TSTP Solution File: GRA009+1 by Leo-III---1.7.15

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.15
% Problem  : GRA009+1 : TPTP v8.2.0. Bugfixed v3.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d THM

% Computer : n002.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 Jun 24 06:36:13 EDT 2024

% Result   : Theorem 9.98s 3.14s
% Output   : Refutation 9.98s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   14
%            Number of leaves      :   47
% Syntax   : Number of formulae    :  140 (  27 unt;  29 typ;   0 def)
%            Number of atoms       :  452 ( 122 equ;   0 cnn)
%            Maximal formula atoms :   14 (   4 avg)
%            Number of connectives : 1395 ( 146   ~; 115   |; 134   &; 938   @)
%                                         (   3 <=>;  54  =>;   2  <=;   3 <~>)
%            Maximal formula depth :   20 (   9 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   34 (  34   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   32 (  29 usr;  14 con; 0-3 aty)
%            Number of variables   :  298 (   0   ^ 273   !;  25   ?; 298   :)

% Comments : 
%------------------------------------------------------------------------------
thf(complete_type,type,
    complete: $o ).

thf(shortest_path_type,type,
    shortest_path: $i > $i > $i > $o ).

thf(on_path_type,type,
    on_path: $i > $i > $o ).

thf(sequential_type,type,
    sequential: $i > $i > $o ).

thf(triangle_type,type,
    triangle: $i > $i > $i > $o ).

thf(path_type,type,
    path: $i > $i > $i > $o ).

thf(vertex_type,type,
    vertex: $i > $o ).

thf(edge_type,type,
    edge: $i > $o ).

thf(tail_of_type,type,
    tail_of: $i > $i ).

thf(head_of_type,type,
    head_of: $i > $i ).

thf(path_cons_type,type,
    path_cons: $i > $i > $i ).

thf(empty_type,type,
    empty: $i ).

thf(number_of_in_type,type,
    number_of_in: $i > $i > $i ).

thf(sequential_pairs_type,type,
    sequential_pairs: $i ).

thf(minus_type,type,
    minus: $i > $i > $i ).

thf(length_of_type,type,
    length_of: $i > $i ).

thf(n1_type,type,
    n1: $i ).

thf(less_or_equal_type,type,
    less_or_equal: $i > $i > $o ).

thf(graph_type,type,
    graph: $i ).

thf(edges_type,type,
    edges: $i ).

thf(precedes_type,type,
    precedes: $i > $i > $i > $o ).

thf(in_path_type,type,
    in_path: $i > $i > $o ).

thf(triangles_type,type,
    triangles: $i ).

thf(sk1_type,type,
    sk1: $i ).

thf(sk2_type,type,
    sk2: $i ).

thf(sk3_type,type,
    sk3: $i ).

thf(sk4_type,type,
    sk4: $i ).

thf(sk5_type,type,
    sk5: $i ).

thf(sk9_type,type,
    sk9: $i > $i > $i > $i ).

thf(10,axiom,
    ! [A: $i,B: $i] :
      ( ( sequential @ A @ B )
    <=> ( ( edge @ A )
        & ( edge @ B )
        & ( A != B )
        & ( ( head_of @ A )
          = ( tail_of @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',sequential_defn) ).

thf(65,plain,
    ! [A: $i,B: $i] :
      ( ( ( sequential @ A @ B )
       => ( ( edge @ A )
          & ( edge @ B )
          & ( A != B )
          & ( ( head_of @ A )
            = ( tail_of @ B ) ) ) )
      & ( ( ( edge @ A )
          & ( edge @ B )
          & ( A != B )
          & ( ( head_of @ A )
            = ( tail_of @ B ) ) )
       => ( sequential @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(19,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( path @ B @ C @ A )
        & ! [D: $i,E: $i] :
            ( ( ( on_path @ D @ A )
              & ( on_path @ E @ A )
              & ( sequential @ D @ E ) )
           => ? [F: $i] : ( triangle @ D @ E @ F ) ) )
     => ( ( number_of_in @ sequential_pairs @ A )
        = ( number_of_in @ triangles @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',sequential_pairs_and_triangles) ).

thf(141,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( path @ B @ C @ A )
        & ! [D: $i,E: $i] :
            ( ( ( on_path @ D @ A )
              & ( on_path @ E @ A )
              & ( sequential @ D @ E ) )
           => ? [F: $i] : ( triangle @ D @ E @ F ) ) )
     => ( ( number_of_in @ sequential_pairs @ A )
        = ( number_of_in @ triangles @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( shortest_path @ A @ B @ E )
        & ( precedes @ C @ D @ E ) )
     => ( ~ ? [F: $i] :
              ( ( ( tail_of @ F )
                = ( tail_of @ C ) )
              & ( ( head_of @ F )
                = ( head_of @ D ) ) )
        & ~ ( precedes @ D @ C @ E ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',shortest_path_properties) ).

