TSTP Solution File: ITP048^1 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP048^1 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n010.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 29.27s 9.18s
% Output   : Refutation 30.84s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  315
% Syntax   : Number of formulae    :  591 ( 222 unt;  41 typ;   0 def)
%            Number of atoms       : 1414 ( 362 equ;   6 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 4655 ( 122   ~;  20   |;  82   &;3803   @)
%                                         (   0 <=>; 628  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   7 avg)
%            Number of types       :    7 (   6 usr)
%            Number of type conns  :  241 ( 241   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   40 (  36 usr;  16 con; 0-4 aty)
%            Number of variables   : 1636 ( 160   ^1410   !;  66   ?;1636   :)

% Comments : 
%------------------------------------------------------------------------------
thf(list_P559422087at_nat_type,type,
    list_P559422087at_nat: $tType ).

thf(product_prod_nat_nat_type,type,
    product_prod_nat_nat: $tType ).

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

thf(capacity_type,type,
    capacity: $tType ).

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

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

thf(connected_a_type,type,
    connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).

thf(connected_capacity_type,type,
    connected_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > $o ).

thf(dist_a_type,type,
    dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat > $o ).

thf(dist_capacity_type,type,
    dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat > $o ).

thf(isPath_a_type,type,
    isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).

thf(isPath_capacity_type,type,
    isPath_capacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).

thf(isShortestPath_a_type,type,
    isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).

thf(isShor1936442771pacity_type,type,
    isShor1936442771pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).

thf(isSimplePath_a_type,type,
    isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).

thf(isSimp1359852763pacity_type,type,
    isSimp1359852763pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).

thf(min_dist_a_type,type,
    min_dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).

thf(min_dist_capacity_type,type,
    min_dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat ).

thf(reachableNodes_a_type,type,
    reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).

thf(reacha1693770334pacity_type,type,
    reacha1693770334pacity: ( product_prod_nat_nat > capacity ) > nat > set_nat ).

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

thf(size_s1990949619at_nat_type,type,
    size_s1990949619at_nat: list_P559422087at_nat > nat ).

thf(ord_Least_nat_type,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(ord_less_nat_type,type,
    ord_less_nat: nat > nat > $o ).

thf(ord_less_eq_nat_type,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(collect_nat_type,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(member_nat_type,type,
    member_nat: nat > set_nat > $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_P559422087at_nat ).

thf(p1_type,type,
    p1: list_P559422087at_nat ).

thf(p1a_type,type,
    p1a: list_P559422087at_nat ).

thf(p2_type,type,
    p2: list_P559422087at_nat ).

thf(p2_a_type,type,
    p2_a: list_P559422087at_nat ).

thf(p3_type,type,
    p3: list_P559422087at_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(217,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_49_less__eqE) ).

thf(892,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)],[217]) ).

thf(188,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ( dist_a @ A @ D @ E @ F )
       => ( ( ( min_dist_a @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_a @ A @ B @ D )
            = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_Graph_Omin__dist__split_I1_J) ).

thf(809,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ( dist_a @ A @ D @ E @ F )
       => ( ( ( min_dist_a @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_a @ A @ B @ D )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(125,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_268_order_Onot__eq__order__implies__strict) ).

thf(629,plain,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(201,axiom,
    ( ord_less_eq_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_nat @ A @ B )
          | ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_271_dual__order_Oorder__iff__strict) ).

thf(846,plain,
    ( ord_less_eq_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_nat @ A @ B )
          | ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(240,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ( ( ( min_dist_capacity @ A @ B @ C )
          = D )
       => ( ( ord_less_nat @ E @ D )
         => ? [F: nat] :
              ( ( connected_capacity @ A @ B @ F )
              & ( ( min_dist_capacity @ A @ B @ F )
                = E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_Graph_Omin__dist__less) ).

thf(962,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ( ( ( min_dist_capacity @ A @ B @ C )
          = D )
       => ( ( ord_less_nat @ E @ D )
         => ? [F: nat] :
              ( ( connected_capacity @ A @ B @ F )
              & ( ( min_dist_capacity @ A @ B @ F )
                = E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(36,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_215_linear) ).

thf(352,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)],[36]) ).

thf(53,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_le__refl) ).

thf(408,plain,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(1,conjecture,
    ord_less_nat @ ( min_dist_capacity @ c @ s @ t ) @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1a ) @ ( size_s1990949619at_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_s1990949619at_nat @ p1a ) @ ( size_s1990949619at_nat @ p2_a ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(276,plain,
    ~ ( ord_less_nat @ ( min_dist_capacity @ c @ s @ t ) @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1a ) @ ( size_s1990949619at_nat @ p2_a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(249,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ( dist_capacity @ A @ B @ ( min_dist_capacity @ A @ B @ C ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_Graph_Omin__dist__is__dist) ).

thf(994,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ( dist_capacity @ A @ B @ ( min_dist_capacity @ A @ B @ C ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(16,axiom,
    ( reacha1693770334pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat] : ( collect_nat @ ( connected_capacity @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_Graph_OreachableNodes__def) ).

thf(308,plain,
    ( reacha1693770334pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat] : ( collect_nat @ ( connected_capacity @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(43,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(379,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)],[43]) ).

thf(135,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_102_add__mono__thms__linordered__field_I5_J) ).

thf(655,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)],[135]) ).

thf(46,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(385,plain,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(94,axiom,
    ! [A: nat,B: nat] :
      ( ( connected_capacity @ c @ A @ B )
     => ~ ! [C: list_P559422087at_nat] :
            ~ ( isShor1936442771pacity @ c @ A @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_obtain__shortest__path) ).

thf(533,plain,
    ! [A: nat,B: nat] :
      ( ( connected_capacity @ c @ A @ B )
     => ~ ! [C: list_P559422087at_nat] :
            ~ ( isShor1936442771pacity @ c @ A @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(100,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_less__not__refl2) ).

thf(550,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(154,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_269_dual__order_Ostrict__implies__order) ).

thf(715,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(74,axiom,
    ! [A: nat,B: nat] :
      ( ( connected_a @ c2 @ A @ B )
      = ( ? [C: list_P559422087at_nat] : ( isPath_a @ c2 @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_g_H_Oconnected__def) ).

thf(472,plain,
    ! [A: nat,B: nat] :
      ( ( connected_a @ c2 @ A @ B )
      = ( ? [C: list_P559422087at_nat] : ( isPath_a @ c2 @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(32,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_218_order_Otrans) ).

thf(340,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)],[32]) ).

thf(96,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_99_add__strict__mono) ).

thf(538,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)],[96]) ).

thf(229,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_65_add__le__mono) ).

thf(930,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)],[229]) ).

thf(141,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_192_g_H_Omin__dist__split_I1_J) ).

thf(670,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)],[141]) ).

thf(219,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_dual__order_Ostrict__implies__not__eq) ).

thf(898,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(57,axiom,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ ( B @ C ) )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_eq_nat @ A @ ( B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_order__subst1) ).

thf(417,plain,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ ( B @ C ) )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_eq_nat @ A @ ( B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(214,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(885,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)],[214]) ).

thf(12,axiom,
    dist_capacity = dist_capacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_Graph_Odist_Ocong) ).

thf(301,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(134,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_less__imp__neq) ).

thf(651,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(22,axiom,
    isSimp1359852763pacity = isSimp1359852763pacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_Graph_OisSimplePath_Ocong) ).

thf(320,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(105,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_less__not__sym) ).

thf(565,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(47,axiom,
    connected_capacity = connected_capacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_Graph_Oconnected_Ocong) ).

thf(389,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(158,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_ord__less__eq__trans) ).

thf(724,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(107,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
      = ( ( isPath_capacity @ c @ A @ B @ C )
        & ! [D: list_P559422087at_nat] :
            ( ( isPath_capacity @ c @ A @ D @ C )
           => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ B ) @ ( size_s1990949619at_nat @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_isShortestPath__def) ).

thf(570,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
      = ( ( isPath_capacity @ c @ A @ B @ C )
        & ! [D: list_P559422087at_nat] :
            ( ( isPath_capacity @ c @ A @ D @ C )
           => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ B ) @ ( size_s1990949619at_nat @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(156,axiom,
    isPath_a @ c2 @ ua @ p2_a @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_P2_H) ).

thf(719,plain,
    isPath_a @ c2 @ ua @ p2_a @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(179,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_capacity @ c @ A @ B @ C )
     => ? [D: list_P559422087at_nat] : ( isSimp1359852763pacity @ c @ A @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_isSPath__pathLE) ).

thf(785,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_capacity @ c @ A @ B @ C )
     => ? [D: list_P559422087at_nat] : ( isSimp1359852763pacity @ c @ A @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(113,axiom,
    min_dist_capacity = min_dist_capacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_Graph_Omin__dist_Ocong) ).

