPath_type,type,
    isPath: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).

thf(isShortestPath_type,type,
    isShortestPath: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).

thf(isSimplePath_type,type,
    isSimplePath: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).

thf(min_dist_type,type,
    min_dist: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > nat ) ).

thf(reachableNodes_type,type,
    reachableNodes: 
      !>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( set @ nat ) ) ).

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

thf(size_size_type,type,
    size_size: 
      !>[TA: $tType] : ( TA > nat ) ).

thf(ord_Least_type,type,
    ord_Least: 
      !>[TA: $tType] : ( ( TA > $o ) > TA ) ).

thf(ord_less_type,type,
    ord_less: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

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(c_type,type,
    c: ( product_prod @ nat @ nat ) > capacity ).

thf(c2_type,type,
    c2: ( product_prod @ nat @ nat ) > a ).

thf(p_type,type,
    p: list @ ( product_prod @ nat @ nat ) ).

thf(p1_type,type,
    p1: list @ ( product_prod @ nat @ nat ) ).

thf(p1a_type,type,
    p1a: list @ ( product_prod @ nat @ nat ) ).

thf(p2_type,type,
    p2: list @ ( product_prod @ nat @ nat ) ).

thf(p2_a_type,type,
    p2_a: list @ ( product_prod @ nat @ nat ) ).

thf(p3_type,type,
    p3: list @ ( product_prod @ nat @ nat ) ).

thf(s_type,type,
    s: nat ).

thf(t_type,type,
    t: nat ).

thf(u_type,type,
    u: nat ).

thf(ua_type,type,
    ua: nat ).

thf(v_type,type,
    v: nat ).

thf(va_type,type,
    va: nat ).

thf(201,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ! [D: nat] :
            ( ( dist @ a @ c2 @ A @ D @ C )
           => ( ord_less_eq @ nat @ B @ D ) )
       => ( ( min_dist @ a @ c2 @ A @ C )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_g_H_Omin__distI__eq) ).

thf(809,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ! [D: nat] :
            ( ( dist @ a @ c2 @ A @ D @ C )
           => ( ord_less_eq @ nat @ B @ D ) )
       => ( ( min_dist @ a @ c2 @ A @ C )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(185,axiom,
    ! [A: nat,B: nat] :
      ( ( min_dist @ capacity @ c @ A @ B )
      = ( ord_Least @ nat
        @ ^ [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_min__dist__def) ).

thf(760,plain,
    ! [A: nat,B: nat] :
      ( ( min_dist @ capacity @ c @ A @ B )
      = ( ord_Least @ nat
        @ ^ [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(296,axiom,
    ! [TA: $tType] :
      ( ( dense_order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ? [C: TA] :
              ( ( ord_less @ TA @ A @ C )
              & ( ord_less @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_dense) ).

thf(1105,plain,
    ! [TA: $tType] :
      ( ( dense_order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ? [C: TA] :
              ( ( ord_less @ TA @ A @ C )
              & ( ord_less @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(103,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_less__trans) ).

thf(555,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(226,axiom,
    ! [TA: $tType] :
      ( ( no_top @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_gt__ex) ).

thf(909,plain,
    ! [TA: $tType] :
      ( ( no_top @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(18,axiom,
    linordered_idom @ a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_1) ).

thf(333,plain,
    linordered_idom @ a,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(249,axiom,
    isShortestPath @ capacity @ c @ s @ p @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_SP) ).

thf(977,plain,
    isShortestPath @ capacity @ c @ s @ p @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(286,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( connected @ capacity @ c @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_connected__distI) ).

thf(1073,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( connected @ capacity @ c @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(220,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA] : ( A @ D @ D )
           => ( ! [D: TA,E: TA] :
                  ( ( A @ E @ D )
                 => ( A @ D @ E ) )
             => ( A @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_linorder__less__wlog) ).

thf(873,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA] : ( A @ D @ D )
           => ( ! [D: TA,E: TA] :
                  ( ( A @ E @ D )
                 => ( A @ D @ E ) )
             => ( A @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(40,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
            = ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_antisym__conv) ).

thf(379,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
            = ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(63,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_191_ord__eq__le__subst) ).

thf(440,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(129,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( dist @ a @ c2 @ A @ ( min_dist @ a @ c2 @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_g_H_Omin__dist__is__dist) ).

thf(614,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( dist @ a @ c2 @ A @ ( min_dist @ a @ c2 @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(121,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ capacity @ c @ A @ B @ C )
     => ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ A @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_isSPath__pathLE) ).

thf(597,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ capacity @ c @ A @ B @ C )
     => ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ A @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(91,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_add_Ocommute) ).

thf(523,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)],[91]) ).

thf(114,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( min_dist @ TA )
        = ( min_dist @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_Graph_Omin__dist_Ocong) ).

thf(582,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(273,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ( dist @ a @ c2 @ C @ D @ E )
       => ( dist @ a @ c2 @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_g_H_Odist__trans) ).

thf(1041,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ( dist @ a @ c2 @ C @ D @ E )
       => ( dist @ a @ c2 @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(217,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( ( ( min_dist @ a @ c2 @ A @ B )
          = C )
       => ( ( ord_less @ nat @ D @ C )
         => ? [E: nat] :
              ( ( connected @ a @ c2 @ A @ E )
              & ( ( min_dist @ a @ c2 @ A @ E )
                = D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_g_H_Omin__dist__less) ).

thf(861,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( ( ( min_dist @ a @ c2 @ A @ B )
          = C )
       => ( ( ord_less @ nat @ D @ C )
         => ? [E: nat] :
              ( ( connected @ a @ c2 @ A @ E )
              & ( ( min_dist @ a @ c2 @ A @ E )
                = D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(95,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ A @ B )
        | ( A = B ) )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_less__or__eq__imp__le) ).

thf(533,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ A @ B )
        | ( A = B ) )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(76,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ B @ A )
              & ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_dual__order_Oeq__iff) ).

thf(472,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ B @ A )
              & ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(202,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_ord__eq__less__trans) ).

thf(814,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(160,axiom,
    semigroup_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add) ).

thf(698,plain,
    semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(181,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_less__imp__not__eq2) ).

thf(747,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(96,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
      = ( ( ord_less @ nat @ A @ B )
        | ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_nat__neq__iff) ).

thf(538,plain,
    ! [A: nat,B: nat] :
      ( ( A != B )
      = ( ( ord_less @ nat @ A @ B )
        | ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(281,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ord_less_eq @ nat @ ( min_dist @ a @ c2 @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_g_H_Omin__dist__minD) ).

thf(1063,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ord_less_eq @ nat @ ( min_dist @ a @ c2 @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(22,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB] :
          ( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
         => ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_le__funI) ).

thf(340,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB] :
          ( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
         => ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(292,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ( dist @ capacity @ c @ C @ D @ E )
       => ( dist @ capacity @ c @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_dist__trans) ).

thf(1095,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ( dist @ capacity @ c @ C @ D @ E )
       => ( dist @ capacity @ c @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(223,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/sandbox2/benchmark/theBenchmark.p',fact_31_add__le__cancel__left) ).

thf(898,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)],[223]) ).

thf(193,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_less__imp__not__eq) ).

thf(782,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(209,axiom,
    ord @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_3) ).

thf(841,plain,
    ord @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(248,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_ord__less__eq__trans) ).

thf(973,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(275,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
     => ( isSimplePath @ capacity @ c @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_shortestPath__is__simple) ).

thf(1046,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
     => ( isSimplePath @ capacity @ c @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(43,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( preorder @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Opreorder) ).

thf(389,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( preorder @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(278,axiom,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( A @ B ) @ C ) @ ( A @ ( plus_plus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_mono__nat__linear__lb) ).

thf(1055,plain,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( A @ B ) @ C ) @ ( A @ ( plus_plus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(110,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 @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_add__right__mono) ).

thf(570,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 @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(208,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_add__less__cancel__right) ).

thf(838,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(24,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o] :
          ( ? [B: TA] : ( A @ B )
         => ( A @ ( ord_Least @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_LeastI__ex) ).

thf(344,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o] :
          ( ? [B: TA] : ( A @ B )
         => ( A @ ( ord_Least @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(30,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( connected @ TA @ A @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_Graph_Oconnected__distI) ).

thf(357,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( connected @ TA @ A @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(279,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/sandbox2/benchmark/theBenchmark.p',fact_42_add__le__imp__le__right) ).

thf(1059,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)],[279]) ).

thf(239,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isPath @ TA @ A @ B @ C @ D )
         => ( dist @ TA @ A @ B @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_Graph_OisPath__distD) ).

thf(949,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isPath @ TA @ A @ B @ C @ D )
         => ( dist @ TA @ A @ B @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(70,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_order__subst1) ).

thf(460,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(162,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/sandbox2/benchmark/theBenchmark.p',fact_51_add__left__mono) ).

thf(702,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)],[162]) ).