thf(77,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( shortest_path @ A @ B @ E )
        & ( precedes @ C @ D @ E ) )
     => ( ~ ? [F: $i] :
              ( ( ( tail_of @ F )
                = ( tail_of @ C ) )
              & ( ( head_of @ F )
                = ( head_of @ D ) ) )
        & ~ ( precedes @ D @ C @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(78,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( shortest_path @ A @ B @ E )
      | ~ ( precedes @ C @ D @ E )
      | ( ( tail_of @ F )
       != ( tail_of @ C ) )
      | ( ( head_of @ F )
       != ( head_of @ D ) ) ),
    inference(cnf,[status(esa)],[77]) ).

thf(80,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( ( tail_of @ F )
       != ( tail_of @ C ) )
      | ( ( head_of @ F )
       != ( head_of @ D ) )
      | ~ ( shortest_path @ A @ B @ E )
      | ~ ( precedes @ C @ D @ E ) ),
    inference(lifteq,[status(thm)],[78]) ).

thf(3,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     <= ( ( vertex @ A )
        & ( vertex @ B )
        & ? [D: $i] :
            ( ( edge @ D )
            & ( A
              = ( tail_of @ D ) )
            & ( ( ( B
                  = ( head_of @ D ) )
                & ( C
                  = ( path_cons @ D @ empty ) ) )
              | ? [E: $i] :
                  ( ( path @ ( head_of @ D ) @ B @ E )
                  & ( C
                    = ( path_cons @ D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',path_defn) ).

thf(27,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
      | ~ ( ( vertex @ A )
          & ( vertex @ B )
          & ? [D: $i] :
              ( ( edge @ D )
              & ( A
                = ( tail_of @ D ) )
              & ( ( ( B
                    = ( head_of @ D ) )
                  & ( C
                    = ( path_cons @ D @ empty ) ) )
                | ? [E: $i] :
                    ( ( path @ ( head_of @ D ) @ B @ E )
                    & ( C
                      = ( path_cons @ D @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(28,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( path @ A @ B @ C )
      | ~ ( vertex @ A )
      | ~ ( vertex @ B )
      | ~ ( edge @ D )
      | ( A
       != ( tail_of @ D ) )
      | ( B
       != ( head_of @ D ) )
      | ( C
       != ( path_cons @ D @ empty ) ) ),
    inference(cnf,[status(esa)],[27]) ).

thf(30,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( A
       != ( tail_of @ D ) )
      | ( B
       != ( head_of @ D ) )
      | ( C
       != ( path_cons @ D @ empty ) )
      | ( path @ A @ B @ C )
      | ~ ( vertex @ A )
      | ~ ( vertex @ B )
      | ~ ( edge @ D ) ),
    inference(lifteq,[status(thm)],[28]) ).

thf(31,plain,
    ! [A: $i] :
      ( ( path @ ( tail_of @ A ) @ ( head_of @ A ) @ ( path_cons @ A @ empty ) )
      | ~ ( vertex @ ( tail_of @ A ) )
      | ~ ( vertex @ ( head_of @ A ) )
      | ~ ( edge @ A ) ),
    inference(simp,[status(thm)],[30]) ).

thf(1,conjecture,
    ( complete
   => ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
        ( ( ( shortest_path @ A @ B @ C )
          & ( on_path @ D @ C )
          & ( on_path @ E @ C )
          & ( sequential @ D @ E ) )
       => ? [F: $i] : ( triangle @ D @ E @ F ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',complete_means_path_and_stuff_means_triangles) ).

thf(2,negated_conjecture,
    ~ ( complete
     => ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
          ( ( ( shortest_path @ A @ B @ C )
            & ( on_path @ D @ C )
            & ( on_path @ E @ C )
            & ( sequential @ D @ E ) )
         => ? [F: $i] : ( triangle @ D @ E @ F ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(20,plain,
    ~ ( complete
     => ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
          ( ( ( shortest_path @ A @ B @ C )
            & ( on_path @ D @ C )
            & ( on_path @ E @ C )
            & ( sequential @ D @ E ) )
         => ? [F: $i] : ( triangle @ D @ E @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(23,plain,
    sequential @ sk4 @ sk5,
    inference(cnf,[status(esa)],[20]) ).

thf(66,plain,
    ( ! [A: $i,B: $i] :
        ( ( sequential @ A @ B )
       => ( ( edge @ A )
          & ( edge @ B )
          & ( A != B )
          & ( ( head_of @ A )
            = ( tail_of @ B ) ) ) )
    & ! [A: $i,B: $i] :
        ( ( ( edge @ A )
          & ( edge @ B )
          & ( A != B )
          & ( ( head_of @ A )
            = ( tail_of @ B ) ) )
       => ( sequential @ A @ B ) ) ),
    inference(miniscope,[status(thm)],[65]) ).