thf(587,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(185,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_256_order_Oasym) ).

thf(802,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(220,axiom,
    isPath_a @ c2 @ ua @ p2_a @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178__C1_Oprems_C_I3_J) ).

thf(902,plain,
    isPath_a @ c2 @ ua @ p2_a @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(128,axiom,
    ( isShortestPath_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_a @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_a @ A @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_Graph_OisShortestPath__min__dist__def) ).

thf(638,plain,
    ( isShortestPath_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_a @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_a @ A @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(58,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ord_less_eq_nat @ ( min_dist_a @ A @ B @ D ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_Graph_Omin__dist__minD) ).

thf(421,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ord_less_eq_nat @ ( min_dist_a @ A @ B @ D ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(56,axiom,
    isPath_capacity = isPath_capacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_Graph_OisPath_Ocong) ).

thf(416,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(25,axiom,
    ! [A: nat > $o,B: nat,C: nat > $o] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( C @ D ) )
       => ( C @ ( ord_Least_nat @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_264_LeastI2) ).

thf(325,plain,
    ! [A: nat > $o,B: nat,C: nat > $o] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( C @ D ) )
       => ( C @ ( ord_Least_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(8,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_a @ c2 @ A @ B @ C )
     => ? [D: list_P559422087at_nat] : ( isSimplePath_a @ c2 @ A @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_g_H_OisSPath__pathLE) ).

thf(289,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_a @ c2 @ A @ B @ C )
     => ? [D: list_P559422087at_nat] : ( isSimplePath_a @ c2 @ A @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(20,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_capacity @ A @ B @ C @ D )
     => ? [E: list_P559422087at_nat] : ( isSimp1359852763pacity @ A @ B @ E @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_Graph_OisSPath__pathLE) ).

thf(316,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_capacity @ A @ B @ C @ D )
     => ? [E: list_P559422087at_nat] : ( isSimp1359852763pacity @ A @ B @ E @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(275,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
      = ( ( isSimp1359852763pacity @ c @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_capacity @ c @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_isShortestPath__alt) ).

thf(1100,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
      = ( ( isSimp1359852763pacity @ c @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_capacity @ c @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(6,axiom,
    ! [A: product_prod_nat_nat > a,B: nat] : ( connected_a @ A @ B @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_Graph_Oconnected__refl) ).

thf(284,plain,
    ! [A: product_prod_nat_nat > a,B: nat] : ( connected_a @ A @ B @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(175,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_244_less__induct) ).

thf(770,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)],[175]) ).

thf(75,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_189_g_H_Omin__dist__le) ).

thf(475,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)],[75]) ).

thf(23,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_order__refl) ).

thf(321,plain,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(131,axiom,
    ( dist_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
        ? [E: list_P559422087at_nat] :
          ( ( isPath_capacity @ A @ B @ E @ D )
          & ( ( size_s1990949619at_nat @ E )
            = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_Graph_Odist__def) ).

thf(646,plain,
    ( dist_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
        ? [E: list_P559422087at_nat] :
          ( ( isPath_capacity @ A @ B @ E @ D )
          & ( ( size_s1990949619at_nat @ E )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(34,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).

thf(348,plain,
    ! [A: nat,B: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(76,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( A @ ( ord_Least_nat @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_267_LeastI) ).

thf(480,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( A @ ( ord_Least_nat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(78,axiom,
    ! [A: nat,B: nat] :
      ( ( connected_a @ c2 @ A @ B )
     => ~ ! [C: list_P559422087at_nat] :
            ~ ( isShortestPath_a @ c2 @ A @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_g_H_Oobtain__shortest__path) ).

thf(485,plain,
    ! [A: nat,B: nat] :
      ( ( connected_a @ c2 @ A @ B )
     => ~ ! [C: list_P559422087at_nat] :
            ~ ( isShortestPath_a @ c2 @ A @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(197,axiom,
    ! [A: nat] : ( connected_capacity @ c @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_connected__refl) ).

thf(834,plain,
    ! [A: nat] : ( connected_capacity @ c @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(245,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_less__irrefl__nat) ).

thf(981,plain,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(50,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat] :
      ( ( connected_a @ A @ B @ C )
     => ~ ! [D: list_P559422087at_nat] :
            ~ ( isShortestPath_a @ A @ B @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_Graph_Oobtain__shortest__path) ).

thf(396,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat] :
      ( ( connected_a @ A @ B @ C )
     => ~ ! [D: list_P559422087at_nat] :
            ~ ( isShortestPath_a @ A @ B @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(137,axiom,
    ord_less_nat @ ( min_dist_capacity @ c @ s @ ua ) @ ( size_s1990949619at_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(660,plain,
    ord_less_nat @ ( min_dist_capacity @ c @ s @ ua ) @ ( size_s1990949619at_nat @ p1a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(251,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat] :
      ( ( connected_a @ A @ B @ C )
     => ( ( ( min_dist_a @ A @ B @ C )
          = D )
       => ( ( ord_less_nat @ E @ D )
         => ? [F: nat] :
              ( ( connected_a @ A @ B @ F )
              & ( ( min_dist_a @ A @ B @ F )
                = E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_Graph_Omin__dist__less) ).

thf(998,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat] :
      ( ( connected_a @ A @ B @ C )
     => ( ( ( min_dist_a @ A @ B @ C )
          = D )
       => ( ( ord_less_nat @ E @ D )
         => ? [F: nat] :
              ( ( connected_a @ A @ B @ F )
              & ( ( min_dist_a @ A @ B @ F )
                = E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(247,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_67_add__leD2) ).

thf(988,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)],[247]) ).

thf(61,axiom,
    ( connected_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat] :
        ? [D: nat] : ( dist_a @ A @ B @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_Graph_Oconnected__by__dist) ).

thf(428,plain,
    ( connected_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat] :
        ? [D: nat] : ( dist_a @ A @ B @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(18,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_dual__order_Orefl) ).

thf(312,plain,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(10,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
        = ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_antisym__conv) ).

thf(293,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
        = ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(237,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_not__add__less1) ).

thf(956,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(267,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_eq_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_93_add__mono__thms__linordered__field_I4_J) ).

thf(1072,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_eq_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)],[267]) ).

thf(67,axiom,
    ( min_dist_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat] :
          ( ord_Least_nat
          @ ^ [D: nat] : ( dist_a @ A @ B @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_Graph_Omin__dist__def) ).

thf(453,plain,
    ( min_dist_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat] :
          ( ord_Least_nat
          @ ^ [D: nat] : ( dist_a @ A @ B @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(268,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_82_infinite__descent) ).

thf(1074,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)],[268]) ).

thf(91,axiom,
    min_dist_a = min_dist_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_Graph_Omin__dist_Ocong) ).

thf(529,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(138,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_less__trans) ).

thf(661,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(98,axiom,
    ( ord_less_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( B != A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_270_dual__order_Ostrict__iff__order) ).

thf(546,plain,
    ( ord_less_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( B != A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(227,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ( ord_less_nat @ B @ A )
        | ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_not__less__iff__gr__or__eq) ).

thf(925,plain,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ( ord_less_nat @ B @ A )
        | ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(171,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_capacity @ A @ B @ C @ D )
     => ( dist_capacity @ A @ B @ ( size_s1990949619at_nat @ C ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_Graph_OisPath__distD) ).

thf(761,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_capacity @ A @ B @ C @ D )
     => ( dist_capacity @ A @ B @ ( size_s1990949619at_nat @ C ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(266,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_83_nat__less__induct) ).

thf(1067,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)],[266]) ).

thf(153,axiom,
    isShor1936442771pacity @ c @ s @ p @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_SP) ).

thf(714,plain,
    isShor1936442771pacity @ c @ s @ p @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(183,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_110_add__lessD1) ).

thf(798,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)],[183]) ).

thf(87,axiom,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( C @ B )
          = D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_eq_nat @ ( C @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_ord__le__eq__subst) ).

thf(512,plain,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( C @ B )
          = D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_eq_nat @ ( C @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(66,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ~ ( ord_less_eq_nat @ B @ C ) )
     => ( ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ A @ C ) )
       => ( ( ( ord_less_eq_nat @ A @ C )
           => ~ ( ord_less_eq_nat @ C @ B ) )
         => ( ( ( ord_less_eq_nat @ C @ B )
             => ~ ( ord_less_eq_nat @ B @ A ) )
           => ( ( ( ord_less_eq_nat @ B @ C )
               => ~ ( ord_less_eq_nat @ C @ A ) )
             => ~ ( ( ord_less_eq_nat @ C @ A )
                 => ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_le__cases3) ).