thf(182,axiom,
    ! [TA: $tType,A: TA > $o,B: TA,C: TA > nat,D: nat] :
      ( ( A @ B )
     => ( ! [E: TA] :
            ( ( A @ E )
           => ? [F: TA] :
                ( ( A @ F )
                & ~ ( ord_less_eq @ nat @ ( C @ F ) @ ( C @ E ) ) ) )
       => ? [E: TA] :
            ( ( A @ E )
            & ~ ( ord_less @ nat @ ( C @ E ) @ ( plus_plus @ nat @ ( C @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_ex__has__greatest__nat__lemma) ).

thf(751,plain,
    ! [TA: $tType,A: TA > $o,B: TA,C: TA > nat,D: nat] :
      ( ( A @ B )
     => ( ! [E: TA] :
            ( ( A @ E )
           => ? [F: TA] :
                ( ( A @ F )
                & ~ ( ord_less_eq @ nat @ ( C @ F ) @ ( C @ E ) ) ) )
       => ? [E: TA] :
            ( ( A @ E )
            & ~ ( ord_less @ nat @ ( C @ E ) @ ( plus_plus @ nat @ ( C @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(54,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_ord__le__eq__subst) ).

thf(416,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(65,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
     => ( isSimplePath @ a @ c2 @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_g_H_OshortestPath__is__simple) ).

thf(448,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
     => ( isSimplePath @ a @ c2 @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(9,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( type @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___HOL_Otype) ).

thf(316,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( type @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(136,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
              ( ( isPath @ TA @ A @ B @ C @ D )
              & ! [E: list @ ( product_prod @ nat @ nat )] :
                  ( ( isPath @ TA @ A @ B @ E @ D )
                 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_Graph_OisShortestPath__def) ).

thf(638,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
              ( ( isPath @ TA @ A @ B @ C @ D )
              & ! [E: list @ ( product_prod @ nat @ nat )] :
                  ( ( isPath @ TA @ A @ B @ E @ D )
                 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(214,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ~ ( ord_less @ TA @ A @ B ) )
          = ( ( ord_less @ TA @ B @ A )
            | ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_not__less__iff__gr__or__eq) ).

thf(853,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ~ ( ord_less @ TA @ A @ B ) )
          = ( ( ord_less @ TA @ B @ A )
            | ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(299,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( min_dist @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
              ( ord_Least @ nat
              @ ^ [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_Graph_Omin__dist__def) ).

thf(1115,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( min_dist @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
              ( ord_Least @ nat
              @ ^ [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(14,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ( A @ B )
         => ( A @ ( ord_Least @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_LeastI) ).

thf(325,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ( A @ B )
         => ( A @ ( ord_Least @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(77,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_190_order__subst2) ).

thf(475,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(183,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/sandbox2/benchmark/theBenchmark.p',fact_30_add__le__cancel__right) ).

thf(756,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)],[183]) ).

thf(195,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_add__mono__thms__linordered__field_I5_J) ).

thf(789,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(12,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ordere236663937imp_le @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(321,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ordere236663937imp_le @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(140,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
              ( ( isSimplePath @ TA @ A @ B @ C @ D )
              & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
                = ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_Graph_OisShortestPath__alt) ).

thf(646,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
              ( ( isSimplePath @ TA @ A @ B @ C @ D )
              & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
                = ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(235,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_add__less__cancel__left) ).

thf(937,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(200,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ? [C: TA] :
              ( B
              = ( plus_plus @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_le__iff__add) ).

thf(806,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ? [C: TA] :
              ( B
              = ( plus_plus @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(28,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( connected @ a @ c2 @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_g_H_Oconnected__distI) ).

thf(353,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( connected @ a @ c2 @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(79,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_eq__iff) ).

thf(480,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(266,axiom,
    ! [TA: $tType,A: TA > $o,B: TA,C: TA > nat] :
      ( ( A @ B )
     => ? [D: TA] :
          ( ( A @ D )
          & ! [E: TA] :
              ( ( A @ E )
             => ( ord_less_eq @ nat @ ( C @ D ) @ ( C @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_ex__has__least__nat) ).

thf(1020,plain,
    ! [TA: $tType,A: TA > $o,B: TA,C: TA > nat] :
      ( ( A @ B )
     => ? [D: TA] :
          ( ( A @ D )
          & ! [E: TA] :
              ( ( A @ E )
             => ( ord_less_eq @ nat @ ( C @ D ) @ ( C @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(26,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_order__class_Oorder_Oantisym) ).

thf(348,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(156,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ A )
           => ( ord_less @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_214_dual__order_Ostrict__trans) ).

thf(687,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ A )
           => ( ord_less @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(158,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ( ( '?' @ TA )
        = ( ^ [A: TA > $o] :
            ? [B: TA] :
              ( ( A @ B )
              & ! [C: TA] :
                  ( ( ord_less @ TA @ C @ B )
                 => ~ ( A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_exists__least__iff) ).

thf(692,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ( ( '?' @ TA )
        = ( ^ [A: TA > $o] :
            ? [B: TA] :
              ( ( A @ B )
              & ! [C: TA] :
                  ( ( ord_less @ TA @ C @ B )
                 => ~ ( A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(176,axiom,
    ordere223160158up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).

thf(734,plain,
    ordere223160158up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(67,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_Collect__cong) ).

thf(453,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)],[67]) ).

thf(259,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_Nat_Oex__has__greatest__nat) ).

thf(998,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(85,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ( dist @ TA @ A @ D @ E @ F )
           => ( ( ( min_dist @ TA @ A @ B @ F )
                = ( plus_plus @ nat @ C @ E ) )
             => ( ( min_dist @ TA @ A @ B @ D )
                = C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_Graph_Omin__dist__split_I1_J) ).

thf(507,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ( dist @ TA @ A @ D @ E @ F )
           => ( ( ( min_dist @ TA @ A @ B @ F )
                = ( plus_plus @ nat @ C @ E ) )
             => ( ( min_dist @ TA @ A @ B @ D )
                = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(7,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_order__class_Oorder_Oeq__iff) ).

thf(312,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(251,axiom,
    ! [TA: $tType] : ( size @ ( list @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_List_Olist___Nat_Osize_11) ).

thf(981,plain,
    ! [TA: $tType] : ( size @ ( list @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(187,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_le__eq__less__or__eq) ).

thf(766,plain,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(5,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_order_Otrans) ).

thf(307,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(62,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( dist @ TA )
        = ( dist @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_Graph_Odist_Ocong) ).

thf(439,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(147,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_less__not__sym) ).

thf(661,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(99,axiom,
    size @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osize) ).

thf(546,plain,
    size @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(283,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less_eq @ nat @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_add__leD2) ).

thf(1067,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less_eq @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(167,axiom,
    ! [TA: $tType] :
      ( ( no_bot @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_lt__ex) ).

thf(714,plain,
    ! [TA: $tType] :
      ( ( no_bot @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(19,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_le__funE) ).

thf(334,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(243,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/sandbox2/benchmark/theBenchmark.p',fact_82_group__cancel_Oadd2) ).

thf(957,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)],[243]) ).

thf(205,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
      = ( ( isSimplePath @ a @ c2 @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ a @ c2 @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_g_H_OisShortestPath__alt) ).

thf(825,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
      = ( ( isSimplePath @ a @ c2 @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ a @ c2 @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(191,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( B != A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_dual__order_Ostrict__iff__order) ).

thf(778,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( B != A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(1,conjecture,
    ord_less @ nat @ ( min_dist @ capacity @ c @ s @ t ) @ ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( ord_less @ nat @ ( min_dist @ capacity @ c @ s @ t ) @ ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(302,plain,
    ~ ( ord_less @ nat @ ( min_dist @ capacity @ c @ s @ t ) @ ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(204,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( ! [D: nat] :
            ( ( dist @ capacity @ c @ A @ D @ B )
           => ( ! [E: nat] :
                  ( ( dist @ capacity @ c @ A @ E @ B )
                 => ( ord_less_eq @ nat @ D @ E ) )
             => ( C @ D ) ) )
       => ( C @ ( min_dist @ capacity @ c @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_min__distI2) ).

thf(820,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( ! [D: nat] :
            ( ( dist @ capacity @ c @ A @ D @ B )
           => ( ! [E: nat] :
                  ( ( dist @ capacity @ c @ A @ E @ B )
                 => ( ord_less_eq @ nat @ D @ E ) )
             => ( C @ D ) ) )
       => ( C @ ( min_dist @ capacity @ c @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(51,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( cancel_semigroup_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add) ).

thf(407,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( cancel_semigroup_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(134,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
      = ( ? [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ a @ c2 @ A @ D @ C )
            & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
              = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_g_H_Odist__def) ).

thf(630,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
      = ( ? [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ a @ c2 @ A @ D @ C )
            & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
              = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(81,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ordere223160158up_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add) ).

thf(484,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ordere223160158up_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(32,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_antisym) ).

thf(361,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(133,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ ( min_dist @ capacity @ c @ A @ B ) )
       => ? [D: nat] :
            ( ( connected @ capacity @ c @ A @ D )
            & ( ( min_dist @ capacity @ c @ A @ D )
              = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_min__dist__le) ).