thf(68,plain,
    ! [B: $i,A: $i] :
      ( ~ ( sequential @ A @ B )
      | ( edge @ A ) ),
    inference(cnf,[status(esa)],[66]) ).

thf(385,plain,
    ! [B: $i,A: $i] :
      ( ( edge @ A )
      | ( ( sequential @ sk4 @ sk5 )
       != ( sequential @ A @ B ) ) ),
    inference(paramod_ordered,[status(thm)],[23,68]) ).

thf(386,plain,
    edge @ sk4,
    inference(pattern_uni,[status(thm)],[385:[bind(A,$thf( sk4 )),bind(B,$thf( sk5 ))]]) ).

thf(4,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     => ( ( number_of_in @ sequential_pairs @ C )
        = ( minus @ ( length_of @ C ) @ n1 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',path_length_sequential_pairs) ).

thf(34,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     => ( ( number_of_in @ sequential_pairs @ C )
        = ( minus @ ( length_of @ C ) @ n1 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(35,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( path @ A @ B @ C )
      | ( ( number_of_in @ sequential_pairs @ C )
        = ( minus @ ( length_of @ C ) @ n1 ) ) ),
    inference(cnf,[status(esa)],[34]) ).

thf(36,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( minus @ ( length_of @ C ) @ n1 )
        = ( number_of_in @ sequential_pairs @ C ) )
      | ~ ( path @ A @ B @ C ) ),
    inference(lifteq,[status(thm)],[35]) ).

thf(17,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ B @ C @ A )
     => ! [D: $i,E: $i] :
          ( ( precedes @ D @ E @ A )
         => ( ( on_path @ D @ A )
            & ( on_path @ E @ A )
            & ( ( sequential @ D @ E )
            <~> ? [F: $i] :
                  ( ( sequential @ D @ F )
                  & ( precedes @ F @ E @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',precedes_properties) ).

thf(122,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ B @ C @ A )
     => ! [D: $i,E: $i] :
          ( ( precedes @ D @ E @ A )
         => ( ( on_path @ D @ A )
            & ( on_path @ E @ A )
            & ~ ( ( ( sequential @ D @ E )
                 => ? [F: $i] :
                      ( ( sequential @ D @ F )
                      & ( precedes @ F @ E @ A ) ) )
                & ( ( sequential @ D @ E )
                  | ~ ? [F: $i] :
                        ( ( sequential @ D @ F )
                        & ( precedes @ F @ E @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(8,axiom,
    ! [A: $i] :
      ( ( edge @ A )
     => ( ( vertex @ ( head_of @ A ) )
        & ( vertex @ ( tail_of @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',edge_ends_are_vertices) ).

thf(59,plain,
    ! [A: $i] :
      ( ( edge @ A )
     => ( ( vertex @ ( head_of @ A ) )
        & ( vertex @ ( tail_of @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(60,plain,
    ! [A: $i] :
      ( ~ ( edge @ A )
      | ( vertex @ ( head_of @ A ) ) ),
    inference(cnf,[status(esa)],[59]) ).

thf(391,plain,
    ! [A: $i] :
      ( ( vertex @ ( head_of @ A ) )
      | ( ( edge @ sk4 )
       != ( edge @ A ) ) ),
    inference(paramod_ordered,[status(thm)],[386,60]) ).

thf(392,plain,
    vertex @ ( head_of @ sk4 ),
    inference(pattern_uni,[status(thm)],[391:[bind(A,$thf( sk4 ))]]) ).

thf(413,plain,
    ! [A: $i] :
      ( ( path @ ( tail_of @ A ) @ ( head_of @ A ) @ ( path_cons @ A @ empty ) )
      | ~ ( vertex @ ( tail_of @ A ) )
      | ~ ( edge @ A )
      | ( ( vertex @ ( head_of @ sk4 ) )
       != ( vertex @ ( head_of @ A ) ) ) ),
    inference(paramod_ordered,[status(thm)],[392,31]) ).

thf(414,plain,
    ( ( path @ ( tail_of @ sk4 ) @ ( head_of @ sk4 ) @ ( path_cons @ sk4 @ empty ) )
    | ~ ( vertex @ ( tail_of @ sk4 ) )
    | ~ ( edge @ sk4 ) ),
    inference(pattern_uni,[status(thm)],[413:[bind(A,$thf( sk4 ))]]) ).

thf(61,plain,
    ! [A: $i] :
      ( ~ ( edge @ A )
      | ( vertex @ ( tail_of @ A ) ) ),
    inference(cnf,[status(esa)],[59]) ).