thf(439,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ~ ( ord_less_eq_nat @ B @ C ) )
     => ( ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ A @ C ) )
       => ( ( ( ord_less_eq_nat @ A @ C )
           => ~ ( ord_less_eq_nat @ C @ B ) )
         => ( ( ( ord_less_eq_nat @ C @ B )
             => ~ ( ord_less_eq_nat @ B @ A ) )
           => ( ( ( ord_less_eq_nat @ B @ C )
               => ~ ( ord_less_eq_nat @ C @ A ) )
             => ~ ( ( ord_less_eq_nat @ C @ A )
                 => ~ ( ord_less_eq_nat @ A @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(29,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShor1936442771pacity @ A @ B @ C @ D )
     => ( isSimp1359852763pacity @ A @ B @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_Graph_OshortestPath__is__simple) ).

thf(334,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShor1936442771pacity @ A @ B @ C @ D )
     => ( isSimp1359852763pacity @ A @ B @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(258,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_105_trans__less__add1) ).

thf(1042,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)],[258]) ).

thf(13,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_Least__le) ).

thf(302,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(146,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ C @ D ) )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_add__mono__thms__linordered__field_I3_J) ).

thf(693,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ C @ D ) )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(159,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_capacity @ c @ A @ B @ C )
     => ( dist_capacity @ c @ A @ ( size_s1990949619at_nat @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_isPath__distD) ).

thf(729,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_capacity @ c @ A @ B @ C )
     => ( dist_capacity @ c @ A @ ( size_s1990949619at_nat @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(212,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist_a @ c2 @ A @ B @ C )
      = ( ? [D: list_P559422087at_nat] :
            ( ( isPath_a @ c2 @ A @ D @ C )
            & ( ( size_s1990949619at_nat @ D )
              = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_g_H_Odist__def) ).

thf(879,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist_a @ c2 @ A @ B @ C )
      = ( ? [D: list_P559422087at_nat] :
            ( ( isPath_a @ c2 @ A @ D @ C )
            & ( ( size_s1990949619at_nat @ D )
              = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(26,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShortestPath_a @ A @ B @ C @ D )
     => ( isSimplePath_a @ A @ B @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_Graph_OshortestPath__is__simple) ).

thf(329,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShortestPath_a @ A @ B @ C @ D )
     => ( isSimplePath_a @ A @ B @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(216,axiom,
    ! [A: nat] :
      ( ( reacha1693770334pacity @ c @ A )
      = ( collect_nat @ ( connected_capacity @ c @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_reachableNodes__def) ).

thf(889,plain,
    ! [A: nat] :
      ( ( reacha1693770334pacity @ c @ A )
      = ( collect_nat @ ( connected_capacity @ c @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(80,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_190_g_H_Omin__dist__minD) ).

thf(493,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)],[80]) ).

thf(143,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(682,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)],[143]) ).

thf(152,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_103_less__add__eq__less) ).

thf(710,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)],[152]) ).

thf(242,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(974,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)],[242]) ).

thf(203,axiom,
    ( isShortestPath_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_a @ A @ B @ C @ D )
          & ! [E: list_P559422087at_nat] :
              ( ( isPath_a @ A @ B @ E @ D )
             => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ C ) @ ( size_s1990949619at_nat @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_Graph_OisShortestPath__def) ).

thf(852,plain,
    ( isShortestPath_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_a @ A @ B @ C @ D )
          & ! [E: list_P559422087at_nat] :
              ( ( isPath_a @ A @ B @ E @ D )
             => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ C ) @ ( size_s1990949619at_nat @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(64,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_a @ A @ B @ C @ D )
     => ? [E: list_P559422087at_nat] : ( isSimplePath_a @ A @ B @ E @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_Graph_OisSPath__pathLE) ).

thf(435,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_a @ A @ B @ C @ D )
     => ? [E: list_P559422087at_nat] : ( isSimplePath_a @ A @ B @ E @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(111,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(583,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)],[111]) ).

thf(89,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(522,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)],[89]) ).

thf(195,axiom,
    ( isShor1936442771pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isSimp1359852763pacity @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_capacity @ A @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_Graph_OisShortestPath__alt) ).

thf(830,plain,
    ( isShor1936442771pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isSimp1359852763pacity @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_capacity @ A @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(269,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ord_less_nat @ A @ ( ord_Least_nat @ B ) )
     => ~ ( B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_not__less__Least) ).

thf(1079,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ord_less_nat @ A @ ( ord_Least_nat @ B ) )
     => ~ ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(254,axiom,
    ( isShor1936442771pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_capacity @ A @ B @ C @ D )
          & ! [E: list_P559422087at_nat] :
              ( ( isPath_capacity @ A @ B @ E @ D )
             => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ C ) @ ( size_s1990949619at_nat @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_Graph_OisShortestPath__def) ).

thf(1027,plain,
    ( isShor1936442771pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_capacity @ A @ B @ C @ D )
          & ! [E: list_P559422087at_nat] :
              ( ( isPath_capacity @ A @ B @ E @ D )
             => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ C ) @ ( size_s1990949619at_nat @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(165,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_104_trans__less__add2) ).

thf(742,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)],[165]) ).

thf(210,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_le__add2) ).

thf(874,plain,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(133,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(650,plain,
    isPath_capacity @ c @ s @ p1 @ v,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(122,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ A )
       => ( ord_less_nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_dual__order_Ostrict__trans) ).

thf(615,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ A )
       => ( ord_less_nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(52,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_187_g_H_Omin__distI2) ).

thf(403,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)],[52]) ).

thf(252,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_56_le__neq__implies__less) ).

thf(1006,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)],[252]) ).

thf(208,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_68_add__leD1) ).

thf(869,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)],[208]) ).

thf(83,axiom,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( C @ B ) @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_eq_nat @ ( C @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_order__subst2) ).

thf(499,plain,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( C @ B ) @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_eq_nat @ ( C @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(205,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( ~ ( ord_less_nat @ B @ A ) )
        = ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_antisym__conv3) ).

thf(857,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( ~ ( ord_less_nat @ B @ A ) )
        = ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(79,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat > $o] :
      ( ( connected_a @ A @ B @ C )
     => ( ! [E: nat] :
            ( ( dist_a @ A @ B @ E @ C )
           => ( ! [F: nat] :
                  ( ( dist_a @ A @ B @ F @ C )
                 => ( ord_less_eq_nat @ E @ F ) )
             => ( D @ E ) ) )
       => ( D @ ( min_dist_a @ A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_Graph_Omin__distI2) ).

thf(488,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat > $o] :
      ( ( connected_a @ A @ B @ C )
     => ( ! [E: nat] :
            ( ( dist_a @ A @ B @ E @ C )
           => ( ! [F: nat] :
                  ( ( dist_a @ A @ B @ F @ C )
                 => ( ord_less_eq_nat @ E @ F ) )
             => ( D @ E ) ) )
       => ( D @ ( min_dist_a @ A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(31,axiom,
    isSimplePath_a = isSimplePath_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_Graph_OisSimplePath_Ocong) ).

thf(339,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(230,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_257_neq__iff) ).

thf(933,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)],[230]) ).

thf(101,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
      = ( ( isPath_capacity @ c @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_capacity @ c @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_isShortestPath__min__dist__def) ).

thf(554,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
      = ( ( isPath_capacity @ c @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_capacity @ c @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(241,axiom,
    ! [A: list_P559422087at_nat,B: list_P559422087at_nat] :
      ( ( ( size_s1990949619at_nat @ A )
       != ( size_s1990949619at_nat @ B ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_size__neq__size__imp__neq) ).

thf(970,plain,
    ! [A: list_P559422087at_nat,B: list_P559422087at_nat] :
      ( ( ( size_s1990949619at_nat @ A )
       != ( size_s1990949619at_nat @ B ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(49,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_225_order__trans) ).

thf(393,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)],[49]) ).

thf(149,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_193_g_H_Odist__trans) ).

thf(701,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)],[149]) ).

thf(177,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_dual__order_Oasym) ).

thf(779,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(264,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_ord__eq__less__trans) ).

thf(1060,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(261,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_51_add__mono) ).

thf(1050,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)],[261]) ).

thf(69,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_ord__eq__le__trans) ).

thf(457,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(246,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(984,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)],[246]) ).

thf(223,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ( dist_capacity @ A @ D @ E @ F )
       => ( ( ( min_dist_capacity @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_capacity @ A @ B @ D )
            = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_Graph_Omin__dist__split_I1_J) ).

thf(911,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ( dist_capacity @ A @ D @ E @ F )
       => ( ( ( min_dist_capacity @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_capacity @ A @ B @ D )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(84,axiom,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( ord_less_eq_nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_order__class_Oorder_Oeq__iff) ).