thf(625,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ ( min_dist @ capacity @ c @ A @ B ) )
       => ? [D: nat] :
            ( ( connected @ capacity @ c @ A @ D )
            & ( ( min_dist @ capacity @ c @ A @ D )
              = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(60,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).

thf(435,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(16,axiom,
    order @ $o,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_9) ).

thf(329,plain,
    order @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(115,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_le__trans) ).

thf(583,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(84,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
           => ~ ( ord_less_eq @ TA @ B @ C ) )
         => ( ( ( ord_less_eq @ TA @ B @ A )
             => ~ ( ord_less_eq @ TA @ A @ C ) )
           => ( ( ( ord_less_eq @ TA @ A @ C )
               => ~ ( ord_less_eq @ TA @ C @ B ) )
             => ( ( ( ord_less_eq @ TA @ C @ B )
                 => ~ ( ord_less_eq @ TA @ B @ A ) )
               => ( ( ( ord_less_eq @ TA @ B @ C )
                   => ~ ( ord_less_eq @ TA @ C @ A ) )
                 => ~ ( ( ord_less_eq @ TA @ C @ A )
                     => ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_le__cases3) ).

thf(493,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
           => ~ ( ord_less_eq @ TA @ B @ C ) )
         => ( ( ( ord_less_eq @ TA @ B @ A )
             => ~ ( ord_less_eq @ TA @ A @ C ) )
           => ( ( ( ord_less_eq @ TA @ A @ C )
               => ~ ( ord_less_eq @ TA @ C @ B ) )
             => ( ( ( ord_less_eq @ TA @ C @ B )
                 => ~ ( ord_less_eq @ TA @ B @ A ) )
               => ( ( ( ord_less_eq @ TA @ B @ C )
                   => ~ ( ord_less_eq @ TA @ C @ A ) )
                 => ~ ( ( ord_less_eq @ TA @ C @ A )
                     => ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(154,axiom,
    ( ( min_dist @ capacity @ c @ s @ t )
    = ( plus_plus @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( min_dist @ capacity @ c @ ua @ t ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0__092_060open_062min__dist_As_At_A_061_Amin__dist_As_Au_A_L_Amin__dist_Au_At_092_060close_062) ).

thf(682,plain,
    ( ( min_dist @ capacity @ c @ s @ t )
    = ( plus_plus @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( min_dist @ capacity @ c @ ua @ t ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(102,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_eq__imp__le) ).

thf(551,plain,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(165,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_not__add__less1) ).

thf(710,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(229,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_nat__le__linear) ).

thf(916,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(73,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( linorder @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Olinorder) ).

thf(467,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( linorder @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(68,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isShortestPath @ TA @ A @ B @ C @ D )
         => ( isPath @ TA @ A @ B @ C @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_Graph_OshortestPath__is__path) ).

thf(456,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isShortestPath @ TA @ A @ B @ C @ D )
         => ( isPath @ TA @ A @ B @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(210,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
      = ( ( isPath @ capacity @ c @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ capacity @ c @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_isShortestPath__min__dist__def) ).

thf(842,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
      = ( ( isPath @ capacity @ c @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ capacity @ c @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(290,axiom,
    ab_semigroup_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add) ).

thf(1089,plain,
    ab_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(142,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_dual__order_Ostrict__implies__order) ).

thf(650,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(298,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_less__not__refl2) ).

thf(1111,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(172,axiom,
    connected @ capacity @ c @ s @ ua,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29__092_060open_062connected_As_Au_092_060close_062) ).

thf(725,plain,
    connected @ capacity @ c @ s @ ua,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(49,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( order @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Oorder) ).

thf(403,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( order @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(218,axiom,
    ! [TA: $tType] :
      ( ( strict2144017051up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_add__strict__mono) ).

thf(869,plain,
    ! [TA: $tType] :
      ( ( strict2144017051up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(127,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: $o] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ A )
           => C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_less__imp__triv) ).

thf(610,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: $o] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ A )
           => C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(300,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ( dist @ capacity @ c @ C @ D @ E )
       => ( ( ( min_dist @ capacity @ c @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ capacity @ c @ C @ E )
            = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_min__dist__split_I2_J) ).

thf(1118,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ( dist @ capacity @ c @ C @ D @ E )
       => ( ( ( min_dist @ capacity @ c @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ capacity @ c @ C @ E )
            = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(89,axiom,
    strict2144017051up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add) ).

thf(519,plain,
    strict2144017051up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(179,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_less__not__refl) ).

thf(742,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(233,axiom,
    ! [A: nat] : ( connected @ capacity @ c @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_connected__refl) ).

thf(933,plain,
    ! [A: nat] : ( connected @ capacity @ c @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(138,axiom,
    isPath @ a @ c2 @ u @ p2 @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062) ).

thf(642,plain,
    isPath @ a @ c2 @ u @ p2 @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(92,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_nat__le__iff__add) ).

thf(526,plain,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(265,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ( dist @ capacity @ c @ C @ D @ E )
       => ( ( ( min_dist @ capacity @ c @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ capacity @ c @ A @ C )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_min__dist__split_I1_J) ).

thf(1016,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ( dist @ capacity @ c @ C @ D @ E )
       => ( ( ( min_dist @ capacity @ c @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ capacity @ c @ A @ C )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(100,axiom,
    ordere236663937imp_le @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(547,plain,
    ordere236663937imp_le @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(224,axiom,
    ! [A: nat,B: nat] :
      ( ( min_dist @ a @ c2 @ A @ B )
      = ( ord_Least @ nat
        @ ^ [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_g_H_Omin__dist__def) ).

thf(901,plain,
    ! [A: nat,B: nat] :
      ( ( min_dist @ a @ c2 @ A @ B )
      = ( ord_Least @ nat
        @ ^ [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(287,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ord_less_eq @ nat @ ( min_dist @ TA @ A @ B @ D ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_Graph_Omin__dist__minD) ).

thf(1075,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ord_less_eq @ nat @ ( min_dist @ TA @ A @ B @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(153,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less_eq @ nat @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_nat__add__left__cancel__le) ).

thf(679,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less_eq @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(247,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/sandbox2/benchmark/theBenchmark.p',fact_79_add_Oright__cancel) ).

thf(970,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)],[247]) ).

thf(132,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_le__Suc__ex) ).

thf(622,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)],[132]) ).

thf(45,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ A )
           => ( ord_less_eq @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_dual__order_Otrans) ).

thf(393,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ A )
           => ( ord_less_eq @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(258,axiom,
    isPath @ a @ c2 @ ua @ p2_a @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_P2_H) ).

thf(997,plain,
    isPath @ a @ c2 @ ua @ p2_a @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(227,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_trans__less__add2) ).

thf(911,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(109,axiom,
    ordere779506340up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).

thf(569,plain,
    ordere779506340up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( group_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ogroup__add) ).

thf(371,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( group_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(135,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ! [D: nat] :
            ( ( dist @ capacity @ c @ A @ D @ C )
           => ( ord_less_eq @ nat @ B @ D ) )
       => ( ( min_dist @ capacity @ c @ A @ C )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_min__distI__eq) ).

thf(633,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ! [D: nat] :
            ( ( dist @ capacity @ c @ A @ D @ C )
           => ( ord_less_eq @ nat @ B @ D ) )
       => ( ( min_dist @ capacity @ c @ A @ C )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(97,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
      = ( ( isPath @ capacity @ c @ A @ B @ C )
        & ! [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ capacity @ c @ A @ D @ C )
           => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_isShortestPath__def) ).

thf(541,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
      = ( ( isPath @ capacity @ c @ A @ B @ C )
        & ! [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ capacity @ c @ A @ D @ C )
           => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(145,axiom,
    comm_monoid_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).

thf(657,plain,
    comm_monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(268,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat > $o] :
          ( ( connected @ TA @ A @ B @ C )
         => ( ! [E: nat] :
                ( ( dist @ TA @ A @ B @ E @ C )
               => ( ! [F: nat] :
                      ( ( dist @ TA @ A @ B @ F @ C )
                     => ( ord_less_eq @ nat @ E @ F ) )
                 => ( D @ E ) ) )
           => ( D @ ( min_dist @ TA @ A @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_Graph_Omin__distI2) ).

thf(1026,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat > $o] :
          ( ( connected @ TA @ A @ B @ C )
         => ( ! [E: nat] :
                ( ( dist @ TA @ A @ B @ E @ C )
               => ( ! [F: nat] :
                      ( ( dist @ TA @ A @ B @ F @ C )
                     => ( ord_less_eq @ nat @ E @ F ) )
                 => ( D @ E ) ) )
           => ( D @ ( min_dist @ TA @ A @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(241,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
         => ( ord_less @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_add__less__imp__less__left) ).

thf(953,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
         => ( ord_less @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(169,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/sandbox2/benchmark/theBenchmark.p',fact_55_add__mono__thms__linordered__semiring_I3_J) ).