thf(395,plain,
    ! [A: $i] :
      ( ( vertex @ ( tail_of @ A ) )
      | ( ( edge @ sk4 )
       != ( edge @ A ) ) ),
    inference(paramod_ordered,[status(thm)],[386,61]) ).

thf(396,plain,
    vertex @ ( tail_of @ sk4 ),
    inference(pattern_uni,[status(thm)],[395:[bind(A,$thf( sk4 ))]]) ).

thf(714,plain,
    ( ( path @ ( tail_of @ sk4 ) @ ( head_of @ sk4 ) @ ( path_cons @ sk4 @ empty ) )
    | ~ $true
    | ~ $true ),
    inference(rewrite,[status(thm)],[414,386,396]) ).

thf(715,plain,
    path @ ( tail_of @ sk4 ) @ ( head_of @ sk4 ) @ ( path_cons @ sk4 @ empty ),
    inference(simp,[status(thm)],[714]) ).

thf(69,plain,
    ! [B: $i,A: $i] :
      ( ~ ( sequential @ A @ B )
      | ( A != B ) ),
    inference(cnf,[status(esa)],[66]) ).

thf(72,plain,
    ! [B: $i,A: $i] :
      ( ( A != B )
      | ~ ( sequential @ A @ B ) ),
    inference(lifteq,[status(thm)],[69]) ).

thf(73,plain,
    ! [A: $i] :
      ~ ( sequential @ A @ A ),
    inference(simp,[status(thm)],[72]) ).

thf(151,plain,
    ! [A: $i] :
      ( ( sequential @ sk4 @ sk5 )
     != ( sequential @ A @ A ) ),
    inference(paramod_ordered,[status(thm)],[23,73]) ).

thf(152,plain,
    ! [A: $i] :
      ( ( sk4 != A )
      | ( sk5 != A ) ),
    inference(simp,[status(thm)],[151]) ).

thf(153,plain,
    sk5 != sk4,
    inference(simp,[status(thm)],[152]) ).

thf(7,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     => ( ( vertex @ A )
        & ( vertex @ B )
        & ? [D: $i] :
            ( ( edge @ D )
            & ( A
              = ( tail_of @ D ) )
            & ( ( ( B
                  = ( head_of @ D ) )
                & ( C
                  = ( path_cons @ D @ empty ) ) )
            <~> ? [E: $i] :
                  ( ( path @ ( head_of @ D ) @ B @ E )
                  & ( C
                    = ( path_cons @ D @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',path_properties) ).

thf(42,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     => ( ( vertex @ A )
        & ( vertex @ B )
        & ? [D: $i] :
            ( ( edge @ D )
            & ( A
              = ( tail_of @ D ) )
            & ~ ( ( ( ( B
                      = ( head_of @ D ) )
                    & ( C
                      = ( path_cons @ D @ empty ) ) )
                 => ? [E: $i] :
                      ( ( path @ ( head_of @ D ) @ B @ E )
                      & ( C
                        = ( path_cons @ D @ E ) ) ) )
                & ( ( ( B
                      = ( head_of @ D ) )
                    & ( C
                      = ( path_cons @ D @ empty ) ) )
                  | ~ ? [E: $i] :
                        ( ( path @ ( head_of @ D ) @ B @ E )
                        & ( C
                          = ( path_cons @ D @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(24,plain,
    on_path @ sk4 @ sk3,
    inference(cnf,[status(esa)],[20]) ).

thf(5,axiom,
    ! [A: $i,B: $i] : ( less_or_equal @ ( number_of_in @ A @ B ) @ ( number_of_in @ A @ graph ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',graph_has_them_all) ).

thf(37,plain,
    ! [A: $i,B: $i] : ( less_or_equal @ ( number_of_in @ A @ B ) @ ( number_of_in @ A @ graph ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(25,plain,
    shortest_path @ sk1 @ sk2 @ sk3,
    inference(cnf,[status(esa)],[20]) ).

thf(12,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( shortest_path @ A @ B @ C )
    <=> ( ( path @ A @ B @ C )
        & ( A != B )
        & ! [D: $i] :
            ( ( path @ A @ B @ D )
           => ( less_or_equal @ ( length_of @ C ) @ ( length_of @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',shortest_path_defn) ).

