%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRA006+1 : TPTP v9.0.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 : Sat Jun 21 05:06:36 AM UTC 2025
% Result : Theorem 21.26s 21.43s
% Output : Assurance 0s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.11 % Problem : GRA006+1 : TPTP v9.0.0. Bugfixed v3.2.0.
% 0.02/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n010.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 15:24:11 EDT 2025
% 0.11/0.32 % CPUTime :
% 21.26/21.43 % SZS status Theorem
% 21.26/21.43 % SZS output start Proof
% 21.26/21.43 tff(head_of_type, type, (
% 21.26/21.43 head_of: $i > $i)).
% 21.26/21.43 tff(tptp_fun_E2_9_type, type, (
% 21.26/21.43 tptp_fun_E2_9: $i)).
% 21.26/21.43 tff(tptp_fun_E_0_type, type, (
% 21.26/21.43 tptp_fun_E_0: ( $i * $i ) > $i)).
% 21.26/21.43 tff(tail_of_type, type, (
% 21.26/21.43 tail_of: $i > $i)).
% 21.26/21.43 tff(tptp_fun_E1_10_type, type, (
% 21.26/21.43 tptp_fun_E1_10: $i)).
% 21.26/21.43 tff(edge_type, type, (
% 21.26/21.43 edge: $i > $o)).
% 21.26/21.43 tff(in_path_type, type, (
% 21.26/21.43 in_path: ( $i * $i ) > $o)).
% 21.26/21.43 tff(tptp_fun_P_8_type, type, (
% 21.26/21.43 tptp_fun_P_8: $i)).
% 21.26/21.43 tff(path_type, type, (
% 21.26/21.43 path: ( $i * $i * $i ) > $o)).
% 21.26/21.43 tff(tptp_fun_V2_11_type, type, (
% 21.26/21.43 tptp_fun_V2_11: $i)).
% 21.26/21.43 tff(tptp_fun_E_1_type, type, (
% 21.26/21.43 tptp_fun_E_1: ( $i * $i * $i ) > $i)).
% 21.26/21.43 tff(tptp_fun_V1_12_type, type, (
% 21.26/21.43 tptp_fun_V1_12: $i)).
% 21.26/21.43 tff(path_cons_type, type, (
% 21.26/21.43 path_cons: ( $i * $i ) > $i)).
% 21.26/21.43 tff(empty_type, type, (
% 21.26/21.43 empty: $i)).
% 21.26/21.43 tff(tptp_fun_TP_2_type, type, (
% 21.26/21.43 tptp_fun_TP_2: ( $i * $i * $i ) > $i)).
% 21.26/21.43 tff(vertex_type, type, (
% 21.26/21.43 vertex: $i > $o)).
% 21.26/21.43 tff(less_or_equal_type, type, (
% 21.26/21.43 less_or_equal: ( $i * $i ) > $o)).
% 21.26/21.43 tff(length_of_type, type, (
% 21.26/21.43 length_of: $i > $i)).
% 21.26/21.43 tff(shortest_path_type, type, (
% 21.26/21.43 shortest_path: ( $i * $i * $i ) > $o)).
% 21.26/21.43 tff(tptp_fun_P_5_type, type, (
% 21.26/21.43 tptp_fun_P_5: ( $i * $i * $i ) > $i)).
% 21.26/21.43 tff(tptp_fun_E3_13_type, type, (
% 21.26/21.43 tptp_fun_E3_13: $i)).
% 21.26/21.43 tff(tptp_fun_E3_14_type, type, (
% 21.26/21.43 tptp_fun_E3_14: $i)).
% 21.26/21.43 tff(precedes_type, type, (
% 21.26/21.43 precedes: ( $i * $i * $i ) > $o)).
% 21.26/21.43 tff(complete_type, type, (
% 21.26/21.43 complete: $o)).
% 21.26/21.43 tff(on_path_type, type, (
% 21.26/21.43 on_path: ( $i * $i ) > $o)).
% 21.26/21.43 tff(sequential_type, type, (
% 21.26/21.43 sequential: ( $i * $i ) > $o)).
% 21.26/21.43 tff(tptp_fun_E3_4_type, type, (
% 21.26/21.43 tptp_fun_E3_4: ( $i * $i * $i ) > $i)).
% 21.26/21.43 tff(1,plain,
% 21.26/21.43 ((head_of(tptp_fun_E_0(tail_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) = tail_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (tail_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(tail_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))))),
% 21.26/21.43 inference(commutativity,[status(thm)],[])).
% 21.26/21.43 tff(2,plain,
% 21.26/21.43 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))))),
% 21.26/21.43 inference(commutativity,[status(thm)],[])).
% 21.26/21.43 tff(3,assumption,(head_of(E2!9) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))), introduced(assumption)).
% 21.26/21.43 tff(4,plain,
% 21.26/21.43 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) = head_of(E2!9)) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))))),
% 21.26/21.43 inference(monotonicity,[status(thm)],[3])).
% 21.26/21.43 tff(5,plain,
% 21.26/21.43 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) = head_of(E2!9)) <=> (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))))),
% 21.26/21.43 inference(transitivity,[status(thm)],[4, 2])).