thf(718,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)],[169]) ).

thf(232,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_ord__less__eq__subst) ).

thf(926,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(47,axiom,
    preorder @ $o,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_7) ).

thf(398,plain,
    preorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(271,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/sandbox2/benchmark/theBenchmark.p',fact_76_add__left__imp__eq) ).

thf(1035,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)],[271]) ).

thf(105,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_218_order_Ostrict__trans) ).

thf(558,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(285,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_le__add2) ).

thf(1071,plain,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(197,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less @ nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_add__lessD1) ).

thf(794,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less @ nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(116,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ( dist @ TA @ A @ D @ E @ F )
           => ( dist @ TA @ A @ B @ ( plus_plus @ nat @ C @ E ) @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_Graph_Odist__trans) ).

thf(586,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ( dist @ TA @ A @ D @ E @ F )
           => ( dist @ TA @ A @ B @ ( plus_plus @ nat @ C @ E ) @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(215,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_order_Oasym) ).

thf(856,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(180,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ capacity @ c @ A @ B @ C )
     => ( dist @ capacity @ c @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_isPath__distD) ).

thf(745,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ capacity @ c @ A @ B @ C )
     => ( dist @ capacity @ c @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(157,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
      = ( ( isPath @ a @ c2 @ A @ B @ C )
        & ! [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ a @ c2 @ A @ D @ C )
           => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_g_H_OisShortestPath__def) ).

thf(689,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
      = ( ( isPath @ a @ c2 @ A @ B @ C )
        & ! [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ a @ c2 @ A @ D @ C )
           => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(38,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ordere779506340up_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add) ).

thf(375,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ordere779506340up_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(123,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
          = ( ( ord_less @ TA @ A @ B )
            | ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_neq__iff) ).

thf(601,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
          = ( ( ord_less @ TA @ A @ B )
            | ( ord_less @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(87,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/sandbox2/benchmark/theBenchmark.p',fact_83_group__cancel_Oadd1) ).

thf(513,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)],[87]) ).

thf(58,axiom,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_6) ).

thf(430,plain,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(225,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ! [E: nat] :
                ( ( dist @ TA @ A @ B @ E @ D )
               => ( ord_less_eq @ nat @ C @ E ) )
           => ( ( min_dist @ TA @ A @ B @ D )
              = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_Graph_Omin__distI__eq) ).

thf(904,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ! [E: nat] :
                ( ( dist @ TA @ A @ B @ E @ D )
               => ( ord_less_eq @ nat @ C @ E ) )
           => ( ( min_dist @ TA @ A @ B @ D )
              = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(55,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ~ ! [C: list @ ( product_prod @ nat @ nat )] :
            ~ ( isShortestPath @ a @ c2 @ A @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_g_H_Oobtain__shortest__path) ).

thf(423,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ~ ! [C: list @ ( product_prod @ nat @ nat )] :
            ~ ( isShortestPath @ a @ c2 @ A @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(113,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ~ ( A @ C )
         => ? [D: nat] :
              ( ( ord_less @ nat @ D @ C )
              & ~ ( A @ D ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_infinite__descent) ).

thf(577,plain,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ~ ( A @ C )
         => ? [D: nat] :
              ( ( ord_less @ nat @ D @ C )
              & ~ ( A @ D ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(137,axiom,
    ord_less_eq @ nat @ ( min_dist @ capacity @ c @ ua @ t ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2__092_060open_062min__dist_Au_At_A_092_060le_062_Alength_Ap2_H_092_060close_062) ).

thf(641,plain,
    ord_less_eq @ nat @ ( min_dist @ capacity @ c @ ua @ t ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(44,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isShortestPath @ TA @ A @ B @ C @ D )
         => ( isSimplePath @ TA @ A @ B @ C @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_Graph_OshortestPath__is__simple) ).

thf(391,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isShortestPath @ TA @ A @ B @ C @ D )
         => ( isSimplePath @ TA @ A @ B @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(41,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_ord__le__eq__trans) ).

thf(382,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(111,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_less__irrefl__nat) ).

thf(572,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(166,axiom,
    linorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).

thf(713,plain,
    linorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(291,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( ! [D: nat] :
            ( ( dist @ a @ c2 @ A @ D @ B )
           => ( ! [E: nat] :
                  ( ( dist @ a @ c2 @ A @ E @ B )
                 => ( ord_less_eq @ nat @ D @ E ) )
             => ( C @ D ) ) )
       => ( C @ ( min_dist @ a @ c2 @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_g_H_Omin__distI2) ).

thf(1090,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( ! [D: nat] :
            ( ( dist @ a @ c2 @ A @ D @ B )
           => ( ! [E: nat] :
                  ( ( dist @ a @ c2 @ A @ E @ B )
                 => ( ord_less_eq @ nat @ D @ E ) )
             => ( C @ D ) ) )
       => ( C @ ( min_dist @ a @ c2 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(163,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
      = ( ? [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ capacity @ c @ A @ D @ C )
            & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
              = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_dist__def) ).

thf(704,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
      = ( ? [D: list @ ( product_prod @ nat @ nat )] :
            ( ( isPath @ capacity @ c @ A @ D @ C )
            & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
              = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(238,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ a @ c2 @ A @ B @ C )
     => ( dist @ a @ c2 @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_g_H_OisPath__distD) ).

thf(947,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ a @ c2 @ A @ B @ C )
     => ( dist @ a @ c2 @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(124,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
      = ( ? [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_connected__by__dist) ).

thf(604,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
      = ( ? [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(207,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ( ( ( min_dist @ TA @ A @ B @ C )
              = D )
           => ( ( ord_less @ nat @ E @ D )
             => ? [F: nat] :
                  ( ( connected @ TA @ A @ B @ F )
                  & ( ( min_dist @ TA @ A @ B @ F )
                    = E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_Graph_Omin__dist__less) ).

thf(831,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ( ( ( min_dist @ TA @ A @ B @ C )
              = D )
           => ( ( ord_less @ nat @ E @ D )
             => ? [F: nat] :
                  ( ( connected @ TA @ A @ B @ F )
                  & ( ( min_dist @ TA @ A @ B @ F )
                    = E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(262,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ~ ! [C: TA] :
                ( B
               != ( plus_plus @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_less__eqE) ).

thf(1011,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ~ ! [C: TA] :
                ( B
               != ( plus_plus @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(276,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ( dist @ a @ c2 @ C @ D @ E )
       => ( ( ( min_dist @ a @ c2 @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ a @ c2 @ C @ E )
            = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_g_H_Omin__dist__split_I2_J) ).

thf(1048,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ( dist @ a @ c2 @ C @ D @ E )
       => ( ( ( min_dist @ a @ c2 @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ a @ c2 @ C @ E )
            = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(108,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less_eq @ nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_add__leD1) ).

thf(567,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less_eq @ nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(10,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( connected @ TA )
        = ( connected @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_Graph_Oconnected_Ocong) ).

thf(318,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(141,axiom,
    isPath @ a @ c2 @ s @ p3 @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_assms_I5_J) ).

thf(649,plain,
    isPath @ a @ c2 @ s @ p3 @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(122,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_less__imp__le__nat) ).

thf(599,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(39,axiom,
    ! [A: nat] : ( connected @ a @ c2 @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_g_H_Oconnected__refl) ).

thf(377,plain,
    ! [A: nat] : ( connected @ a @ c2 @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(254,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_less__irrefl) ).

thf(990,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(257,axiom,
    isPath @ capacity @ c @ s @ p1 @ v,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24__092_060open_062isPath_As_Ap1_____Av_____092_060close_062) ).

thf(996,plain,
    isPath @ capacity @ c @ s @ p1 @ v,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(130,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_add__mono__thms__linordered__field_I1_J) ).

thf(616,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(3,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isSimplePath @ TA )
        = ( isSimplePath @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_Graph_OisSimplePath_Ocong) ).

thf(303,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(301,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_add__mono__thms__linordered__field_I3_J) ).

thf(1122,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(221,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ ( min_dist @ a @ c2 @ A @ B ) )
       => ? [D: nat] :
            ( ( connected @ a @ c2 @ A @ D )
            & ( ( min_dist @ a @ c2 @ A @ D )
              = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_g_H_Omin__dist__le) ).

thf(890,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( connected @ a @ c2 @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ ( min_dist @ a @ c2 @ A @ B ) )
       => ? [D: nat] :
            ( ( connected @ a @ c2 @ A @ D )
            & ( ( min_dist @ a @ c2 @ A @ D )
              = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(126,axiom,
    ord_less @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1__092_060open_062min__dist_As_Au_A_060_Alength_Ap1_092_060close_062) ).

thf(609,plain,
    ord_less @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(13,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_le__cases) ).

thf(323,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(80,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( isShortestPath @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_Graph_OisShortestPath_Ocong) ).

thf(483,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(222,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/sandbox2/benchmark/theBenchmark.p',fact_75_add__right__imp__eq) ).