thf(81,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( shortest_path @ A @ B @ C )
       => ( ( path @ A @ B @ C )
          & ( A != B )
          & ! [D: $i] :
              ( ( path @ A @ B @ D )
             => ( less_or_equal @ ( length_of @ C ) @ ( length_of @ D ) ) ) ) )
      & ( ( ( path @ A @ B @ C )
          & ( A != B )
          & ! [D: $i] :
              ( ( path @ A @ B @ D )
             => ( less_or_equal @ ( length_of @ C ) @ ( length_of @ D ) ) ) )
       => ( shortest_path @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(82,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( shortest_path @ A @ B @ C )
       => ( ( path @ A @ B @ C )
          & ( A != B )
          & ! [D: $i] :
              ( ( path @ A @ B @ D )
             => ( less_or_equal @ ( length_of @ C ) @ ( length_of @ D ) ) ) ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( ( path @ A @ B @ C )
          & ( A != B )
          & ! [D: $i] :
              ( ( path @ A @ B @ D )
             => ( less_or_equal @ ( length_of @ C ) @ ( length_of @ D ) ) ) )
       => ( shortest_path @ A @ B @ C ) ) ),
    inference(miniscope,[status(thm)],[81]) ).

thf(87,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( path @ A @ B @ C )
      | ( A = B )
      | ( path @ A @ B @ ( sk9 @ C @ B @ A ) )
      | ( shortest_path @ A @ B @ C ) ),
    inference(cnf,[status(esa)],[82]) ).

thf(92,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = B )
      | ~ ( path @ A @ B @ C )
      | ( path @ A @ B @ ( sk9 @ C @ B @ A ) )
      | ( shortest_path @ A @ B @ C ) ),
    inference(lifteq,[status(thm)],[87]) ).

thf(93,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A = B )
      | ~ ( path @ A @ B @ C )
      | ( path @ A @ B @ ( sk9 @ C @ B @ A ) )
      | ( shortest_path @ A @ B @ C ) ),
    inference(simp,[status(thm)],[92]) ).

thf(84,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( shortest_path @ A @ B @ C )
      | ( A != B ) ),
    inference(cnf,[status(esa)],[82]) ).

thf(88,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( A != B )
      | ~ ( shortest_path @ A @ B @ C ) ),
    inference(lifteq,[status(thm)],[84]) ).

thf(89,plain,
    ! [B: $i,A: $i] :
      ~ ( shortest_path @ A @ A @ B ),
    inference(simp,[status(thm)],[88]) ).

thf(70,plain,
    ! [B: $i,A: $i] :
      ( ~ ( sequential @ A @ B )
      | ( edge @ B ) ),
    inference(cnf,[status(esa)],[66]) ).

thf(919,plain,
    ! [B: $i,A: $i] :
      ( ( edge @ B )
      | ( ( sequential @ sk4 @ sk5 )
       != ( sequential @ A @ B ) ) ),
    inference(paramod_ordered,[status(thm)],[23,70]) ).

thf(920,plain,
    edge @ sk5,
    inference(pattern_uni,[status(thm)],[919:[bind(A,$thf( sk4 )),bind(B,$thf( sk5 ))]]) ).

thf(15,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( path @ A @ B @ C )
        & ( in_path @ D @ C ) )
     => ( ( vertex @ D )
        & ? [E: $i] :
            ( ( on_path @ E @ C )
            & ( ( D
                = ( head_of @ E ) )
              | ( D
                = ( tail_of @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',in_path_properties) ).

thf(102,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( path @ A @ B @ C )
        & ( in_path @ D @ C ) )
     => ( ( vertex @ D )
        & ? [E: $i] :
            ( ( on_path @ E @ C )
            & ( ( D
                = ( head_of @ E ) )
              | ( D
                = ( tail_of @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(79,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( shortest_path @ A @ B @ E )
      | ~ ( precedes @ C @ D @ E )
      | ~ ( precedes @ D @ C @ E ) ),
    inference(cnf,[status(esa)],[77]) ).

thf(167,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( precedes @ C @ D @ E )
      | ~ ( precedes @ D @ C @ E )
      | ( ( shortest_path @ sk1 @ sk2 @ sk3 )
       != ( shortest_path @ A @ B @ E ) ) ),
    inference(paramod_ordered,[status(thm)],[25,79]) ).

thf(168,plain,
    ! [B: $i,A: $i] :
      ( ~ ( precedes @ A @ B @ sk3 )
      | ~ ( precedes @ B @ A @ sk3 ) ),
    inference(pattern_uni,[status(thm)],[167:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 )),bind(C,$thf( C )),bind(D,$thf( D )),bind(E,$thf( sk3 ))]]) ).

thf(171,plain,
    ! [B: $i,A: $i] :
      ( ~ ( precedes @ A @ B @ sk3 )
      | ~ ( precedes @ B @ A @ sk3 ) ),
    inference(simp,[status(thm)],[168]) ).

thf(343,plain,
    ! [B: $i,A: $i] :
      ( ~ ( precedes @ A @ B @ sk3 )
      | ( ( precedes @ B @ A @ sk3 )
       != ( precedes @ A @ B @ sk3 ) )
      | ~ $true ),
    inference(eqfactor_ordered,[status(thm)],[171]) ).