thf(503,plain,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( ord_less_eq_nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(119,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_add__less__le__mono) ).

thf(605,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(33,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_ord__le__eq__trans) ).

thf(343,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(40,axiom,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( A
        = ( B @ C ) )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_eq_nat @ A @ ( B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_ord__eq__le__subst) ).

thf(360,plain,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( A
        = ( B @ C ) )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_eq_nat @ E @ F )
             => ( ord_less_eq_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_eq_nat @ A @ ( B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(226,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ( dist_a @ A @ D @ E @ F )
       => ( ( ( min_dist_a @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_a @ A @ D @ F )
            = E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_Graph_Omin__dist__split_I2_J) ).

thf(921,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ( dist_a @ A @ D @ E @ F )
       => ( ( ( min_dist_a @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_a @ A @ D @ F )
            = E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(97,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ! [E: nat] :
            ( ( dist_capacity @ A @ B @ E @ D )
           => ( ord_less_eq_nat @ C @ E ) )
       => ( ( min_dist_capacity @ A @ B @ D )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_Graph_Omin__distI__eq) ).

thf(541,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ! [E: nat] :
            ( ( dist_capacity @ A @ B @ E @ D )
           => ( ord_less_eq_nat @ C @ E ) )
       => ( ( min_dist_capacity @ A @ B @ D )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(136,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_4_add__less__cancel__left) ).

thf(657,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)],[136]) ).

thf(173,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_106_add__less__mono1) ).

thf(765,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)],[173]) ).

thf(142,axiom,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( A
        = ( B @ C ) )
     => ( ( ord_less_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_nat @ A @ ( B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_263_ord__eq__less__subst) ).

thf(674,plain,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( A
        = ( B @ C ) )
     => ( ( ord_less_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_nat @ A @ ( B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(92,axiom,
    ( ( '?' @ nat )
    = ( ^ [A: nat > $o] :
        ? [B: nat] :
          ( ( A @ B )
          & ! [C: nat] :
              ( ( ord_less_nat @ C @ B )
             => ~ ( A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_exists__least__iff) ).

thf(530,plain,
    ( ( '?' @ nat )
    = ( ^ [A: nat > $o] :
        ? [B: nat] :
          ( ( A @ B )
          & ! [C: nat] :
              ( ( ord_less_nat @ C @ B )
             => ~ ( A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(236,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_add_Oassoc) ).

thf(953,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(55,axiom,
    ! [A: nat > $o,B: nat > $o] :
      ( ? [C: nat] : ( A @ C )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( B @ C ) )
       => ( B @ ( ord_Least_nat @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_266_LeastI2__ex) ).

thf(412,plain,
    ! [A: nat > $o,B: nat > $o] :
      ( ? [C: nat] : ( A @ C )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( B @ C ) )
       => ( B @ ( ord_Least_nat @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(123,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(618,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)],[123]) ).

thf(62,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(430,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)],[62]) ).

thf(166,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_less__imp__not__eq2) ).

thf(745,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(108,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_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_91_add__le__less__mono) ).

thf(573,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_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)],[108]) ).

thf(11,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( connected_a @ A @ B @ C )
     => ( ( ord_less_eq_nat @ D @ ( min_dist_a @ A @ B @ C ) )
       => ? [E: nat] :
            ( ( connected_a @ A @ B @ E )
            & ( ( min_dist_a @ A @ B @ E )
              = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_Graph_Omin__dist__le) ).

thf(296,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( connected_a @ A @ B @ C )
     => ( ( ord_less_eq_nat @ D @ ( min_dist_a @ A @ B @ C ) )
       => ? [E: nat] :
            ( ( connected_a @ A @ B @ E )
            & ( ( min_dist_a @ A @ B @ E )
              = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(161,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_order_Ostrict__trans) ).

thf(733,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(104,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_62_trans__le__add2) ).

thf(562,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)],[104]) ).

thf(59,axiom,
    ! [A: nat > $o,B: nat > $o] :
      ( ! [C: nat] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect_nat @ A )
        = ( collect_nat @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_Collect__cong) ).

thf(423,plain,
    ! [A: nat > $o,B: nat > $o] :
      ( ! [C: nat] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect_nat @ A )
        = ( collect_nat @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(200,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(843,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)],[200]) ).

thf(7,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_198_g_H_Omin__dist__def) ).

thf(286,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)],[7]) ).

thf(9,axiom,
    ( reachableNodes_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat] : ( collect_nat @ ( connected_a @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_Graph_OreachableNodes__def) ).

thf(291,plain,
    ( reachableNodes_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat] : ( collect_nat @ ( connected_a @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(5,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_181_g_H_Oconnected__by__dist) ).

thf(281,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)],[5]) ).

thf(231,axiom,
    ! [A: nat,B: nat,C: nat,D: 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_53_add__mono__thms__linordered__semiring_I2_J) ).

thf(936,plain,
    ! [A: nat,B: nat,C: nat,D: 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)],[231]) ).

thf(44,axiom,
    dist_a = dist_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_Graph_Odist_Ocong) ).

thf(382,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(180,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_dual__order_Oirrefl) ).

thf(787,plain,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(273,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_a @ A @ B @ C @ D )
     => ( dist_a @ A @ B @ ( size_s1990949619at_nat @ C ) @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_Graph_OisPath__distD) ).

thf(1090,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isPath_a @ A @ B @ C @ D )
     => ( dist_a @ A @ B @ ( size_s1990949619at_nat @ C ) @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(186,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_add__le__cancel__right) ).

thf(804,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(150,axiom,
    isPath_a @ c2 @ s @ p3 @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_assms_I5_J) ).

thf(704,plain,
    isPath_a @ c2 @ s @ p3 @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(72,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [B: nat] : ( member_nat @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).

thf(466,plain,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [B: nat] : ( member_nat @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(274,axiom,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( C @ B )
          = D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_nat @ ( C @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_262_ord__less__eq__subst) ).

thf(1092,plain,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( C @ B )
          = D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_nat @ ( C @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(118,axiom,
    isPath_capacity @ c @ s @ p1a @ va,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6__C1_Oprems_C_I2_J) ).

thf(604,plain,
    isPath_capacity @ c @ s @ p1a @ va,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(106,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(567,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)],[106]) ).

thf(21,axiom,
    ( connected_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
        ? [D: list_P559422087at_nat] : ( isPath_capacity @ A @ B @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_Graph_Oconnected__def) ).

thf(318,plain,
    ( connected_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
        ? [D: list_P559422087at_nat] : ( isPath_capacity @ A @ B @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(260,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_add__le__imp__le__right) ).

thf(1048,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(162,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(736,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)],[162]) ).

thf(248,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_55_less__mono__imp__le__mono) ).

thf(990,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)],[248]) ).

thf(250,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_a @ c2 @ A @ B @ C )
     => ( dist_a @ c2 @ A @ ( size_s1990949619at_nat @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_g_H_OisPath__distD) ).

thf(996,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isPath_a @ c2 @ A @ B @ C )
     => ( dist_a @ c2 @ A @ ( size_s1990949619at_nat @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(170,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_add__left__mono) ).

thf(758,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(190,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_61_nat__le__iff__add) ).

thf(816,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)],[190]) ).

thf(24,axiom,
    ( connected_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
        ? [D: nat] : ( dist_capacity @ A @ B @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_Graph_Oconnected__by__dist) ).

thf(323,plain,
    ( connected_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
        ? [D: nat] : ( dist_capacity @ A @ B @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(77,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_217_le__cases) ).

thf(483,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)],[77]) ).

thf(255,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(1029,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)],[255]) ).

thf(70,axiom,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( ord_less_eq_nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_eq__iff) ).

thf(462,plain,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( ord_less_eq_nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(202,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_less__imp__not__eq) ).

thf(848,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(88,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShortestPath_a @ A @ B @ C @ D )
     => ( isPath_a @ A @ B @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_Graph_OshortestPath__is__path) ).

thf(520,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShortestPath_a @ A @ B @ C @ D )
     => ( isPath_a @ A @ B @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(124,axiom,
    ! [A: nat,B: nat,C: $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ A )
       => C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_less__imp__triv) ).

thf(626,plain,
    ! [A: nat,B: nat,C: $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ A )
       => C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(224,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_less__irrefl) ).

thf(915,plain,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(144,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ( ( ord_less_eq_nat @ D @ ( min_dist_capacity @ A @ B @ C ) )
       => ? [E: nat] :
            ( ( connected_capacity @ A @ B @ E )
            & ( ( min_dist_capacity @ A @ B @ E )
              = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_Graph_Omin__dist__le) ).