thf(895,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)],[222]) ).

thf(17,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( ord_less_eq @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB] :
            ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_le__fun__def) ).

thf(330,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( ord_less_eq @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB] :
            ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(280,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/sandbox2/benchmark/theBenchmark.p',fact_43_add__le__imp__le__left) ).

thf(1061,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)],[280]) ).

thf(216,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_trans__less__add1) ).

thf(858,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(34,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_dual__order_Orefl) ).

thf(367,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(199,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_ord__eq__less__subst) ).

thf(799,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(173,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_add__le__less__mono) ).

thf(726,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(212,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/sandbox2/benchmark/theBenchmark.p',fact_80_add_Oleft__cancel) ).

thf(848,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)],[212]) ).

thf(90,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( A != B )
       => ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_le__neq__implies__less) ).

thf(520,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( A != B )
       => ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(252,axiom,
    ! [A: nat] :
      ( ( reachableNodes @ a @ c2 @ A )
      = ( collect @ nat @ ( connected @ a @ c2 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_g_H_OreachableNodes__def) ).

thf(983,plain,
    ! [A: nat] :
      ( ( reachableNodes @ a @ c2 @ A )
      = ( collect @ nat @ ( connected @ a @ c2 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(48,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_ord__eq__le__trans) ).

thf(399,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(188,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less @ TA @ A @ B )
              | ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_254_dual__order_Oorder__iff__strict) ).

thf(768,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less @ TA @ A @ B )
              | ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(264,axiom,
    isPath @ capacity @ c @ ua @ p2_a @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13__092_060open_062isPath_Au_Ap2_H_At_092_060close_062) ).

thf(1015,plain,
    isPath @ capacity @ c @ ua @ p2_a @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(253,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ( ( ord_less_eq @ nat @ D @ ( min_dist @ TA @ A @ B @ C ) )
           => ? [E: nat] :
                ( ( connected @ TA @ A @ B @ E )
                & ( ( min_dist @ TA @ A @ B @ E )
                  = D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_Graph_Omin__dist__le) ).

thf(986,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ( ( ord_less_eq @ nat @ D @ ( min_dist @ TA @ A @ B @ C ) )
           => ? [E: nat] :
                ( ( connected @ TA @ A @ B @ E )
                & ( ( min_dist @ TA @ A @ B @ E )
                  = D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(144,axiom,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_less__mono__imp__le__mono) ).

thf(653,plain,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(231,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_add__mono__thms__linordered__field_I2_J) ).

thf(922,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(146,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
      ( ! [D: TA] :
          ( ~ ( A @ D )
         => ? [E: TA] :
              ( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
              & ~ ( A @ E ) ) )
     => ( A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_infinite__descent__measure) ).

thf(658,plain,
    ! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
      ( ! [D: TA] :
          ( ~ ( A @ D )
         => ? [E: TA] :
              ( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
              & ~ ( A @ E ) ) )
     => ( A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(274,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_add__strict__right__mono) ).

thf(1044,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(159,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
      = ( ( isSimplePath @ capacity @ c @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ capacity @ c @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_isShortestPath__alt) ).

thf(695,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
      = ( ( isSimplePath @ capacity @ c @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ capacity @ c @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(213,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_add__mono__thms__linordered__field_I4_J) ).

thf(851,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(295,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ~ ! [C: list @ ( product_prod @ nat @ nat )] :
            ~ ( isShortestPath @ capacity @ c @ A @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_obtain__shortest__path) ).

thf(1102,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ~ ! [C: list @ ( product_prod @ nat @ nat )] :
            ~ ( isShortestPath @ capacity @ c @ A @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(170,axiom,
    no_top @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Ono__top) ).

thf(722,plain,
    no_top @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(83,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( connected @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
            ? [D: list @ ( product_prod @ nat @ nat )] : ( isPath @ TA @ A @ B @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_Graph_Oconnected__def) ).

thf(490,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( connected @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
            ? [D: list @ ( product_prod @ nat @ nat )] : ( isPath @ TA @ A @ B @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(56,axiom,
    ord @ $o,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_10) ).

thf(426,plain,
    ord @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(118,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_trans__le__add2) ).

thf(589,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(289,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( ( ( min_dist @ capacity @ c @ A @ B )
          = C )
       => ( ( ord_less @ nat @ D @ C )
         => ? [E: nat] :
              ( ( connected @ capacity @ c @ A @ E )
              & ( ( min_dist @ capacity @ c @ A @ E )
                = D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_min__dist__less) ).

thf(1081,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( ( ( min_dist @ capacity @ c @ A @ B )
          = C )
       => ( ( ord_less @ nat @ D @ C )
         => ? [E: nat] :
              ( ( connected @ capacity @ c @ A @ E )
              & ( ( min_dist @ capacity @ c @ A @ E )
                = D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(23,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isPath @ TA @ A @ B @ C @ D )
         => ? [E: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ TA @ A @ B @ E @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_Graph_OisSPath__pathLE) ).

thf(342,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
          ( ( isPath @ TA @ A @ B @ C @ D )
         => ? [E: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ TA @ A @ B @ E @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(29,axiom,
    ! [TA: $tType] : ( preorder @ ( set @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_4) ).

thf(355,plain,
    ! [TA: $tType] : ( preorder @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(69,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ( A @ B )
         => ( ord_less_eq @ TA @ ( ord_Least @ TA @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_Least__le) ).

thf(458,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ( A @ B )
         => ( ord_less_eq @ TA @ ( ord_Least @ TA @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(94,axiom,
    ! [TA: $tType,TB: $tType] : ( size @ ( product_prod @ TB @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Nat_Osize_12) ).

thf(531,plain,
    ! [TA: $tType,TB: $tType] : ( size @ ( product_prod @ TB @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(237,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_measure__induct__rule) ).

thf(944,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(148,axiom,
    isPath @ capacity @ c @ s @ p1a @ va,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6__C1_Oprems_C_I2_J) ).

thf(663,plain,
    isPath @ capacity @ c @ s @ p1a @ va,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(101,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/sandbox2/benchmark/theBenchmark.p',fact_85_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(548,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)],[101]) ).

thf(104,axiom,
    cancel_semigroup_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add) ).

thf(557,plain,
    cancel_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(164,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_trans__le__add1) ).

thf(707,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(53,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ab_semigroup_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add) ).

thf(414,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ab_semigroup_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(15,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
          | ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_linear) ).

thf(327,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
          | ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(178,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ( dist @ TA @ A @ D @ E @ F )
           => ( ( ( min_dist @ TA @ A @ B @ F )
                = ( plus_plus @ nat @ C @ E ) )
             => ( ( min_dist @ TA @ A @ D @ F )
                = E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_Graph_Omin__dist__split_I2_J) ).

thf(739,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
          ( ( dist @ TA @ A @ B @ C @ D )
         => ( ( dist @ TA @ A @ D @ E @ F )
           => ( ( ( min_dist @ TA @ A @ B @ F )
                = ( plus_plus @ nat @ C @ E ) )
             => ( ( min_dist @ TA @ A @ D @ F )
                = E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(52,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less_eq @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA,E: TA] :
                ( ( A @ E @ D )
               => ( A @ D @ E ) )
           => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_linorder__wlog) ).

thf(409,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less_eq @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA,E: TA] :
                ( ( A @ E @ D )
               => ( A @ D @ E ) )
           => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(293,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
          | ( A = B )
          | ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_less__linear) ).

thf(1098,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
          | ( A = B )
          | ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(196,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( A != B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_linorder__cases) ).

thf(791,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( A != B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(272,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_measure__induct) ).

thf(1038,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(71,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( strict2144017051up_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add) ).

thf(463,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( strict2144017051up_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( comm_monoid_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add) ).

thf(338,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( comm_monoid_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(75,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_order__refl) ).

thf(470,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(240,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_add__less__le__mono) ).

thf(951,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(31,axiom,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_5) ).

thf(359,plain,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(139,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less @ TB @ A @ ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_order__less__subst1) ).

thf(643,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less @ TB @ A @ ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(98,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ( dist @ TA @ A @ B @ ( min_dist @ TA @ A @ B @ C ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_Graph_Omin__dist__is__dist) ).

thf(544,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ( dist @ TA @ A @ B @ ( min_dist @ TA @ A @ B @ C ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(20,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( semigroup_add @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Osemigroup__add) ).

thf(336,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( semigroup_add @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(267,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/sandbox2/benchmark/theBenchmark.p',fact_11_add__left__cancel) ).

thf(1023,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)],[267]) ).

thf(277,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ! [C: TA] :
              ( ! [D: TA] :
                  ( ( ord_less @ TA @ D @ C )
                 => ( A @ D ) )
             => ( A @ C ) )
         => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_less__induct) ).

thf(1052,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ! [C: TA] :
              ( ! [D: TA] :
                  ( ( ord_less @ TA @ D @ C )
                 => ( A @ D ) )
             => ( A @ C ) )
         => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(261,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ord_less @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_order__less__subst2) ).