% 21.26/21.43 tff(6,plain,
% 21.26/21.43 ((head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))))) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))), head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) = head_of(E2!9))),
% 21.26/21.43 inference(symmetry,[status(thm)],[5])).
% 21.26/21.43 tff(7,plain,
% 21.26/21.43 ((tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(commutativity,[status(thm)],[])).
% 21.26/21.44 tff(8,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))))),
% 21.26/21.44 inference(symmetry,[status(thm)],[7])).
% 21.26/21.44 tff(9,assumption,(~((~(head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) | (~(tail_of(E1!10) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))))), introduced(assumption)).
% 21.26/21.44 tff(10,plain,
% 21.26/21.44 (((~(head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) | (~(tail_of(E1!10) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))))) | (tail_of(E1!10) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(tautology,[status(thm)],[])).
% 21.26/21.44 tff(11,plain,
% 21.26/21.44 (tail_of(E1!10) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))),
% 21.26/21.44 inference(unit_resolution,[status(thm)],[10, 9])).
% 21.26/21.44 tff(12,plain,
% 21.26/21.44 (((~(head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) | (~(tail_of(E1!10) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))))) | (head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(tautology,[status(thm)],[])).
% 21.26/21.44 tff(13,plain,
% 21.26/21.44 (head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))),
% 21.26/21.44 inference(unit_resolution,[status(thm)],[12, 9])).
% 21.26/21.44 tff(14,assumption,(tail_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))), introduced(assumption)).
% 21.26/21.44 tff(15,assumption,(head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))), introduced(assumption)).
% 21.26/21.44 tff(16,plain,
% 21.26/21.44 (tptp_fun_E_0(head_of(E1!10), head_of(E2!9)) = tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[15])).
% 21.26/21.44 tff(17,plain,
% 21.26/21.44 (tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9)) = tptp_fun_E_0(head_of(E1!10), head_of(E2!9))),
% 21.26/21.44 inference(symmetry,[status(thm)],[16])).
% 21.26/21.44 tff(18,plain,
% 21.26/21.44 (head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[17])).
% 21.26/21.44 tff(19,plain,
% 21.26/21.44 (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9)))),
% 21.26/21.44 inference(symmetry,[status(thm)],[18])).
% 21.26/21.44 tff(20,plain,
% 21.26/21.44 ((head_of(E2!9) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (head_of(E2!9) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[19])).
% 21.26/21.44 tff(21,plain,
% 21.26/21.44 ((head_of(E2!9) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9)))) <=> (head_of(E2!9) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))))),
% 21.26/21.44 inference(symmetry,[status(thm)],[20])).
% 21.26/21.44 tff(22,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))) = head_of(E2!9)) <=> (head_of(E2!9) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))))),
% 21.26/21.44 inference(commutativity,[status(thm)],[])).
% 21.26/21.44 tff(23,assumption,(tail_of(E1!10) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))), introduced(assumption)).
% 21.26/21.44 tff(24,assumption,(tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))), introduced(assumption)).
% 21.26/21.44 tff(25,plain,
% 21.26/21.44 (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))),
% 21.26/21.44 inference(symmetry,[status(thm)],[24])).
% 21.26/21.44 tff(26,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(commutativity,[status(thm)],[])).
% 21.26/21.44 tff(27,plain,
% 21.26/21.44 ((head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[15])).
% 21.26/21.44 tff(28,plain,
% 21.26/21.44 ((head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(transitivity,[status(thm)],[27, 26])).
% 21.26/21.44 tff(29,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))))),
% 21.26/21.44 inference(symmetry,[status(thm)],[28])).
% 21.26/21.44 tff(30,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))))),
% 21.26/21.44 inference(commutativity,[status(thm)],[])).
% 21.26/21.44 tff(31,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[19])).
% 21.26/21.44 tff(32,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))))),
% 21.26/21.44 inference(transitivity,[status(thm)],[31, 30])).
% 21.26/21.44 tff(33,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(symmetry,[status(thm)],[32])).
% 21.26/21.44 tff(34,plain,
% 21.26/21.44 (head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))),
% 21.26/21.44 inference(transitivity,[status(thm)],[18, 25])).
% 21.26/21.44 tff(35,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[34])).
% 21.26/21.44 tff(36,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))), head_of(E2!9))))),
% 21.26/21.44 inference(symmetry,[status(thm)],[35])).
% 21.26/21.44 tff(37,plain,
% 21.26/21.44 ((head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10)))) <=> (head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9))))),
% 21.26/21.44 inference(transitivity,[status(thm)],[36, 33, 29])).
% 21.26/21.44 tff(38,plain,
% 21.26/21.44 ((~(head_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))) = tail_of(tptp_fun_E_0(head_of(E1!10), tail_of(E1!10))))) <=> (~(head_of(E1!10) = head_of(tptp_fun_E_0(head_of(E1!10), head_of(E2!9)))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[37])).