thf(685,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ( ( ord_less_eq_nat @ D @ ( min_dist_capacity @ A @ B @ C ) )
       => ? [E: nat] :
            ( ( connected_capacity @ A @ B @ E )
            & ( ( min_dist_capacity @ A @ B @ E )
              = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(51,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_eq__refl) ).

thf(399,plain,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(238,axiom,
    ord_less_eq_nat @ ( min_dist_capacity @ c @ ua @ t ) @ ( size_s1990949619at_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(959,plain,
    ord_less_eq_nat @ ( min_dist_capacity @ c @ ua @ t ) @ ( size_s1990949619at_nat @ p2_a ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(174,axiom,
    ( ord_less_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( A != B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_nat__less__le) ).

thf(768,plain,
    ( ord_less_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
          & ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(147,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_94_mono__nat__linear__lb) ).

thf(695,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)],[147]) ).

thf(4,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_dual__order_Otrans) ).

thf(278,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(68,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_182_g_H_Omin__dist__is__dist) ).

thf(455,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)],[68]) ).

thf(120,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_191_g_H_Omin__dist__split_I2_J) ).

thf(608,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)],[120]) ).

thf(85,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ! [E: nat] :
            ( ( dist_a @ A @ B @ E @ D )
           => ( ord_less_eq_nat @ C @ E ) )
       => ( ( min_dist_a @ A @ B @ D )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_Graph_Omin__distI__eq) ).

thf(505,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ! [E: nat] :
            ( ( dist_a @ A @ B @ E @ D )
           => ( ord_less_eq_nat @ C @ E ) )
       => ( ( min_dist_a @ A @ B @ D )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(256,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_97_add__strict__right__mono) ).

thf(1034,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)],[256]) ).

thf(60,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_206_g_H_Oconnected__distI) ).

thf(426,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)],[60]) ).

thf(233,axiom,
    ( plus_plus_nat
    = ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_add_Ocommute) ).

thf(944,plain,
    ( plus_plus_nat
    = ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(182,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_not__add__less2) ).

thf(795,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(99,axiom,
    ! [A: nat] :
    ? [B: nat] : ( ord_less_nat @ A @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_259_gt__ex) ).

thf(548,plain,
    ! [A: nat] :
    ? [B: nat] : ( ord_less_nat @ A @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(132,axiom,
    ( isShortestPath_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isSimplePath_a @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_a @ A @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_Graph_OisShortestPath__alt) ).

thf(648,plain,
    ( isShortestPath_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isSimplePath_a @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_a @ A @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(38,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_214_antisym) ).

thf(355,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)],[38]) ).

thf(193,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_57_less__or__eq__imp__le) ).

thf(823,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)],[193]) ).

thf(198,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
      = ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_add__right__cancel) ).

thf(836,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
      = ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(270,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_258_neqE) ).

thf(1081,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)],[270]) ).

thf(167,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ( dist_a @ A @ D @ E @ F )
       => ( dist_a @ A @ B @ ( plus_plus_nat @ C @ E ) @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_Graph_Odist__trans) ).

thf(749,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( ( dist_a @ A @ D @ E @ F )
       => ( dist_a @ A @ B @ ( plus_plus_nat @ C @ E ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(102,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_48_add__right__mono) ).

thf(557,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)],[102]) ).

thf(110,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
      | ( A = B )
      | ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_less__linear) ).

thf(580,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
      | ( A = B )
      | ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(145,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_31_add__le__cancel__left) ).

thf(690,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)],[145]) ).

thf(129,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_66_le__Suc__ex) ).

thf(640,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)],[129]) ).

thf(151,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat > $o] :
      ( ( connected_capacity @ A @ B @ C )
     => ( ! [E: nat] :
            ( ( dist_capacity @ A @ B @ E @ C )
           => ( ! [F: nat] :
                  ( ( dist_capacity @ A @ B @ F @ C )
                 => ( ord_less_eq_nat @ E @ F ) )
             => ( D @ E ) ) )
       => ( D @ ( min_dist_capacity @ A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_Graph_Omin__distI2) ).

thf(705,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat > $o] :
      ( ( connected_capacity @ A @ B @ C )
     => ( ! [E: nat] :
            ( ( dist_capacity @ A @ B @ E @ C )
           => ( ! [F: nat] :
                  ( ( dist_capacity @ A @ B @ F @ C )
                 => ( ord_less_eq_nat @ E @ F ) )
             => ( D @ E ) ) )
       => ( D @ ( min_dist_capacity @ A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(215,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_43_add__le__imp__le__left) ).

thf(887,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)],[215]) ).

thf(163,axiom,
    isPath_a @ c2 @ u @ p2 @ t,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062) ).

thf(739,plain,
    isPath_a @ c2 @ u @ p2 @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(17,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
     => ( isPath_a @ c2 @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_g_H_OshortestPath__is__path) ).

thf(310,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
     => ( isPath_a @ c2 @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(262,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_add__less__cancel__right) ).

thf(1053,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(27,axiom,
    isShor1936442771pacity = isShor1936442771pacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_Graph_OisShortestPath_Ocong) ).

thf(331,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(112,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_less__asym_H) ).

thf(585,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(155,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_less__imp__not__less) ).

thf(717,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(109,axiom,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( C @ B ) @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_nat @ ( C @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_260_order__less__subst2) ).

thf(576,plain,
    ! [A: nat,B: nat,C: nat > nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( C @ B ) @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( C @ E ) @ ( C @ F ) ) )
         => ( ord_less_nat @ ( C @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(130,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_109_add__less__mono) ).

thf(643,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)],[130]) ).

thf(30,axiom,
    ! [A: nat] :
      ( ( reachableNodes_a @ c2 @ A )
      = ( collect_nat @ ( connected_a @ c2 @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_g_H_OreachableNodes__def) ).

thf(336,plain,
    ! [A: nat] :
      ( ( reachableNodes_a @ c2 @ A )
      = ( collect_nat @ ( connected_a @ c2 @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(244,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_less__not__refl) ).

thf(978,plain,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(19,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( connected_capacity @ A @ B @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_Graph_Oconnected__distI) ).

thf(314,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( connected_capacity @ A @ B @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(116,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_186_g_H_Omin__dist__less) ).

thf(595,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)],[116]) ).

thf(121,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_linorder__cases) ).

thf(612,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(82,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShor1936442771pacity @ A @ B @ C @ D )
     => ( isPath_capacity @ A @ B @ C @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_Graph_OshortestPath__is__path) ).

thf(497,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
      ( ( isShor1936442771pacity @ A @ B @ C @ D )
     => ( isPath_capacity @ A @ B @ C @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(211,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_add__left__cancel) ).

thf(876,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(271,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( C = D ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_add__mono__thms__linordered__semiring_I3_J) ).

thf(1084,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( C = D ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(81,axiom,
    ! [A: nat] : ( connected_a @ c2 @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_g_H_Oconnected__refl) ).

thf(495,plain,
    ! [A: nat] : ( connected_a @ c2 @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(41,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_188_g_H_Omin__distI__eq) ).

thf(368,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)],[41]) ).

thf(243,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_272_order_Ostrict__implies__order) ).

thf(976,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(221,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ A @ D )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_group__cancel_Oadd1) ).

thf(903,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ A @ D )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(189,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
      = ( ( isSimplePath_a @ c2 @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_a @ c2 @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_g_H_OisShortestPath__alt) ).

thf(813,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
      = ( ( isSimplePath_a @ c2 @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_a @ c2 @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(218,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_44_le__iff__add) ).

thf(896,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)],[218]) ).

thf(14,axiom,
    connected_a = connected_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_Graph_Oconnected_Ocong) ).

thf(304,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(234,axiom,
    ! [A: nat,B: nat,C: nat,D: 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_101_add__mono__thms__linordered__field_I2_J) ).

thf(946,plain,
    ! [A: nat,B: nat,C: nat,D: 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)],[234]) ).

thf(35,axiom,
    isPath_a = isPath_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_Graph_OisPath_Ocong) ).

thf(351,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(48,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_dual__order_Oantisym) ).

thf(390,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(126,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
      = ( ( isPath_a @ c2 @ A @ B @ C )
        & ! [D: list_P559422087at_nat] :
            ( ( isPath_a @ c2 @ A @ D @ C )
           => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ B ) @ ( size_s1990949619at_nat @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_g_H_OisShortestPath__def) ).

thf(632,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
      = ( ( isPath_a @ c2 @ A @ B @ C )
        & ! [D: list_P559422087at_nat] :
            ( ( isPath_a @ c2 @ A @ D @ C )
           => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ B ) @ ( size_s1990949619at_nat @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(160,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
     => ( isPath_capacity @ c @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_shortestPath__is__path) ).