thf(1008,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ord_less @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(120,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_dual__order_Oirrefl) ).

thf(595,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(128,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
          & ( A != B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_nat__less__le) ).

thf(612,plain,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
          & ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(250,axiom,
    ! [A: nat] :
      ( ( reachableNodes @ capacity @ c @ A )
      = ( collect @ nat @ ( connected @ capacity @ c @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_reachableNodes__def) ).

thf(978,plain,
    ! [A: nat] :
      ( ( reachableNodes @ capacity @ c @ A )
      = ( collect @ nat @ ( connected @ capacity @ c @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(42,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
      = ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c2 @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_g_H_Oconnected__def) ).

thf(386,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
      = ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c2 @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(211,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less @ nat @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_nat__add__left__cancel__less) ).

thf(845,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(72,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ a @ c2 @ A @ B @ C )
     => ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c2 @ A @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_g_H_OisSPath__pathLE) ).

thf(465,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isPath @ a @ c2 @ A @ B @ C )
     => ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c2 @ A @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(66,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_dual__order_Oantisym) ).

thf(450,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(151,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_nat__less__induct) ).

thf(671,plain,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(184,axiom,
    isPath @ a @ c2 @ ua @ p2_a @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153__C1_Oprems_C_I3_J) ).

thf(759,plain,
    isPath @ a @ c2 @ ua @ p2_a @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(189,axiom,
    ! [TA: $tType] :
      ( ( size @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( size_size @ TA @ A )
           != ( size_size @ TA @ B ) )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_size__neq__size__imp__neq) ).

thf(771,plain,
    ! [TA: $tType] :
      ( ( size @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( size_size @ TA @ A )
           != ( size_size @ TA @ B ) )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(297,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( reachableNodes @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( collect @ nat @ ( connected @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_Graph_OreachableNodes__def) ).

thf(1108,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( reachableNodes @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( collect @ nat @ ( connected @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(244,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( dist @ capacity @ c @ A @ ( min_dist @ capacity @ c @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_min__dist__is__dist) ).

thf(961,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
     => ( dist @ capacity @ c @ A @ ( min_dist @ capacity @ c @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(11,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).

thf(319,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(4,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA > $o] :
          ( ? [C: TA] : ( A @ C )
         => ( ! [C: TA] :
                ( ( A @ C )
               => ( B @ C ) )
           => ( B @ ( ord_Least @ TA @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_LeastI2__ex) ).

thf(304,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA > $o] :
          ( ? [C: TA] : ( A @ C )
         => ( ! [C: TA] :
                ( ( A @ C )
               => ( B @ C ) )
           => ( B @ ( ord_Least @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(294,axiom,
    order @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_2) ).

thf(1101,plain,
    order @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(192,axiom,
    isPath @ capacity @ c @ s @ p1a @ va,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_P1) ).

thf(781,plain,
    isPath @ capacity @ c @ s @ p1a @ va,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(256,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(995,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(117,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(588,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(161,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_add__less__mono1) ).

thf(699,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(245,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/sandbox2/benchmark/theBenchmark.p',fact_54_add__mono__thms__linordered__semiring_I2_J) ).

thf(963,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)],[245]) ).

thf(88,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_less__asym) ).

thf(517,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(143,axiom,
    linordered_idom @ capacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_0) ).

thf(652,plain,
    linordered_idom @ capacity,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(27,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_le__funD) ).

thf(351,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(175,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_add__le__mono1) ).

thf(731,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(194,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
      = ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_connected__def) ).

thf(786,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ capacity @ c @ A @ B )
      = ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(269,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
      = ( ( isPath @ a @ c2 @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ a @ c2 @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_g_H_OisShortestPath__min__dist__def) ).

thf(1030,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
      = ( ( isPath @ a @ c2 @ A @ B @ C )
        & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
          = ( min_dist @ a @ c2 @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(228,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_order_Ostrict__implies__order) ).

thf(914,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(86,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_neqE) ).

thf(510,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(150,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ( dist @ a @ c2 @ C @ D @ E )
       => ( ( ( min_dist @ a @ c2 @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ a @ c2 @ A @ C )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_g_H_Omin__dist__split_I1_J) ).

thf(667,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( dist @ a @ c2 @ A @ B @ C )
     => ( ( dist @ a @ c2 @ C @ D @ E )
       => ( ( ( min_dist @ a @ c2 @ A @ E )
            = ( plus_plus @ nat @ B @ D ) )
         => ( ( min_dist @ a @ c2 @ A @ C )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(186,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/sandbox2/benchmark/theBenchmark.p',fact_12_add__right__cancel) ).

thf(763,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)],[186]) ).

thf(155,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ C @ D )
       => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_add__less__mono) ).

thf(684,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ C @ D )
       => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(25,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).

thf(346,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(168,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_add__strict__left__mono) ).

thf(716,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(288,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_less__not__refl3) ).

thf(1077,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(78,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
     => ( isPath @ a @ c2 @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_g_H_OshortestPath__is__path) ).

thf(478,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ a @ c2 @ A @ B @ C )
     => ( isPath @ a @ c2 @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(131,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA,B: TA > $o] :
          ( ( ord_less @ TA @ A @ ( ord_Least @ TA @ B ) )
         => ~ ( B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_not__less__Least) ).

thf(620,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA,B: TA > $o] :
          ( ( ord_less @ TA @ A @ ( ord_Least @ TA @ B ) )
         => ~ ( B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(64,axiom,
    linorder @ $o,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_8) ).

thf(447,plain,
    linorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(8,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( reachableNodes @ TA )
        = ( reachableNodes @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_Graph_OreachableNodes_Ocong) ).

thf(315,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(59,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA,C: TA > $o] :
          ( ( A @ B )
         => ( ! [D: TA] :
                ( ( A @ D )
               => ( C @ D ) )
           => ( C @ ( ord_Least @ TA @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_LeastI2) ).

thf(432,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA,C: TA > $o] :
          ( ( A @ B )
         => ( ! [D: TA] :
                ( ( A @ D )
               => ( C @ D ) )
           => ( C @ ( ord_Least @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(246,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( dist @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
            ? [E: list @ ( product_prod @ nat @ nat )] :
              ( ( isPath @ TA @ A @ B @ E @ D )
              & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E )
                = C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_Graph_Odist__def) ).

thf(967,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( dist @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
            ? [E: list @ ( product_prod @ nat @ nat )] :
              ( ( isPath @ TA @ A @ B @ E @ D )
              & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E )
                = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(203,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
         => ( ord_less @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_add__less__imp__less__right) ).

thf(818,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
         => ( ord_less @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(50,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ~ ! [D: list @ ( product_prod @ nat @ nat )] :
                ~ ( isShortestPath @ TA @ A @ B @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_Graph_Oobtain__shortest__path) ).

thf(405,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
          ( ( connected @ TA @ A @ B @ C )
         => ~ ! [D: list @ ( product_prod @ nat @ nat )] :
                ~ ( isShortestPath @ TA @ A @ B @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(37,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_order__trans) ).

thf(373,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(177,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_order_Ostrict__implies__not__eq) ).

thf(735,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(107,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( ~ ( ord_less @ TA @ B @ A ) )
            = ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_antisym__conv3) ).

thf(564,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( ~ ( ord_less @ TA @ B @ A ) )
            = ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(230,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ C @ B )
          = ( plus_plus @ nat @ A @ D ) )
       => ( ord_less @ nat @ C @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_less__add__eq__less) ).

thf(918,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ C @ B )
          = ( plus_plus @ nat @ A @ D ) )
       => ( ord_less @ nat @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(61,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ord @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Oord) ).

thf(437,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ord @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(149,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_le__antisym) ).

thf(664,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(270,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_le__refl) ).

thf(1033,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(206,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_not__add__less2) ).

thf(828,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(282,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/sandbox2/benchmark/theBenchmark.p',fact_52_add__mono) ).

thf(1065,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)],[282]) ).

thf(242,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_less__imp__not__less) ).

thf(955,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(198,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
              ( ( isPath @ TA @ A @ B @ C @ D )
              & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
                = ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_Graph_OisShortestPath__min__dist__def) ).

thf(796,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isShortestPath @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
              ( ( isPath @ TA @ A @ B @ C @ D )
              & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
                = ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(46,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).

thf(395,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(174,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ~ ( ( ord_less_eq @ nat @ A @ C )
         => ~ ( ord_less_eq @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_add__leE) ).

thf(728,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ~ ( ( ord_less_eq @ nat @ A @ C )
         => ~ ( ord_less_eq @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(112,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_dual__order_Oasym) ).

thf(575,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(57,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( connected @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
            ? [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_Graph_Oconnected__by__dist) ).

thf(427,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( connected @ TA )
        = ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
            ? [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(93,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_linorder__neqE__nat) ).

thf(528,plain,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(255,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/sandbox2/benchmark/theBenchmark.p',fact_81_add_Oassoc) ).

thf(992,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)],[255]) ).