% 21.26/21.44 tff(39,plain,
% 21.26/21.44 ((tail_of(E1!10) = head_of(E1!10)) <=> (head_of(E1!10) = tail_of(E1!10))),
% 21.26/21.44 inference(commutativity,[status(thm)],[])).
% 21.26/21.44 tff(40,plain,
% 21.26/21.44 ((head_of(E1!10) = tail_of(E1!10)) <=> (tail_of(E1!10) = head_of(E1!10))),
% 21.26/21.44 inference(symmetry,[status(thm)],[39])).
% 21.26/21.44 tff(41,plain,
% 21.26/21.44 ((~(head_of(E1!10) = tail_of(E1!10))) <=> (~(tail_of(E1!10) = head_of(E1!10)))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[40])).
% 21.26/21.44 tff(42,plain,
% 21.26/21.44 (^[V1: $i, V2: $i, SP: $i] : rewrite((~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))))))),
% 21.26/21.44 inference(bind,[status(th)],[])).
% 21.26/21.44 tff(43,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))),
% 21.26/21.44 inference(quant_intro,[status(thm)],[42])).
% 21.26/21.44 tff(44,plain,
% 21.26/21.44 (^[V1: $i, V2: $i, SP: $i] : refl((~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))))),
% 21.26/21.44 inference(bind,[status(th)],[])).
% 21.26/21.44 tff(45,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 21.26/21.44 inference(quant_intro,[status(thm)],[44])).
% 21.26/21.44 tff(46,plain,
% 21.26/21.44 (^[V1: $i, V2: $i, SP: $i] : rewrite((~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))))),
% 21.26/21.44 inference(bind,[status(th)],[])).
% 21.26/21.44 tff(47,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 21.26/21.44 inference(quant_intro,[status(thm)],[46])).
% 21.26/21.44 tff(48,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 21.26/21.44 inference(transitivity,[status(thm)],[47, 45])).
% 21.26/21.44 tff(49,plain,
% 21.26/21.44 (^[V1: $i, V2: $i, SP: $i] : trans(monotonicity(rewrite(((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) <=> ((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))), rewrite(((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))) <=> ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))), ((((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> (((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))), rewrite((((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))), ((((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))))),
% 21.26/21.44 inference(bind,[status(th)],[])).
% 21.26/21.44 tff(50,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 21.26/21.44 inference(quant_intro,[status(thm)],[49])).
% 21.26/21.44 tff(51,plain,
% 21.26/21.44 (^[V1: $i, V2: $i, SP: $i] : rewrite((((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & (shortest_path(V1, V2, SP) | ((~path(V1, V2, SP)) | (V1 = V2) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))) <=> (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))),
% 21.26/21.44 inference(bind,[status(th)],[])).
% 21.26/21.44 tff(52,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & (shortest_path(V1, V2, SP) | ((~path(V1, V2, SP)) | (V1 = V2) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))) <=> ![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))),
% 21.26/21.44 inference(quant_intro,[status(thm)],[51])).
% 21.26/21.44 tff(53,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) <=> ![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 21.26/21.44 inference(rewrite,[status(thm)],[])).
% 21.26/21.44 tff(54,plain,
% 21.26/21.44 (^[V1: $i, V2: $i, SP: $i] : rewrite((shortest_path(V1, V2, SP) <=> ((path(V1, V2, SP) & (~(V1 = V2))) & ![P: $i] : (path(V1, V2, P) => less_or_equal(length_of(SP), length_of(P))))) <=> (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))),
% 21.26/21.44 inference(bind,[status(th)],[])).
% 21.26/21.44 tff(55,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> ((path(V1, V2, SP) & (~(V1 = V2))) & ![P: $i] : (path(V1, V2, P) => less_or_equal(length_of(SP), length_of(P))))) <=> ![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 21.26/21.44 inference(quant_intro,[status(thm)],[54])).
% 21.26/21.44 tff(56,axiom,(![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> ((path(V1, V2, SP) & (~(V1 = V2))) & ![P: $i] : (path(V1, V2, P) => less_or_equal(length_of(SP), length_of(P)))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','shortest_path_defn')).
% 21.26/21.44 tff(57,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[56, 55])).
% 21.26/21.44 tff(58,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[57, 53])).
% 21.26/21.44 tff(59,plain,(
% 21.26/21.44 ![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & (shortest_path(V1, V2, SP) | ((~path(V1, V2, SP)) | (V1 = V2) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))),
% 21.26/21.44 inference(skolemize,[status(sab)],[58])).
% 21.26/21.44 tff(60,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[59, 52])).
% 21.26/21.44 tff(61,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[60, 50])).
% 21.26/21.44 tff(62,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[61, 48])).
% 21.26/21.44 tff(63,plain,
% 21.26/21.44 (![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[62, 43])).
% 21.26/21.44 tff(64,plain,
% 21.26/21.44 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))))))) <=> ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))))))),
% 21.26/21.44 inference(rewrite,[status(thm)],[])).
% 21.26/21.44 tff(65,plain,
% 21.26/21.44 ((~((~((V1!12 = V2!11) | shortest_path(V1!12, V2!11, P!8) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))))))))) <=> (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))))))),
% 21.26/21.44 inference(rewrite,[status(thm)],[])).