thf(731,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
     => ( isPath_capacity @ c @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(178,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_less__not__refl3) ).

thf(781,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(114,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_order_Ostrict__implies__not__eq) ).

thf(588,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(148,axiom,
    ( min_dist_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
          ( ord_Least_nat
          @ ^ [D: nat] : ( dist_capacity @ A @ B @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_Graph_Omin__dist__def) ).

thf(699,plain,
    ( min_dist_capacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
          ( ord_Least_nat
          @ ^ [D: nat] : ( dist_capacity @ A @ B @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(140,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
      = ( ( isPath_a @ c2 @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_a @ c2 @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_g_H_OisShortestPath__min__dist__def) ).

thf(667,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
      = ( ( isPath_a @ c2 @ A @ B @ C )
        & ( ( size_s1990949619at_nat @ B )
          = ( min_dist_a @ c2 @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(172,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
     => ( isSimp1359852763pacity @ c @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_shortestPath__is__simple) ).

thf(763,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShor1936442771pacity @ c @ A @ B @ C )
     => ( isSimp1359852763pacity @ c @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(45,axiom,
    ! [A: nat > $o] :
      ( ? [B: nat] : ( A @ B )
     => ( A @ ( ord_Least_nat @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_265_LeastI__ex) ).

thf(383,plain,
    ! [A: nat > $o] :
      ( ? [B: nat] : ( A @ B )
     => ( A @ ( ord_Least_nat @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(196,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ord_less_eq_nat @ ( min_dist_capacity @ A @ B @ D ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_Graph_Omin__dist__minD) ).

thf(832,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ord_less_eq_nat @ ( min_dist_capacity @ A @ B @ D ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(95,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_52_add__mono__thms__linordered__semiring_I1_J) ).

thf(536,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)],[95]) ).

thf(259,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_63_trans__le__add1) ).

thf(1045,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)],[259]) ).

thf(184,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_254_less__asym) ).

thf(800,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(63,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat] : ( connected_capacity @ A @ B @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_Graph_Oconnected__refl) ).

thf(433,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat] : ( connected_capacity @ A @ B @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(86,axiom,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ B @ A )
          & ( ord_less_eq_nat @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_dual__order_Oeq__iff) ).

thf(510,plain,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ B @ A )
          & ( ord_less_eq_nat @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(168,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_88_nat__neq__iff) ).

thf(752,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)],[168]) ).

thf(54,axiom,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
     => ( isSimplePath_a @ c2 @ A @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_g_H_OshortestPath__is__simple) ).

thf(410,plain,
    ! [A: nat,B: list_P559422087at_nat,C: nat] :
      ( ( isShortestPath_a @ c2 @ A @ B @ C )
     => ( isSimplePath_a @ c2 @ A @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(235,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ( dist_capacity @ A @ D @ E @ F )
       => ( dist_capacity @ A @ B @ ( plus_plus_nat @ C @ E ) @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_Graph_Odist__trans) ).

thf(950,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ( dist_capacity @ A @ D @ E @ F )
       => ( dist_capacity @ A @ B @ ( plus_plus_nat @ C @ E ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(191,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_nat @ A @ C )
        = ( plus_plus_nat @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_add__mono__thms__linordered__semiring_I4_J) ).

thf(818,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_nat @ A @ C )
        = ( plus_plus_nat @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(204,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_64_add__le__mono1) ).

thf(854,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)],[204]) ).

thf(181,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(790,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)],[181]) ).

thf(207,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_add__left__imp__eq) ).

thf(865,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(42,axiom,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less_eq_nat @ D @ E )
         => ( A @ D @ E ) )
     => ( ! [D: nat,E: nat] :
            ( ( A @ E @ D )
           => ( A @ D @ E ) )
       => ( A @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_linorder__wlog) ).

thf(373,plain,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less_eq_nat @ D @ E )
         => ( A @ D @ E ) )
     => ( ! [D: nat,E: nat] :
            ( ( A @ E @ D )
           => ( A @ D @ E ) )
       => ( A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(117,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(603,plain,
    isPath_capacity @ c @ ua @ p2_a @ t,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(15,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ~ ! [D: list_P559422087at_nat] :
            ~ ( isShor1936442771pacity @ A @ B @ D @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_Graph_Oobtain__shortest__path) ).

thf(305,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat] :
      ( ( connected_capacity @ A @ B @ C )
     => ~ ! [D: list_P559422087at_nat] :
            ~ ( isShor1936442771pacity @ A @ B @ D @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(225,axiom,
    ! [A: nat,B: nat] :
      ( ( connected_capacity @ c @ A @ B )
      = ( ? [C: list_P559422087at_nat] : ( isPath_capacity @ c @ A @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_connected__def) ).

thf(918,plain,
    ! [A: nat,B: nat] :
      ( ( connected_capacity @ c @ A @ B )
      = ( ? [C: list_P559422087at_nat] : ( isPath_capacity @ c @ A @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(65,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat] :
      ( ( connected_a @ A @ B @ C )
     => ( dist_a @ A @ B @ ( min_dist_a @ A @ B @ C ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_Graph_Omin__dist__is__dist) ).

thf(437,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat] :
      ( ( connected_a @ A @ B @ C )
     => ( dist_a @ A @ B @ ( min_dist_a @ A @ B @ C ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(194,axiom,
    ( isShor1936442771pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_capacity @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_capacity @ A @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_Graph_OisShortestPath__min__dist__def) ).

thf(828,plain,
    ( isShor1936442771pacity
    = ( ^ [A: product_prod_nat_nat > capacity,B: nat,C: list_P559422087at_nat,D: nat] :
          ( ( isPath_capacity @ A @ B @ C @ D )
          & ( ( size_s1990949619at_nat @ C )
            = ( min_dist_capacity @ A @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(157,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(720,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)],[157]) ).

thf(228,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_add__less__imp__less__right) ).

thf(928,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(39,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(358,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)],[39]) ).

thf(187,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_le__add1) ).

thf(807,plain,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(263,axiom,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ ( B @ C ) )
     => ( ( ord_less_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_nat @ A @ ( B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_261_order__less__subst1) ).

thf(1056,plain,
    ! [A: nat,B: nat > nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ ( B @ C ) )
     => ( ( ord_less_nat @ C @ D )
       => ( ! [E: nat,F: nat] :
              ( ( ord_less_nat @ E @ F )
             => ( ord_less_nat @ ( B @ E ) @ ( B @ F ) ) )
         => ( ord_less_nat @ A @ ( B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(139,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_add__strict__left__mono) ).

thf(664,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(239,axiom,
    ( ord_less_eq_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_nat @ A @ B )
          | ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_le__eq__less__or__eq) ).

thf(960,plain,
    ( ord_less_eq_nat
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_nat @ A @ B )
          | ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(127,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_71_add__leE) ).

thf(635,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)],[127]) ).

thf(93,axiom,
    connected_capacity @ c @ s @ ua,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29__092_060open_062connected_As_Au_092_060close_062) ).

thf(532,plain,
    connected_capacity @ c @ s @ ua,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(265,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_less__imp__le__nat) ).

thf(1065,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(199,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_add__right__imp__eq) ).

thf(839,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(164,axiom,
    ( dist_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
        ? [E: list_P559422087at_nat] :
          ( ( isPath_a @ A @ B @ E @ D )
          & ( ( size_s1990949619at_nat @ E )
            = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_Graph_Odist__def) ).

thf(740,plain,
    ( dist_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
        ? [E: list_P559422087at_nat] :
          ( ( isPath_a @ A @ B @ E @ D )
          & ( ( size_s1990949619at_nat @ E )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(71,axiom,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( connected_a @ A @ B @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_Graph_Oconnected__distI) ).

thf(464,plain,
    ! [A: product_prod_nat_nat > a,B: nat,C: nat,D: nat] :
      ( ( dist_a @ A @ B @ C @ D )
     => ( connected_a @ A @ B @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(28,axiom,
    ( connected_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat] :
        ? [D: list_P559422087at_nat] : ( isPath_a @ A @ B @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_Graph_Oconnected__def) ).

thf(332,plain,
    ( connected_a
    = ( ^ [A: product_prod_nat_nat > a,B: nat,C: nat] :
        ? [D: list_P559422087at_nat] : ( isPath_a @ A @ B @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(206,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(860,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)],[206]) ).

thf(90,axiom,
    isShortestPath_a = isShortestPath_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_Graph_OisShortestPath_Ocong) ).

thf(528,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(222,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist_capacity @ c @ A @ B @ C )
      = ( ? [D: list_P559422087at_nat] :
            ( ( isPath_capacity @ c @ A @ D @ C )
            & ( ( size_s1990949619at_nat @ D )
              = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_dist__def) ).