thf(344,plain,
    ! [A: $i] :
      ~ ( precedes @ A @ A @ sk3 ),
    inference(pattern_uni,[status(thm)],[343:[bind(A,$thf( B ))]]) ).

thf(349,plain,
    ! [A: $i] :
      ~ ( precedes @ A @ A @ sk3 ),
    inference(simp,[status(thm)],[344]) ).

thf(38,plain,
    ! [B: $i,A: $i] : ( less_or_equal @ ( number_of_in @ A @ B ) @ ( number_of_in @ A @ graph ) ),
    inference(cnf,[status(esa)],[37]) ).

thf(21,plain,
    complete,
    inference(cnf,[status(esa)],[20]) ).

thf(29,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( path @ A @ B @ C )
      | ~ ( vertex @ A )
      | ~ ( vertex @ B )
      | ~ ( edge @ D )
      | ( A
       != ( tail_of @ D ) )
      | ~ ( path @ ( head_of @ D ) @ B @ E )
      | ( C
       != ( path_cons @ D @ E ) ) ),
    inference(cnf,[status(esa)],[27]) ).

thf(32,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A
       != ( tail_of @ D ) )
      | ( C
       != ( path_cons @ D @ E ) )
      | ( path @ A @ B @ C )
      | ~ ( vertex @ A )
      | ~ ( vertex @ B )
      | ~ ( edge @ D )
      | ~ ( path @ ( head_of @ D ) @ B @ E ) ),
    inference(lifteq,[status(thm)],[29]) ).

thf(33,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( path @ ( tail_of @ B ) @ A @ ( path_cons @ B @ C ) )
      | ~ ( vertex @ ( tail_of @ B ) )
      | ~ ( vertex @ A )
      | ~ ( edge @ B )
      | ~ ( path @ ( head_of @ B ) @ A @ C ) ),
    inference(simp,[status(thm)],[32]) ).

thf(9,axiom,
    ! [A: $i] :
      ( ( edge @ A )
     => ( ( head_of @ A )
       != ( tail_of @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',no_loops) ).

thf(62,plain,
    ! [A: $i] :
      ( ( edge @ A )
     => ( ( head_of @ A )
       != ( tail_of @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(63,plain,
    ! [A: $i] :
      ( ~ ( edge @ A )
      | ( ( head_of @ A )
       != ( tail_of @ A ) ) ),
    inference(cnf,[status(esa)],[62]) ).

thf(64,plain,
    ! [A: $i] :
      ( ( ( head_of @ A )
       != ( tail_of @ A ) )
      | ~ ( edge @ A ) ),
    inference(lifteq,[status(thm)],[63]) ).

thf(937,plain,
    ! [A: $i] :
      ( ( ( head_of @ A )
       != ( tail_of @ A ) )
      | ( ( edge @ sk5 )
       != ( edge @ A ) ) ),
    inference(paramod_ordered,[status(thm)],[920,64]) ).

thf(938,plain,
    ( ( head_of @ sk5 )
   != ( tail_of @ sk5 ) ),
    inference(pattern_uni,[status(thm)],[937:[bind(A,$thf( sk5 ))]]) ).

thf(181,plain,
    ! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ( A = B )
      | ~ ( path @ A @ B @ C )
      | ( shortest_path @ A @ B @ C )
      | ( D = E )
      | ( path @ D @ E @ ( sk9 @ F @ E @ D ) )
      | ( shortest_path @ D @ E @ F )
      | ( ( path @ A @ B @ ( sk9 @ C @ B @ A ) )
       != ( path @ D @ E @ F ) ) ),
    inference(paramod_ordered,[status(thm)],[93,93]) ).

thf(182,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( path @ C @ B @ A )
      | ( shortest_path @ C @ B @ A )
      | ( C = B )
      | ( path @ C @ B @ ( sk9 @ ( sk9 @ A @ B @ C ) @ B @ C ) )
      | ( shortest_path @ C @ B @ ( sk9 @ A @ B @ C ) ) ),
    inference(pattern_uni,[status(thm)],[181:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( G )),bind(D,$thf( I )),bind(E,$thf( H )),bind(F,$thf( sk9 @ G @ H @ I ))]]) ).

thf(298,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C = B )
      | ~ ( path @ C @ B @ A )
      | ( shortest_path @ C @ B @ A )
      | ( path @ C @ B @ ( sk9 @ ( sk9 @ A @ B @ C ) @ B @ C ) )
      | ( shortest_path @ C @ B @ ( sk9 @ A @ B @ C ) ) ),
    inference(simp,[status(thm)],[182]) ).