% 21.26/21.44 tff(66,plain,
% 21.26/21.44 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~((V1!12 = V2!11) | shortest_path(V1!12, V2!11, P!8) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P))))))))))) <=> ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[65])).
% 21.26/21.44 tff(67,plain,
% 21.26/21.44 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~((V1!12 = V2!11) | shortest_path(V1!12, V2!11, P!8) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P))))))))))) <=> ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))))))),
% 21.26/21.44 inference(transitivity,[status(thm)],[66, 64])).
% 21.26/21.44 tff(68,plain,
% 21.26/21.44 ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~((V1!12 = V2!11) | shortest_path(V1!12, V2!11, P!8) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P))))))))))),
% 21.26/21.44 inference(quant_inst,[status(thm)],[])).
% 21.26/21.44 tff(69,plain,
% 21.26/21.44 ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))))))),
% 21.26/21.44 inference(modus_ponens,[status(thm)],[68, 67])).
% 21.26/21.44 tff(70,plain,
% 21.26/21.44 (~((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))))),
% 21.26/21.44 inference(unit_resolution,[status(thm)],[69, 63])).
% 21.26/21.44 tff(71,plain,
% 21.26/21.44 (((~(shortest_path(V1!12, V2!11, P!8) | (V1!12 = V2!11) | (~path(V1!12, V2!11, P!8)) | (~((~path(V1!12, V2!11, tptp_fun_P_5(P!8, V2!11, V1!12))) | less_or_equal(length_of(P!8), length_of(tptp_fun_P_5(P!8, V2!11, V1!12))))))) | (~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))))) | ((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))),
% 21.26/21.44 inference(tautology,[status(thm)],[])).
% 21.26/21.44 tff(72,plain,
% 21.26/21.44 ((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))),
% 21.26/21.44 inference(unit_resolution,[status(thm)],[71, 70])).
% 21.26/21.44 tff(73,plain,
% 21.26/21.44 (((shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8)) & ((![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))))) <=> (shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8) & (![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))))),
% 21.26/21.44 inference(rewrite,[status(thm)],[])).
% 21.26/21.44 tff(74,plain,
% 21.26/21.44 (((![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10)))))) <=> ((![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))))),
% 21.26/21.44 inference(rewrite,[status(thm)],[])).
% 21.26/21.44 tff(75,plain,
% 21.26/21.44 (((shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8)) & ((![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))))) <=> ((shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8)) & ((![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10)))))))),
% 21.26/21.44 inference(monotonicity,[status(thm)],[74])).
% 21.26/21.44 tff(76,plain,
% 21.26/21.44 (((shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8)) & ((![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))))) <=> (shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8) & (![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))))),
% 21.26/21.44 inference(transitivity,[status(thm)],[75, 73])).
% 21.26/21.44 tff(77,plain,
% 21.26/21.44 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))) <=> (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1)))))))),
% 21.26/21.45 inference(rewrite,[status(thm)],[])).
% 21.26/21.45 tff(78,plain,
% 21.26/21.45 ((~(complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => (~(?[E3: $i] : ((edge(E3) & (tail_of(E3) = tail_of(E1))) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : ((edge(E3) & (tail_of(E3) = head_of(E2))) & (head_of(E3) = tail_of(E1)))))))) <=> (~((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))))),
% 21.26/21.45 inference(rewrite,[status(thm)],[])).
% 21.26/21.45 tff(79,axiom,(~(complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => (~(?[E3: $i] : ((edge(E3) & (tail_of(E3) = tail_of(E1))) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : ((edge(E3) & (tail_of(E3) = head_of(E2))) & (head_of(E3) = tail_of(E1)))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','edge_some_way')).
% 21.26/21.45 tff(80,plain,
% 21.26/21.45 (~((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1)))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[79, 78])).
% 21.26/21.45 tff(81,plain,
% 21.26/21.45 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))),
% 21.26/21.45 inference(or_elim,[status(thm)],[80])).
% 21.26/21.45 tff(82,plain,
% 21.26/21.45 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[81, 77])).
% 21.26/21.45 tff(83,plain,
% 21.26/21.45 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[82, 77])).
% 21.26/21.45 tff(84,plain,
% 21.26/21.45 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (~(?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> ?[E3: $i] : (edge(E3) & (tail_of(E3) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[83, 77])).
% 21.26/21.45 tff(85,plain,
% 21.26/21.45 (shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8) & (![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1!10)) & (head_of(E3) = head_of(E2!9)))) | (edge(E3!14) & (tail_of(E3!14) = head_of(E2!9)) & (head_of(E3!14) = tail_of(E1!10)))) & ((edge(E3!13) & (tail_of(E3!13) = tail_of(E1!10)) & (head_of(E3!13) = head_of(E2!9))) | ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10)))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[84, 76])).
% 21.26/21.45 tff(86,plain,
% 21.26/21.45 (shortest_path(V1!12, V2!11, P!8)),
% 21.26/21.45 inference(and_elim,[status(thm)],[85])).