thf(908,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( dist_capacity @ c @ A @ B @ C )
      = ( ? [D: list_P559422087at_nat] :
            ( ( isPath_capacity @ c @ A @ D @ C )
            & ( ( size_s1990949619at_nat @ D )
              = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

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

thf(822,plain,
    isPath_capacity @ c @ s @ p1a @ va,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(3,axiom,
    reachableNodes_a = reachableNodes_a,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_Graph_OreachableNodes_Ocong) ).

thf(277,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(209,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_112_min__dist__def) ).

thf(871,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)],[209]) ).

thf(37,axiom,
    reacha1693770334pacity = reacha1693770334pacity,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_Graph_OreachableNodes_Ocong) ).

thf(354,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(253,axiom,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less_nat @ D @ E )
         => ( A @ D @ E ) )
     => ( ! [D: nat] : ( A @ D @ D )
       => ( ! [D: nat,E: nat] :
              ( ( A @ E @ D )
             => ( A @ D @ E ) )
         => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_linorder__less__wlog) ).

thf(1009,plain,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less_nat @ D @ E )
         => ( A @ D @ E ) )
     => ( ! [D: nat] : ( A @ D @ D )
       => ( ! [D: nat,E: nat] :
              ( ( A @ E @ D )
             => ( A @ D @ E ) )
         => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(169,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(755,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(272,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_96_add__less__imp__less__left) ).

thf(1088,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)],[272]) ).

thf(115,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_add_Oleft__commute) ).

thf(592,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(213,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_81_linorder__neqE__nat) ).

thf(882,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)],[213]) ).

thf(73,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_224_order__class_Oorder_Oantisym) ).

thf(469,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)],[73]) ).

thf(232,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( C = D ) )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_add__mono__thms__linordered__field_I1_J) ).

thf(940,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( C = D ) )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(257,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ D @ A )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_group__cancel_Oadd2) ).

thf(1037,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ D @ A )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(176,axiom,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ( dist_capacity @ A @ D @ E @ F )
       => ( ( ( min_dist_capacity @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_capacity @ A @ D @ F )
            = E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_Graph_Omin__dist__split_I2_J) ).

thf(775,plain,
    ! [A: product_prod_nat_nat > capacity,B: nat,C: nat,D: nat,E: nat,F: nat] :
      ( ( dist_capacity @ A @ B @ C @ D )
     => ( ( dist_capacity @ A @ D @ E @ F )
       => ( ( ( min_dist_capacity @ A @ B @ F )
            = ( plus_plus_nat @ C @ E ) )
         => ( ( min_dist_capacity @ A @ D @ F )
            = E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(103,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(560,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)],[103]) ).