thf(159,plain,
    ! [B: $i,A: $i] :
      ( ( shortest_path @ sk1 @ sk2 @ sk3 )
     != ( shortest_path @ A @ A @ B ) ),
    inference(paramod_ordered,[status(thm)],[25,89]) ).

thf(160,plain,
    ! [B: $i,A: $i] :
      ( ( sk1 != A )
      | ( sk2 != A )
      | ( sk3 != B ) ),
    inference(simp,[status(thm)],[159]) ).

thf(161,plain,
    sk2 != sk1,
    inference(simp,[status(thm)],[160]) ).

thf(818,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( minus @ ( length_of @ C ) @ n1 )
        = ( number_of_in @ sequential_pairs @ C ) )
      | ( ( path @ ( tail_of @ sk4 ) @ ( head_of @ sk4 ) @ ( path_cons @ sk4 @ empty ) )
       != ( path @ A @ B @ C ) ) ),
    inference(paramod_ordered,[status(thm)],[715,36]) ).

thf(819,plain,
    ( ( minus @ ( length_of @ ( path_cons @ sk4 @ empty ) ) @ n1 )
    = ( number_of_in @ sequential_pairs @ ( path_cons @ sk4 @ empty ) ) ),
    inference(pattern_uni,[status(thm)],[818:[bind(A,$thf( tail_of @ sk4 )),bind(B,$thf( head_of @ sk4 )),bind(C,$thf( path_cons @ sk4 @ empty ))]]) ).

thf(22,plain,
    ! [A: $i] :
      ~ ( triangle @ sk4 @ sk5 @ A ),
    inference(cnf,[status(esa)],[20]) ).

thf(169,plain,
    ! [E: $i,D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( shortest_path @ A @ B @ E )
      | ~ ( precedes @ C @ D @ E )
      | ( ( precedes @ D @ C @ E )
       != ( precedes @ C @ D @ E ) )
      | ~ $true ),
    inference(eqfactor_ordered,[status(thm)],[79]) ).

thf(170,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( shortest_path @ A @ B @ D )
      | ~ ( precedes @ C @ C @ D ) ),
    inference(pattern_uni,[status(thm)],[169:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( D ))]]) ).

thf(172,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( shortest_path @ A @ B @ D )
      | ~ ( precedes @ C @ C @ D ) ),
    inference(simp,[status(thm)],[170]) ).

thf(6,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     => ( ( length_of @ C )
        = ( number_of_in @ edges @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',length_defn) ).

thf(39,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ A @ B @ C )
     => ( ( length_of @ C )
        = ( number_of_in @ edges @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(14,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( path @ A @ B @ C )
        & ( on_path @ D @ C ) )
     => ( ( edge @ D )
        & ( in_path @ ( head_of @ D ) @ C )
        & ( in_path @ ( tail_of @ D ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',on_path_properties) ).

thf(98,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( path @ A @ B @ C )
        & ( on_path @ D @ C ) )
     => ( ( edge @ D )
        & ( in_path @ ( head_of @ D ) @ C )
        & ( in_path @ ( tail_of @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(26,plain,
    on_path @ sk5 @ sk3,
    inference(cnf,[status(esa)],[20]) ).

thf(16,axiom,
    ( complete
   => ! [A: $i,B: $i] :
        ( ( ( vertex @ A )
          & ( vertex @ B )
          & ( A != B ) )
       => ? [C: $i] :
            ( ( edge @ C )
            & ( ( ( A
                  = ( head_of @ C ) )
                & ( B
                  = ( tail_of @ C ) ) )
            <~> ( ( B
                  = ( head_of @ C ) )
                & ( A
                  = ( tail_of @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',complete_properties) ).

thf(107,plain,
    ( complete
   => ! [A: $i,B: $i] :
        ( ( ( vertex @ A )
          & ( vertex @ B )
          & ( A != B ) )
       => ? [C: $i] :
            ( ( edge @ C )
            & ~ ( ( ( ( A
                      = ( head_of @ C ) )
                    & ( B
                      = ( tail_of @ C ) ) )
                 => ( ( B
                      = ( head_of @ C ) )
                    & ( A
                      = ( tail_of @ C ) ) ) )
                & ( ( ( A
                      = ( head_of @ C ) )
                    & ( B
                      = ( tail_of @ C ) ) )
                  | ~ ( ( B
                        = ( head_of @ C ) )
                      & ( A
                        = ( tail_of @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(399,plain,
    ! [A: $i] :
      ( ( ( head_of @ A )
       != ( tail_of @ A ) )
      | ( ( edge @ sk4 )
       != ( edge @ A ) ) ),
    inference(paramod_ordered,[status(thm)],[386,64]) ).