% 21.26/21.45 tff(87,plain,
% 21.26/21.45 ((~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))) | (~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))),
% 21.26/21.45 inference(tautology,[status(thm)],[])).
% 21.26/21.45 tff(88,plain,
% 21.26/21.45 ((~((~shortest_path(V1!12, V2!11, P!8)) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))))) | (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))))),
% 21.26/21.45 inference(unit_resolution,[status(thm)],[87, 86])).
% 21.26/21.45 tff(89,plain,
% 21.26/21.45 (~((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8)))),
% 21.26/21.45 inference(unit_resolution,[status(thm)],[88, 72])).
% 21.26/21.45 tff(90,plain,
% 21.26/21.45 (((V1!12 = V2!11) | (~![P: $i] : ((~path(V1!12, V2!11, P)) | less_or_equal(length_of(P!8), length_of(P)))) | (~path(V1!12, V2!11, P!8))) | path(V1!12, V2!11, P!8)),
% 21.26/21.45 inference(tautology,[status(thm)],[])).
% 21.26/21.45 tff(91,plain,
% 21.26/21.45 (path(V1!12, V2!11, P!8)),
% 21.26/21.45 inference(unit_resolution,[status(thm)],[90, 89])).
% 21.26/21.45 tff(92,plain,
% 21.26/21.45 (^[V1: $i, V2: $i, P: $i] : refl(((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))))),
% 21.26/21.45 inference(bind,[status(th)],[])).
% 21.26/21.45 tff(93,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 21.26/21.45 inference(quant_intro,[status(thm)],[92])).
% 21.26/21.45 tff(94,plain,
% 21.26/21.45 (^[V1: $i, V2: $i, P: $i] : rewrite(((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))) <=> ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))))),
% 21.26/21.45 inference(bind,[status(th)],[])).
% 21.26/21.45 tff(95,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 21.26/21.45 inference(quant_intro,[status(thm)],[94])).
% 21.26/21.45 tff(96,plain,
% 21.26/21.45 (^[V1: $i, V2: $i, P: $i] : rewrite(((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & ((((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))),
% 21.26/21.45 inference(bind,[status(th)],[])).
% 21.26/21.45 tff(97,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & ((((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))),
% 21.26/21.45 inference(quant_intro,[status(thm)],[96])).
% 21.26/21.45 tff(98,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))),
% 21.26/21.45 inference(rewrite,[status(thm)],[])).
% 21.26/21.45 tff(99,plain,
% 21.26/21.45 (^[V1: $i, V2: $i, P: $i] : trans(monotonicity(trans(monotonicity(quant_intro(proof_bind(^[E: $i] : trans(monotonicity(trans(monotonicity(rewrite((((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))) <=> (((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))), ((~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))) <=> (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))), rewrite((~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))) <=> (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))), ((~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))) <=> (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))), (((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))) <=> ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))), rewrite(((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))) <=> (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))), (((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))) <=> (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))), (?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))) <=> ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))), (((vertex(V1) & vertex(V2)) & ?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))) <=> ((vertex(V1) & vertex(V2)) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))), rewrite(((vertex(V1) & vertex(V2)) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))) <=> (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))), (((vertex(V1) & vertex(V2)) & ?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))) <=> (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))), ((path(V1, V2, P) => ((vertex(V1) & vertex(V2)) & ?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))) <=> (path(V1, V2, P) => (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))))), rewrite((path(V1, V2, P) => (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))) <=> ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))), ((path(V1, V2, P) => ((vertex(V1) & vertex(V2)) & ?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))) <=> ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))))),
% 21.26/21.45 inference(bind,[status(th)],[])).
% 21.26/21.45 tff(100,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : (path(V1, V2, P) => ((vertex(V1) & vertex(V2)) & ?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP)))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))),
% 21.26/21.45 inference(quant_intro,[status(thm)],[99])).
% 21.26/21.45 tff(101,axiom,(![V1: $i, V2: $i, P: $i] : (path(V1, V2, P) => ((vertex(V1) & vertex(V2)) & ?[E: $i] : ((edge(E) & (V1 = tail_of(E))) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','path_properties')).
% 21.26/21.45 tff(102,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[101, 100])).
% 21.26/21.45 tff(103,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & ?[E: $i] : (edge(E) & (V1 = tail_of(E)) & (~(((V2 = head_of(E)) & (P = path_cons(E, empty))) <=> ?[TP: $i] : (path(head_of(E), V2, TP) & (P = path_cons(E, TP))))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[102, 98])).
% 21.26/21.45 tff(104,plain,(
% 21.26/21.45 ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & ((((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 21.26/21.45 inference(skolemize,[status(sab)],[103])).
% 21.26/21.45 tff(105,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[104, 97])).
% 21.26/21.45 tff(106,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[105, 95])).
% 21.26/21.45 tff(107,plain,
% 21.26/21.45 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 21.26/21.45 inference(modus_ponens,[status(thm)],[106, 93])).
% 21.26/21.45 tff(108,plain,
% 21.26/21.45 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | (~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))))))),
% 21.26/21.45 inference(rewrite,[status(thm)],[])).