thf(1233,plain,
    $false,
    inference(cvc4,[status(thm)],[892,809,629,846,962,352,408,276,994,308,379,655,385,533,550,715,472,340,538,930,670,898,417,885,301,651,320,565,389,724,570,719,785,587,802,902,638,421,416,325,289,316,1100,284,770,475,321,646,348,480,485,834,981,396,660,998,988,428,312,293,956,1072,453,1074,529,661,546,925,761,1067,714,798,512,439,334,1042,302,693,729,879,329,889,493,682,710,974,852,435,583,522,830,1079,1027,742,874,650,615,403,1006,869,499,857,488,339,933,554,970,393,701,779,1060,1050,457,984,911,503,605,343,360,921,541,657,765,674,530,953,412,618,430,745,573,296,733,562,423,843,286,291,281,936,382,787,1090,804,704,466,1092,604,567,318,1048,736,990,996,758,816,323,483,1029,462,848,520,626,915,685,399,959,768,695,278,455,608,505,1034,426,944,795,548,648,355,823,836,1081,749,557,580,690,640,705,887,739,310,1053,331,585,717,576,643,336,978,314,595,612,497,876,1084,495,368,976,903,813,896,304,946,351,390,632,731,781,588,699,667,763,383,832,536,1045,800,433,510,752,410,950,818,854,790,865,373,603,305,918,437,828,720,928,358,807,1056,664,960,635,532,1065,839,740,464,332,860,528,908,822,277,871,354,1009,755,1088,592,882,469,940,1037,775,560]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13  % Problem  : ITP048^1 : TPTP v8.2.0. Released v7.5.0.
% 0.07/0.17  % Command  : run_Leo-III %s %d
% 0.18/0.39  % Computer : n010.cluster.edu
% 0.18/0.39  % Model    : x86_64 x86_64
% 0.18/0.39  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.39  % Memory   : 8042.1875MB
% 0.18/0.39  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.18/0.39  % CPULimit : 300
% 0.18/0.39  % WCLimit  : 300
% 0.18/0.39  % DateTime : Sat May 18 16:09:39 EDT 2024
% 0.18/0.39  % CPUTime  : 
% 1.41/1.30  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 2.60/1.75  % [INFO] 	 Parsing done (445ms). 
% 2.60/1.77  % [INFO] 	 Running in sequential loop mode. 
% 3.96/2.28  % [INFO] 	 eprover registered as external prover. 
% 3.96/2.29  % [INFO] 	 cvc4 registered as external prover. 
% 4.12/2.30  % [INFO] 	 Scanning for conjecture ... 
% 4.40/2.47  % [INFO] 	 Found a conjecture (or negated_conjecture) and 273 axioms. Running axiom selection ... 
% 4.78/2.66  % [INFO] 	 Axiom selection finished. Selected 273 axioms (removed 0 axioms). 
% 6.06/3.00  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 6.10/3.03  % [INFO] 	 Type checking passed. 
% 6.10/3.04  % [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 ... 
% 29.27/9.17  % External prover 'cvc4' found a proof!
% 29.27/9.17  % [INFO] 	 Killing All external provers ... 
% 29.27/9.17  % Time passed: 8584ms (effective reasoning time: 7400ms)
% 29.27/9.18  % 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)>
% 29.27/9.18  % Axioms used in derivation (273): fact_172_Graph_Omin__distI__eq, fact_80_ab__semigroup__add__class_Oadd__ac_I1_J, fact_37_le__trans, fact_22_isPath__distD, fact_2__092_060open_062min__dist_Au_At_A_092_060le_062_Alength_Ap2_H_092_060close_062, fact_142_Graph_OisShortestPath__alt, fact_229_dual__order_Oeq__iff, fact_124_Graph_Omin__dist__is__dist, fact_79_add__mono__thms__linordered__semiring_I4_J, fact_227_linorder__wlog, fact_42_add__le__imp__le__right, fact_36_le__refl, fact_30_add__le__cancel__right, fact_77_group__cancel_Oadd2, fact_6__C1_Oprems_C_I2_J, fact_258_neqE, fact_198_g_H_Omin__dist__def, fact_18_obtain__shortest__path, fact_168_Graph_Omin__dist__less, fact_170_Graph_Omin__dist__le, fact_146_Graph_Omin__dist_Ocong, fact_271_dual__order_Oorder__iff__strict, fact_268_order_Onot__eq__order__implies__strict, fact_221_order__class_Oorder_Oeq__iff, fact_251_less__linear, fact_29__092_060open_062connected_As_Au_092_060close_062, fact_3_nat__add__left__cancel__less, fact_269_dual__order_Ostrict__implies__order, fact_66_le__Suc__ex, fact_127_Graph_Omin__dist__split_I1_J, fact_157_Graph_Odist__trans, fact_50_add__left__mono, fact_93_add__mono__thms__linordered__field_I4_J, fact_112_min__dist__def, fact_71_add__leE, fact_11_add__left__cancel, fact_115_Graph_OisShortestPath__def, fact_95_add__less__imp__less__right, fact_234_dual__order_Ostrict__trans, fact_214_antisym, fact_74_add_Oleft__commute, fact_215_linear, fact_68_add__leD1, fact_184_g_H_OshortestPath__is__simple, fact_134_Graph_OisSPath__pathLE, fact_118_Graph_Omin__distI2, fact_149_Graph_Oconnected__refl, fact_262_ord__less__eq__subst, fact_106_add__less__mono1, fact_140_Graph_Omin__dist__def, fact_200_g_H_OisPath__distD, fact_223_ord__le__eq__trans, fact_97_add__strict__right__mono, fact_180_Least__le, fact_104_trans__less__add2, fact_49_less__eqE, fact_14_dist__trans, fact_150_Graph_Oconnected_Ocong, fact_108_not__add__less1, fact_8_min__dist__split_I2_J, fact_110_add__lessD1, fact_92_add__mono__thms__linordered__field_I3_J, fact_190_g_H_Omin__dist__minD, fact_246_less__imp__not__eq, fact_130_Graph_OreachableNodes_Ocong, fact_117_Graph_Omin__distI2, fact_33_P1, fact_87_less__not__refl, fact_167_Graph_Oobtain__shortest__path, fact_105_trans__less__add1, fact_114_Graph_OisShortestPath__min__dist__def, fact_60_nat__less__le, fact_169_Graph_Omin__dist__less, fact_94_mono__nat__linear__lb, fact_185_g_H_OreachableNodes__def, fact_84_less__irrefl__nat, fact_107_not__add__less2, fact_186_g_H_Omin__dist__less, fact_52_add__mono__thms__linordered__semiring_I1_J, fact_203_g_H_OisShortestPath__def, fact_56_le__neq__implies__less, fact_178__C1_Oprems_C_I3_J, fact_244_less__induct, fact_183_g_H_Oobtain__shortest__path, fact_211_ord__eq__le__subst, fact_236_exists__least__iff, fact_171_Graph_Omin__dist__le, fact_191_g_H_Omin__dist__split_I2_J, fact_23_dist__def, fact_96_add__less__imp__less__left, fact_90_add__less__le__mono, fact_125_Graph_Omin__dist__is__dist, fact_254_less__asym, fact_1__092_060open_062min__dist_As_Au_A_060_Alength_Ap1_092_060close_062, fact_272_order_Ostrict__implies__order, fact_151_Graph_Oconnected_Ocong, fact_131_Graph_OreachableNodes_Ocong, fact_135_Graph_OisSPath__pathLE, fact_250_less__irrefl, fact_69_le__add2, fact_205__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062, fact_41_Nat_Oex__has__greatest__nat, fact_152_Graph_Odist_Ocong, fact_46_Collect__mem__eq, fact_165_Graph_Oconnected__distI, fact_13__092_060open_062isPath_Au_Ap2_H_At_092_060close_062, fact_255_less__imp__neq, fact_109_add__less__mono, fact_55_less__mono__imp__le__mono, fact_72_add__right__imp__eq, fact_88_nat__neq__iff, fact_240_linorder__cases, fact_31_add__le__cancel__left, fact_164_Graph_Oconnected__distI, fact_133_Graph_OisSimplePath_Ocong, fact_225_order__trans, fact_62_trans__le__add2, fact_162_Graph_Oconnected__by__dist, fact_85_less__not__refl3, fact_195_g_H_OisSPath__pathLE, fact_179_not__less__Least, fact_81_linorder__neqE__nat, fact_76_add_Oassoc, fact_256_order_Oasym, fact_34_connected__refl, fact_245_less__not__sym, fact_182_g_H_Omin__dist__is__dist, fact_181_g_H_Oconnected__by__dist, fact_91_add__le__less__mono, fact_120_Graph_Odist__def, fact_228_dual__order_Otrans, fact_216_eq__refl, fact_202_order__refl, fact_148_Graph_Oconnected__refl, fact_261_order__less__subst1, fact_86_less__not__refl2, fact_158_Graph_Oconnected__def, fact_192_g_H_Omin__dist__split_I1_J, fact_243_antisym__conv3, fact_32_nat__add__left__cancel__le, fact_210_order__subst2, fact_99_add__strict__mono, fact_61_nat__le__iff__add, fact_231_dual__order_Ostrict__implies__not__eq, fact_98_add__strict__left__mono, fact_173_Graph_Omin__distI__eq, fact_206_g_H_Oconnected__distI, fact_270_dual__order_Ostrict__iff__order, fact_45_mem__Collect__eq, fact_73_add__left__imp__eq, fact_154_Graph_OisShortestPath_Ocong, fact_132_Graph_OisSimplePath_Ocong, fact_20_min__dist__minD, fact_4_add__less__cancel__left, fact_116_Graph_OisShortestPath__def, fact_103_less__add__eq__less, fact_51_add__mono, fact_64_add__le__mono1, fact_267_LeastI, fact_239_dual__order_Oirrefl, fact_226_dual__order_Orefl, fact_238_order_Ostrict__trans, fact_189_g_H_Omin__dist__le, fact_144_Graph_OisPath_Ocong, fact_137_Graph_OshortestPath__is__simple, fact_57_less__or__eq__imp__le, fact_242_less__imp__not__eq2, fact_197_g_H_OisShortestPath__alt, fact_208_P2_H, fact_204_assms_I5_J, fact_47_Collect__cong, fact_253_less__asym_H, fact_156_Graph_Odist__trans, fact_166_Graph_Oobtain__shortest__path, fact_16_shortestPath__is__path, fact_5_add__less__cancel__right, fact_175_Graph_Omin__dist__minD, fact_237_less__imp__not__less, fact_141_Graph_Omin__dist__def, fact_128_isSPath__pathLE, fact_143_Graph_OisShortestPath__alt, fact_75_add_Ocommute, fact_123_Graph_OisPath__distD, fact_59_less__imp__le__nat, fact_83_nat__less__induct, fact_27_min__distI2, fact_230_dual__order_Oantisym, fact_257_neq__iff, fact_122_Graph_OisPath__distD, fact_136_Graph_OshortestPath__is__simple, fact_266_LeastI2__ex, fact_247_dual__order_Oasym, fact_89_size__neq__size__imp__neq, fact_241_less__imp__triv, fact_264_LeastI2, fact_217_le__cases, fact_7_SP, fact_161_Graph_OshortestPath__is__path, fact_201_g_H_OisShortestPath__min__dist__def, fact_252_less__trans, fact_38_eq__imp__le, fact_194_g_H_Oconnected__def, fact_259_gt__ex, fact_39_le__antisym, fact_160_Graph_OshortestPath__is__path, fact_63_trans__le__add1, fact_220_antisym__conv, fact_67_add__leD2, fact_58_le__eq__less__or__eq, fact_213_eq__iff, fact_65_add__le__mono, fact_222_ord__eq__le__trans, fact_78_group__cancel_Oadd1, fact_19_min__dist__le, fact_249_ord__eq__less__trans, fact_163_Graph_Oconnected__by__dist, fact_224_order__class_Oorder_Oantisym, fact_196_g_H_OshortestPath__is__path, fact_176_Graph_Omin__dist__split_I2_J, fact_232_order_Ostrict__implies__not__eq, fact_119_isShortestPath__alt, fact_15_connected__def, fact_145_Graph_OisPath_Ocong, fact_129_shortestPath__is__simple, fact_219_le__cases3, fact_153_Graph_Odist_Ocong, fact_17_connected__by__dist, fact_25_min__dist__is__dist, fact_44_le__iff__add, fact_193_g_H_Odist__trans, fact_147_Graph_Omin__dist_Ocong, fact_121_Graph_Odist__def, fact_188_g_H_Omin__distI__eq, fact_101_add__mono__thms__linordered__field_I2_J, fact_48_add__right__mono, fact_174_Graph_Omin__dist__minD, fact_24__092_060open_062isPath_As_Ap1_____Av_____092_060close_062, fact_199_g_H_Odist__def, fact_155_Graph_OisShortestPath_Ocong, fact_218_order_Otrans, fact_21_min__distI__eq, fact_159_Graph_Oconnected__def, fact_111_reachableNodes__def, fact_138_Graph_OreachableNodes__def, fact_26_isShortestPath__def, fact_139_Graph_OreachableNodes__def, fact_82_infinite__descent, fact_207_g_H_Oconnected__refl, fact_177_Graph_Omin__dist__split_I2_J, fact_43_add__le__imp__le__left, fact_263_ord__eq__less__subst, fact_10_min__dist__less, fact_54_add__mono__thms__linordered__semiring_I3_J, fact_212_ord__le__eq__subst, fact_53_add__mono__thms__linordered__semiring_I2_J, fact_28_isShortestPath__min__dist__def, fact_235_linorder__less__wlog, fact_35_connected__distI, fact_248_ord__less__eq__trans, fact_9_min__dist__split_I1_J, fact_260_order__less__subst2, fact_102_add__mono__thms__linordered__field_I5_J, fact_0__092_060open_062min__dist_As_At_A_061_Amin__dist_As_Au_A_L_Amin__dist_Au_At_092_060close_062, fact_233_not__less__iff__gr__or__eq, fact_265_LeastI__ex, fact_40_nat__le__linear, fact_70_le__add1, fact_113_Graph_OisShortestPath__min__dist__def, fact_12_add__right__cancel, fact_126_Graph_Omin__dist__split_I1_J, fact_209_order__subst1, fact_187_g_H_Omin__distI2, fact_100_add__mono__thms__linordered__field_I1_J
% 29.27/9.18  % No. of inferences in proof: 550
% 29.27/9.18  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 8584 ms resp. 7400 ms w/o parsing
% 30.84/9.48  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 30.84/9.49  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------