thf(400,plain,
    ( ( head_of @ sk4 )
   != ( tail_of @ sk4 ) ),
    inference(pattern_uni,[status(thm)],[399:[bind(A,$thf( sk4 ))]]) ).

thf(13,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ B @ C @ A )
     => ! [D: $i,E: $i] :
          ( ( precedes @ D @ E @ A )
         <= ( ( on_path @ D @ A )
            & ( on_path @ E @ A )
            & ( ( sequential @ D @ E )
              | ? [F: $i] :
                  ( ( sequential @ D @ F )
                  & ( precedes @ F @ E @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',precedes_defn) ).

thf(94,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( path @ B @ C @ A )
     => ! [D: $i,E: $i] :
          ( ( precedes @ D @ E @ A )
          | ~ ( ( on_path @ D @ A )
              & ( on_path @ E @ A )
              & ( ( sequential @ D @ E )
                | ? [F: $i] :
                    ( ( sequential @ D @ F )
                    & ( precedes @ F @ E @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(927,plain,
    ! [A: $i] :
      ( ( vertex @ ( tail_of @ A ) )
      | ( ( edge @ sk5 )
       != ( edge @ A ) ) ),
    inference(paramod_ordered,[status(thm)],[920,61]) ).

thf(928,plain,
    vertex @ ( tail_of @ sk5 ),
    inference(pattern_uni,[status(thm)],[927:[bind(A,$thf( sk5 ))]]) ).

thf(18,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( triangle @ A @ B @ C )
    <=> ( ( edge @ A )
        & ( edge @ B )
        & ( edge @ C )
        & ( sequential @ A @ B )
        & ( sequential @ B @ C )
        & ( sequential @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',triangle_defn) ).

thf(131,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( triangle @ A @ B @ C )
       => ( ( edge @ A )
          & ( edge @ B )
          & ( edge @ C )
          & ( sequential @ A @ B )
          & ( sequential @ B @ C )
          & ( sequential @ C @ A ) ) )
      & ( ( ( edge @ A )
          & ( edge @ B )
          & ( edge @ C )
          & ( sequential @ A @ B )
          & ( sequential @ B @ C )
          & ( sequential @ C @ A ) )
       => ( triangle @ A @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(1440,plain,
    $false,
    inference(e,[status(thm)],[65,141,77,80,31,386,36,122,715,153,42,24,37,25,20,93,61,396,89,920,60,102,349,38,392,70,21,33,73,938,298,34,161,64,819,22,59,27,172,81,39,98,26,171,23,107,79,400,94,928,131,68,62]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11  % Problem  : GRA009+1 : TPTP v8.2.0. Bugfixed v3.2.0.
% 0.00/0.11  % Command  : run_Leo-III %s %d THM
% 0.10/0.30  % Computer : n002.cluster.edu
% 0.10/0.30  % Model    : x86_64 x86_64
% 0.10/0.30  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.30  % Memory   : 8042.1875MB
% 0.10/0.30  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.10/0.30  % CPULimit : 300
% 0.10/0.30  % WCLimit  : 300
% 0.10/0.30  % DateTime : Tue Jun 18 15:02:55 EDT 2024
% 0.10/0.30  % CPUTime  : 
% 0.84/0.84  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.22/0.97  % [INFO] 	 Parsing done (130ms). 
% 1.22/0.98  % [INFO] 	 Running in sequential loop mode. 
% 1.86/1.21  % [INFO] 	 eprover registered as external prover. 
% 1.86/1.21  % [INFO] 	 Scanning for conjecture ... 
% 1.93/1.29  % [INFO] 	 Found a conjecture (or negated_conjecture) and 17 axioms. Running axiom selection ... 
% 2.19/1.34  % [INFO] 	 Axiom selection finished. Selected 17 axioms (removed 0 axioms). 
% 2.43/1.39  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 2.43/1.39  % [INFO] 	 Type checking passed. 
% 2.43/1.40  % [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 ... 
% 9.98/3.14  % External prover 'e' found a proof!
% 9.98/3.14  % [INFO] 	 Killing All external provers ... 
% 9.98/3.14  % Time passed: 2656ms (effective reasoning time: 2157ms)
% 9.98/3.14  % 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)>
% 9.98/3.14  % Axioms used in derivation (17): path_defn, sequential_defn, graph_has_them_all, path_properties, sequential_pairs_and_triangles, edge_ends_are_vertices, no_loops, length_defn, triangle_defn, complete_properties, shortest_path_defn, in_path_properties, precedes_properties, on_path_properties, path_length_sequential_pairs, shortest_path_properties, precedes_defn
% 9.98/3.14  % No. of inferences in proof: 111
% 9.98/3.14  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 2656 ms resp. 2157 ms w/o parsing
% 9.98/3.20  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 9.98/3.20  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------