% 21.26/21.45 tff(109,plain,
% 21.26/21.45 (((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))))))) | (~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP))))))))) <=> ((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))))))),
% 21.26/21.45 inference(rewrite,[status(thm)],[])).
% 21.26/21.45 tff(110,plain,
% 21.26/21.45 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))))))) | (~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12))))))))))),
% 21.26/21.45 inference(monotonicity,[status(thm)],[109])).
% 21.26/21.45 tff(111,plain,
% 21.26/21.45 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))))))) | (~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | (~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))))))),
% 21.26/21.45 inference(transitivity,[status(thm)],[110, 108])).
% 21.26/21.45 tff(112,plain,
% 21.26/21.45 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))))))) | (~((~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))))))))),
% 21.26/21.46 inference(quant_inst,[status(thm)],[])).
% 21.26/21.46 tff(113,plain,
% 21.26/21.46 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | (~path(V1!12, V2!11, P!8)) | (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12))))))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[112, 111])).
% 21.26/21.46 tff(114,plain,
% 21.26/21.46 (~((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))))),
% 21.26/21.46 inference(unit_resolution,[status(thm)],[113, 107, 91])).
% 21.26/21.46 tff(115,plain,
% 21.26/21.46 (((~vertex(V1!12)) | (~vertex(V2!11)) | (~edge(tptp_fun_E_1(P!8, V2!11, V1!12))) | (~(V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))) | (~((~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12)))))) | (~((~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), tptp_fun_TP_2(P!8, V2!11, V1!12)))) | (~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, tptp_fun_TP_2(P!8, V2!11, V1!12))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, TP)) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), TP)))) | (~(P!8 = path_cons(tptp_fun_E_1(P!8, V2!11, V1!12), empty))) | (~(V2!11 = head_of(tptp_fun_E_1(P!8, V2!11, V1!12))))))) | (V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)))),
% 21.26/21.46 inference(tautology,[status(thm)],[])).
% 21.26/21.46 tff(116,plain,
% 21.26/21.46 (V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12))),
% 21.26/21.46 inference(unit_resolution,[status(thm)],[115, 114])).
% 21.26/21.46 tff(117,plain,
% 21.26/21.46 (tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)) = V1!12),
% 21.26/21.46 inference(symmetry,[status(thm)],[116])).
% 21.26/21.46 tff(118,plain,
% 21.26/21.46 (path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8) <=> path(V1!12, V2!11, P!8)),
% 21.26/21.46 inference(monotonicity,[status(thm)],[117])).
% 21.26/21.46 tff(119,plain,
% 21.26/21.46 (path(V1!12, V2!11, P!8) <=> path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)),
% 21.26/21.46 inference(symmetry,[status(thm)],[118])).
% 21.26/21.46 tff(120,plain,
% 21.26/21.46 (path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[91, 119])).
% 21.26/21.46 tff(121,plain,
% 21.26/21.46 (^[P: $i, V1: $i, V2: $i] : refl(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))))),
% 21.26/21.46 inference(bind,[status(th)],[])).
% 21.26/21.46 tff(122,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(quant_intro,[status(thm)],[121])).
% 21.26/21.46 tff(123,plain,
% 21.26/21.46 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))))),
% 21.26/21.46 inference(bind,[status(th)],[])).
% 21.26/21.46 tff(124,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(quant_intro,[status(thm)],[123])).
% 21.26/21.46 tff(125,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(transitivity,[status(thm)],[124, 122])).
% 21.26/21.46 tff(126,plain,
% 21.26/21.46 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))))),
% 21.26/21.46 inference(bind,[status(th)],[])).
% 21.26/21.46 tff(127,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(quant_intro,[status(thm)],[126])).
% 21.26/21.46 tff(128,plain,
% 21.26/21.46 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & ((sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))))),
% 21.26/21.46 inference(bind,[status(th)],[])).
% 21.26/21.46 tff(129,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & ((sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 21.26/21.46 inference(quant_intro,[status(thm)],[128])).
% 21.26/21.46 tff(130,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 21.26/21.46 inference(rewrite,[status(thm)],[])).
% 21.26/21.46 tff(131,plain,
% 21.26/21.46 (^[P: $i, V1: $i, V2: $i] : trans(monotonicity(quant_intro(proof_bind(^[E1: $i, E2: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))) <=> (sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))), ((~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) <=> (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))), rewrite((~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) <=> (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))), ((~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) <=> (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))), (((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) <=> ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))), rewrite(((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) <=> (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))), (((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) <=> (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))), ((precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))) <=> (precedes(E1, E2, P) => (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))), rewrite((precedes(E1, E2, P) => (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))) <=> ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))), ((precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))) <=> ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))))), (![E1: $i, E2: $i] : (precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))) <=> ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))), ((path(V1, V2, P) => ![E1: $i, E2: $i] : (precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))) <=> (path(V1, V2, P) => ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))))), rewrite((path(V1, V2, P) => ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))), ((path(V1, V2, P) => ![E1: $i, E2: $i] : (precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(bind,[status(th)],[])).