thf(171,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_less__asym_H) ).

thf(723,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(260,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_dual__order_Ostrict__implies__not__eq) ).

thf(1004,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(219,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/sandbox2/benchmark/theBenchmark.p',fact_53_add__mono__thms__linordered__semiring_I1_J) ).

thf(871,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)],[219]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( connected @ TA @ A @ B @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_Graph_Oconnected__refl) ).

thf(369,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( connected @ TA @ A @ B @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(125,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ord_less_eq @ nat @ ( min_dist @ capacity @ c @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_min__dist__minD) ).

thf(607,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist @ capacity @ c @ A @ B @ C )
     => ( ord_less_eq @ nat @ ( min_dist @ capacity @ c @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(82,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A = B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_eq__refl) ).

thf(486,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A = B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(234,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_le__add1) ).

thf(935,plain,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(6,axiom,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).

thf(309,plain,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(152,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ( ord_less_eq @ TA @ A @ B )
           => ( ord_less @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_order_Onot__eq__order__implies__strict) ).

thf(676,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ( ord_less_eq @ TA @ A @ B )
           => ( ord_less @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(119,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ D )
       => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_add__le__mono) ).

thf(592,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ D )
       => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(284,axiom,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
     => ( isPath @ capacity @ c @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_shortestPath__is__path) ).

thf(1069,plain,
    ! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
      ( ( isShortestPath @ capacity @ c @ A @ B @ C )
     => ( isPath @ capacity @ c @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(33,axiom,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
      = ( ? [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_g_H_Oconnected__by__dist) ).

thf(364,plain,
    ! [A: nat,B: nat] :
      ( ( connected @ a @ c2 @ A @ B )
      = ( ? [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(263,axiom,
    preorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_1) ).

thf(1014,plain,
    preorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(74,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ( ( isPath @ TA )
        = ( isPath @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_Graph_OisPath_Ocong) ).

thf(469,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(236,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_less__imp__neq) ).

thf(940,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(190,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/sandbox2/benchmark/theBenchmark.p',fact_77_add_Oleft__commute) ).

thf(775,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)],[190]) ).

thf(106,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/sandbox2/benchmark/theBenchmark.p',fact_84_add__mono__thms__linordered__semiring_I4_J) ).

thf(560,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)],[106]) ).