% 21.26/21.46 tff(132,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : (path(V1, V2, P) => ![E1: $i, E2: $i] : (precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 21.26/21.46 inference(quant_intro,[status(thm)],[131])).
% 21.26/21.46 tff(133,axiom,(![P: $i, V1: $i, V2: $i] : (path(V1, V2, P) => ![E1: $i, E2: $i] : (precedes(E1, E2, P) => ((on_path(E1, P) & on_path(E2, P)) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','precedes_properties')).
% 21.26/21.46 tff(134,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[133, 132])).
% 21.26/21.46 tff(135,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (~(sequential(E1, E2) <=> ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[134, 130])).
% 21.26/21.46 tff(136,plain,(
% 21.26/21.46 ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & ((sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))))),
% 21.26/21.46 inference(skolemize,[status(sab)],[135])).
% 21.26/21.46 tff(137,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | (sequential(E1, tptp_fun_E3_4(E2, E1, P)) & precedes(tptp_fun_E3_4(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[136, 129])).
% 21.26/21.46 tff(138,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[137, 127])).
% 21.26/21.46 tff(139,plain,
% 21.26/21.46 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[138, 125])).
% 21.26/21.46 tff(140,plain,
% 21.26/21.46 (((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | (~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8))))))),
% 21.26/21.46 inference(rewrite,[status(thm)],[])).
% 21.26/21.46 tff(141,plain,
% 21.26/21.46 (((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~on_path(E1, P!8)) | (~on_path(E2, P!8)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))))))) <=> ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8))))))),
% 21.26/21.46 inference(rewrite,[status(thm)],[])).
% 21.26/21.46 tff(142,plain,
% 21.26/21.46 (((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~on_path(E1, P!8)) | (~on_path(E2, P!8)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))))),
% 21.26/21.46 inference(monotonicity,[status(thm)],[141])).
% 21.26/21.46 tff(143,plain,
% 21.26/21.46 (((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~on_path(E1, P!8)) | (~on_path(E2, P!8)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | (~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8))))))),
% 21.26/21.46 inference(transitivity,[status(thm)],[142, 140])).
% 21.26/21.46 tff(144,plain,
% 21.26/21.46 ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~on_path(E1, P!8)) | (~on_path(E2, P!8)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))))),
% 21.26/21.46 inference(quant_inst,[status(thm)],[])).
% 21.26/21.46 tff(145,plain,
% 21.26/21.46 ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P)) | (~((~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P))) | (~precedes(tptp_fun_E3_4(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | (~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[144, 143])).
% 21.26/21.46 tff(146,plain,
% 21.26/21.46 (![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))),
% 21.26/21.46 inference(unit_resolution,[status(thm)],[145, 139, 91])).
% 21.26/21.46 tff(147,plain,
% 21.26/21.46 (precedes(E1!10, E2!9, P!8)),
% 21.26/21.46 inference(and_elim,[status(thm)],[85])).
% 21.26/21.46 tff(148,plain,
% 21.26/21.46 (((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | ((~precedes(E1!10, E2!9, P!8)) | (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9))))))) <=> ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | (~precedes(E1!10, E2!9, P!8)) | (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9))))))),
% 21.26/21.46 inference(rewrite,[status(thm)],[])).
% 21.26/21.46 tff(149,plain,
% 21.26/21.46 (((~precedes(E1!10, E2!9, P!8)) | (~((~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~(sequential(E1!10, E2!9) | (~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))))) | (~on_path(E2!9, P!8)) | (~on_path(E1!10, P!8))))) <=> ((~precedes(E1!10, E2!9, P!8)) | (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9))))))),
% 21.26/21.46 inference(rewrite,[status(thm)],[])).
% 21.26/21.46 tff(150,plain,
% 21.26/21.46 (((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | ((~precedes(E1!10, E2!9, P!8)) | (~((~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~(sequential(E1!10, E2!9) | (~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))))) | (~on_path(E2!9, P!8)) | (~on_path(E1!10, P!8)))))) <=> ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | ((~precedes(E1!10, E2!9, P!8)) | (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9)))))))),
% 21.26/21.46 inference(monotonicity,[status(thm)],[149])).
% 21.26/21.46 tff(151,plain,
% 21.26/21.46 (((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | ((~precedes(E1!10, E2!9, P!8)) | (~((~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~(sequential(E1!10, E2!9) | (~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))))) | (~on_path(E2!9, P!8)) | (~on_path(E1!10, P!8)))))) <=> ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | (~precedes(E1!10, E2!9, P!8)) | (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9))))))),
% 21.26/21.46 inference(transitivity,[status(thm)],[150, 148])).
% 21.26/21.46 tff(152,plain,
% 21.26/21.46 ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | ((~precedes(E1!10, E2!9, P!8)) | (~((~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~(sequential(E1!10, E2!9) | (~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))))) | (~on_path(E2!9, P!8)) | (~on_path(E1!10, P!8)))))),
% 21.26/21.46 inference(quant_inst,[status(thm)],[])).