thf(1124,plain,
    $false,
    inference(cvc4,[status(thm)],[809,760,1105,555,909,333,977,1073,873,379,440,614,597,523,582,1041,861,533,472,814,698,747,538,1063,340,1095,898,782,841,973,1046,389,1055,570,838,344,357,1059,949,460,702,751,416,448,316,638,853,1115,325,475,756,789,321,646,937,806,353,480,1020,348,687,692,734,453,998,507,312,981,766,307,439,661,546,1067,714,334,957,825,778,302,820,407,630,484,361,625,435,329,583,493,682,551,710,916,467,456,842,1089,650,1111,725,403,869,610,1118,519,742,933,642,526,1016,547,901,1075,679,970,622,393,997,911,569,371,633,541,657,1026,953,718,926,398,1035,558,1071,794,586,856,745,689,375,601,513,430,904,423,577,641,391,382,572,713,1090,704,947,604,831,1011,1048,567,318,649,599,377,990,996,616,303,1122,890,609,323,483,895,330,1061,858,367,799,726,848,520,983,399,768,1015,986,653,922,658,1044,695,851,1102,722,490,426,589,1081,342,355,458,531,944,663,548,557,707,414,327,739,409,1098,791,1038,463,338,470,951,359,643,544,336,1023,1052,1008,595,612,978,386,845,465,450,671,759,771,1108,961,319,304,1101,781,995,588,699,963,517,652,351,731,786,1030,914,510,667,763,684,346,716,1077,478,620,447,315,432,967,818,405,373,735,564,918,437,664,1033,828,1065,955,796,395,728,575,427,528,992,723,1004,871,369,607,486,935,309,676,592,1069,364,1014,469,940,775,560]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.14  % Problem  : ITP048^2 : TPTP v8.2.0. Released v7.5.0.
% 0.13/0.17  % Command  : run_Leo-III %s %d
% 0.18/0.38  % Computer : n018.cluster.edu
% 0.18/0.38  % Model    : x86_64 x86_64
% 0.18/0.38  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.38  % Memory   : 8042.1875MB
% 0.18/0.38  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.18/0.38  % CPULimit : 300
% 0.18/0.38  % WCLimit  : 300
% 0.18/0.38  % DateTime : Sat May 18 17:05:09 EDT 2024
% 0.18/0.39  % CPUTime  : 
% 1.00/0.90  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.63/1.15  % [INFO] 	 Parsing done (246ms). 
% 1.99/1.16  % [INFO] 	 Running in sequential loop mode. 
% 2.67/1.38  % [INFO] 	 eprover registered as external prover. 
% 2.67/1.38  % [INFO] 	 cvc4 registered as external prover. 
% 2.67/1.39  % [INFO] 	 Scanning for conjecture ... 
% 2.89/1.48  % [INFO] 	 Found a conjecture (or negated_conjecture) and 300 axioms. Running axiom selection ... 
% 3.17/1.58  % [INFO] 	 Axiom selection finished. Selected 299 axioms (removed 1 axioms). 
% 3.75/1.77  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 4.07/1.78  % [INFO] 	 Type checking passed. 
% 4.07/1.79  % [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 ... 
% 65.45/42.57  % External prover 'cvc4' found a proof!
% 65.45/42.57  % [INFO] 	 Killing All external provers ... 
% 65.45/42.58  % Time passed: 42024ms (effective reasoning time: 41413ms)
% 65.45/42.58  % 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)>
% 65.45/42.58  % Axioms used in derivation (299): fact_53_add__mono__thms__linordered__semiring_I1_J, fact_37_le__trans, fact_249_LeastI2__ex, fact_22_isPath__distD, fact_2__092_060open_062min__dist_Au_At_A_092_060le_062_Alength_Ap2_H_092_060close_062, fact_156_Least__le, subcl_Rings_Olinordered__idom___Groups_Ogroup__add, fact_42_add__le__imp__le__right, fact_36_le__refl, fact_79_add_Oright__cancel, fact_128_shortestPath__is__simple, subcl_Rings_Olinordered__idom___Orderings_Olinorder, tcon_HOL_Obool___Orderings_Opreorder_7, fact_30_add__le__cancel__right, tfree_0, tcon_Nat_Onat___Orderings_Oorder_2, fact_197_le__cases, fact_18_obtain__shortest__path, fact_204_order__class_Oorder_Oantisym, fact_242_order__less__subst2, fact_149_Graph_Omin__dist__le, fact_51_add__left__mono, fact_200_antisym__conv, fact_80_add_Oleft__cancel, fact_6__C1_Oprems_C_I2_J, fact_92_less__not__refl2, fact_248_LeastI__ex, fact_118_min__dist__def, tcon_Nat_Onat___Orderings_Ono__top, fact_190_order__subst2, subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add, fact_133_Graph_OreachableNodes__def, fact_185_le__funD, tcon_Nat_Onat___Orderings_Olinorder, fact_251_order_Onot__eq__order__implies__strict, fact_75_add__right__imp__eq, tcon_Nat_Onat___Nat_Osize, fact_217_less__imp__not__less, fact_129_Graph_OreachableNodes_Ocong, fact_213_not__less__iff__gr__or__eq, fact_255_order_Ostrict__implies__order, fact_29__092_060open_062connected_As_Au_092_060close_062, fact_3_nat__add__left__cancel__less, fact_66_add__le__mono, fact_96_add__less__le__mono, fact_176_g_H_OisPath__distD, fact_11_add__left__cancel, fact_175_g_H_Odist__def, fact_221_less__imp__triv, fact_178_order__refl, fact_134_Graph_Omin__dist__def, fact_64_trans__le__add1, fact_209_dual__order_Oeq__iff, fact_247_LeastI2, fact_150_Graph_Omin__distI__eq, fact_90_less__irrefl__nat, fact_76_add__left__imp__eq, fact_73_measure__induct__rule, fact_74_measure__induct, fact_218_order_Ostrict__trans, fact_244_ord__less__eq__subst, fact_194_antisym, fact_104_add__strict__left__mono, fact_208_dual__order_Otrans, fact_50_less__eqE, fact_122_isShortestPath__alt, fact_121_Graph_Omin__distI2, tcon_fun___Orderings_Oord, fact_224_less__induct, fact_227_dual__order_Oasym, fact_233_less__asym_H, fact_8_min__dist__split_I2_J, fact_220_linorder__cases, fact_151_Graph_Omin__dist__minD, fact_216_exists__least__iff, fact_14_dist__trans, fact_85_ab__semigroup__add__class_Oadd__ac_I1_J, fact_157_g_H_Oconnected__by__dist, fact_139_Graph_Oconnected_Ocong, subcl_Rings_Olinordered__idom___Orderings_Opreorder, fact_105_add__strict__mono, fact_33_P1, fact_155_not__less__Least, fact_52_add__mono, fact_97_add__le__less__mono, fact_237_order_Oasym, fact_161_g_H_OreachableNodes__def, fact_254_dual__order_Oorder__iff__strict, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add, fact_108_add__mono__thms__linordered__field_I5_J, fact_187_le__funI, fact_241_lt__ex, fact_136_Graph_OisPath_Ocong, fact_192_ord__le__eq__subst, fact_60_less__imp__le__nat, tcon_HOL_Obool___Orderings_Oorder_9, fact_253_dual__order_Ostrict__iff__order, fact_56_less__mono__imp__le__mono, tcon_Product__Type_Oprod___Nat_Osize_12, fact_57_le__neq__implies__less, tcon_List_Olist___Nat_Osize_11, fact_169_g_H_Odist__trans, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_23_dist__def, fact_49_add__right__mono, fact_189_order__subst1, fact_145_Graph_Oconnected__by__dist, fact_87_linorder__neqE__nat, subcl_Rings_Olinordered__idom___HOL_Otype, tcon_Nat_Onat___Groups_Ocancel__semigroup__add, fact_83_group__cancel_Oadd1, fact_1__092_060open_062min__dist_As_Au_A_060_Alength_Ap1_092_060close_062, fact_88_infinite__descent, fact_99_add__mono__thms__linordered__field_I4_J, fact_125_Graph_Omin__dist__is__dist, fact_231_less__linear, fact_182_g_H_Oconnected__distI, fact_127_isSPath__pathLE, subcl_Rings_Olinordered__idom___Orderings_Oorder, fact_153__C1_Oprems_C_I3_J, fact_142_Graph_Odist__trans, fact_184_P2_H, fact_41_Nat_Oex__has__greatest__nat, fact_183_g_H_Oconnected__refl, fact_46_Collect__mem__eq, fact_31_add__le__cancel__left, fact_132_Graph_OshortestPath__is__simple, fact_13__092_060open_062isPath_Au_Ap2_H_At_092_060close_062, fact_67_le__Suc__ex, fact_165_g_H_Omin__dist__le, fact_225_less__not__sym, fact_226_less__imp__not__eq, fact_205_order__trans, subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add, fact_34_connected__refl, fact_69_add__leD1, fact_93_less__not__refl, fact_54_add__mono__thms__linordered__semiring_I2_J, tcon_Set_Oset___Orderings_Oord_6, fact_103_add__strict__right__mono, fact_94_nat__neq__iff, fact_68_add__leD2, fact_186_le__funE, fact_65_add__le__mono1, fact_81_add_Oassoc, fact_32_nat__add__left__cancel__le, fact_130_Graph_OisSimplePath_Ocong, fact_235_less__imp__neq, fact_91_less__not__refl3, fact_163_g_H_Omin__distI2, tcon_Nat_Onat___Orderings_Owellorder, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_206_dual__order_Orefl, fact_246_ex__has__least__nat, fact_230_less__irrefl, fact_131_Graph_OisSPath__pathLE, fact_196_eq__refl, fact_102_add__less__imp__less__left, fact_111_trans__less__add1, subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add, fact_215_linorder__less__wlog, fact_55_add__mono__thms__linordered__semiring_I3_J, tcon_HOL_Obool___Orderings_Oord_10, fact_45_mem__Collect__eq, fact_84_add__mono__thms__linordered__semiring_I4_J, fact_78_add_Ocommute, fact_164_g_H_Omin__distI__eq, fact_82_group__cancel_Oadd2, fact_77_add_Oleft__commute, fact_20_min__dist__minD, fact_101_add__less__imp__less__right, fact_4_add__less__cancel__left, fact_229_ord__eq__less__trans, fact_243_order__less__subst1, fact_71_le__add1, fact_199_le__cases3, fact_239_neqE, fact_119_Graph_OisShortestPath__min__dist__def, subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le, fact_106_add__mono__thms__linordered__field_I1_J, fact_70_le__add2, fact_143_Graph_Oconnected__def, fact_202_ord__eq__le__trans, fact_232_less__trans, fact_148_Graph_Omin__dist__less, tcon_Nat_Onat___Orderings_Oord_3, subcl_Rings_Olinordered__idom___Groups_Osemigroup__add, fact_100_mono__nat__linear__lb, fact_89_nat__less__induct, fact_47_Collect__cong, fact_171_g_H_OisSPath__pathLE, fact_5_add__less__cancel__right, fact_252_dual__order_Ostrict__implies__order, fact_110_trans__less__add2, fact_203_ord__le__eq__trans, fact_172_g_H_OshortestPath__is__path, fact_112_add__less__mono1, fact_16_shortestPath__is__path, tcon_Set_Oset___Orderings_Opreorder_4, fact_98_add__mono__thms__linordered__field_I3_J, fact_135_Graph_OisShortestPath__alt, fact_198_order_Otrans, fact_27_min__distI2, fact_158_g_H_Omin__dist__is__dist, fact_228_ord__less__eq__trans, subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add, fact_214_dual__order_Ostrict__trans, fact_62_nat__le__iff__add, fact_159_g_H_Oobtain__shortest__path, fact_236_dense, fact_191_ord__eq__le__subst, fact_181__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062, fact_38_eq__imp__le, fact_7_SP, fact_146_Graph_Oconnected__distI, fact_39_le__antisym, fact_207_linorder__wlog, tcon_fun___Orderings_Opreorder, tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, fact_124_Graph_OisPath__distD, fact_250_LeastI, tcon_Set_Oset___Orderings_Oorder_5, fact_116_add__lessD1, fact_120_Graph_OisShortestPath__def, fact_245_ord__eq__less__subst, fact_173_g_H_OisShortestPath__alt, fact_154_ex__has__greatest__nat__lemma, fact_19_min__dist__le, fact_109_less__add__eq__less, fact_137_Graph_Omin__dist_Ocong, tfree_1, tcon_Nat_Onat___Orderings_Opreorder_1, fact_144_Graph_OshortestPath__is__path, fact_210_dual__order_Oantisym, fact_123_Graph_Odist__def, fact_168_g_H_Omin__dist__split_I1_J, fact_15_connected__def, tcon_HOL_Obool___Orderings_Olinorder_8, fact_195_linear, fact_107_add__mono__thms__linordered__field_I2_J, fact_238_neq__iff, fact_25_min__dist__is__dist, fact_44_le__iff__add, fact_59_le__eq__less__or__eq, fact_17_connected__by__dist, fact_219_dual__order_Oirrefl, fact_179_g_H_OisShortestPath__def, fact_113_not__add__less2, tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, tcon_fun___Orderings_Oorder, fact_170_g_H_Oconnected__def, fact_24__092_060open_062isPath_As_Ap1_____Av_____092_060close_062, tcon_Nat_Onat___Groups_Osemigroup__add, subcl_Rings_Olinordered__idom___Orderings_Oord, fact_21_min__distI__eq, fact_115_add__less__mono, fact_61_nat__less__le, fact_26_isShortestPath__def, fact_162_g_H_Omin__dist__less, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_86_infinite__descent__measure, fact_147_Graph_Oobtain__shortest__path, fact_166_g_H_Omin__dist__minD, fact_201_order__class_Oorder_Oeq__iff, fact_43_add__le__imp__le__left, fact_152_Graph_Omin__dist__split_I2_J, fact_140_Graph_Odist_Ocong, fact_167_g_H_Omin__dist__split_I2_J, fact_10_min__dist__less, tcon_Nat_Onat___Groups_Oab__semigroup__add, subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add, fact_138_Graph_Oconnected__refl, fact_63_trans__le__add2, fact_212_order_Ostrict__implies__not__eq, fact_35_connected__distI, fact_141_Graph_OisShortestPath_Ocong, fact_28_isShortestPath__min__dist__def, fact_177_g_H_OisShortestPath__min__dist__def, fact_9_min__dist__split_I1_J, fact_117_reachableNodes__def, fact_0__092_060open_062min__dist_As_At_A_061_Amin__dist_As_Au_A_L_Amin__dist_Au_At_092_060close_062, fact_180_assms_I5_J, fact_223_antisym__conv3, fact_40_nat__le__linear, fact_222_less__imp__not__eq2, fact_95_size__neq__size__imp__neq, fact_114_not__add__less1, fact_12_add__right__cancel, fact_126_Graph_Omin__dist__split_I1_J, fact_160_g_H_OshortestPath__is__simple, fact_193_eq__iff, fact_58_less__or__eq__imp__le, fact_174_g_H_Omin__dist__def, fact_72_add__leE, fact_240_gt__ex, fact_234_less__asym, fact_188_le__fun__def, fact_211_dual__order_Ostrict__implies__not__eq
% 65.45/42.58  % No. of inferences in proof: 602
% 65.45/42.58  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 42024 ms resp. 41413 ms w/o parsing
% 66.10/42.86  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 66.10/42.86  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------