% 21.26/21.46 tff(153,plain,
% 21.26/21.46 ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!8)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_4(E2, E1, P!8))) | (~precedes(tptp_fun_E3_4(E2, E1, P!8), E2, P!8)))))) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)))))) | (~precedes(E1!10, E2!9, P!8)) | (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9)))))),
% 21.26/21.46 inference(modus_ponens,[status(thm)],[152, 151])).
% 21.26/21.47 tff(154,plain,
% 21.26/21.47 (~((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9))))),
% 21.26/21.47 inference(unit_resolution,[status(thm)],[153, 147, 146])).
% 21.26/21.47 tff(155,plain,
% 21.26/21.47 (((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9)))) | on_path(E1!10, P!8)),
% 21.26/21.47 inference(tautology,[status(thm)],[])).
% 21.26/21.47 tff(156,plain,
% 21.26/21.47 (on_path(E1!10, P!8)),
% 21.26/21.47 inference(unit_resolution,[status(thm)],[155, 154])).
% 21.26/21.47 tff(157,plain,
% 21.26/21.47 (^[V1: $i, V2: $i, P: $i, E: $i] : refl(((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P))))) <=> ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P))))))),
% 21.26/21.47 inference(bind,[status(th)],[])).
% 21.26/21.47 tff(158,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P))))) <=> ![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))),
% 21.26/21.47 inference(quant_intro,[status(thm)],[157])).
% 21.26/21.47 tff(159,plain,
% 21.26/21.47 (^[V1: $i, V2: $i, P: $i, E: $i] : trans(monotonicity(trans(monotonicity(rewrite((path(V1, V2, P) & on_path(E, P)) <=> (~((~on_path(E, P)) | (~path(V1, V2, P))))), ((~(path(V1, V2, P) & on_path(E, P))) <=> (~(~((~on_path(E, P)) | (~path(V1, V2, P))))))), rewrite((~(~((~on_path(E, P)) | (~path(V1, V2, P))))) <=> ((~on_path(E, P)) | (~path(V1, V2, P)))), ((~(path(V1, V2, P) & on_path(E, P))) <=> ((~on_path(E, P)) | (~path(V1, V2, P))))), rewrite((edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)) <=> (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P))))), (((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))) <=> (((~on_path(E, P)) | (~path(V1, V2, P))) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P))))))), rewrite((((~on_path(E, P)) | (~path(V1, V2, P))) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P))))) <=> ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))), (((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))) <=> ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))))),
% 21.26/21.47 inference(bind,[status(th)],[])).
% 21.26/21.47 tff(160,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))) <=> ![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))),
% 21.26/21.47 inference(quant_intro,[status(thm)],[159])).
% 21.26/21.47 tff(161,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))) <=> ![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))),
% 21.26/21.47 inference(rewrite,[status(thm)],[])).
% 21.26/21.47 tff(162,plain,
% 21.26/21.47 (^[V1: $i, V2: $i, P: $i, E: $i] : trans(monotonicity(rewrite(((edge(E) & in_path(head_of(E), P)) & in_path(tail_of(E), P)) <=> (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))), (((path(V1, V2, P) & on_path(E, P)) => ((edge(E) & in_path(head_of(E), P)) & in_path(tail_of(E), P))) <=> ((path(V1, V2, P) & on_path(E, P)) => (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))))), rewrite(((path(V1, V2, P) & on_path(E, P)) => (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))) <=> ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))), (((path(V1, V2, P) & on_path(E, P)) => ((edge(E) & in_path(head_of(E), P)) & in_path(tail_of(E), P))) <=> ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))))),
% 21.26/21.47 inference(bind,[status(th)],[])).
% 21.26/21.47 tff(163,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((path(V1, V2, P) & on_path(E, P)) => ((edge(E) & in_path(head_of(E), P)) & in_path(tail_of(E), P))) <=> ![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))),
% 21.26/21.47 inference(quant_intro,[status(thm)],[162])).
% 21.26/21.47 tff(164,axiom,(![V1: $i, V2: $i, P: $i, E: $i] : ((path(V1, V2, P) & on_path(E, P)) => ((edge(E) & in_path(head_of(E), P)) & in_path(tail_of(E), P)))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','on_path_properties')).
% 21.26/21.47 tff(165,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))),
% 21.26/21.47 inference(modus_ponens,[status(thm)],[164, 163])).
% 21.26/21.47 tff(166,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))),
% 21.26/21.47 inference(modus_ponens,[status(thm)],[165, 161])).
% 21.26/21.47 tff(167,plain,(
% 21.26/21.47 ![V1: $i, V2: $i, P: $i, E: $i] : ((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P)))),
% 21.26/21.47 inference(skolemize,[status(sab)],[166])).
% 21.26/21.47 tff(168,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))),
% 21.26/21.47 inference(modus_ponens,[status(thm)],[167, 160])).
% 21.26/21.47 tff(169,plain,
% 21.26/21.47 (![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))),
% 21.26/21.47 inference(modus_ponens,[status(thm)],[168, 158])).
% 21.26/21.47 tff(170,plain,
% 21.26/21.47 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | ((~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | (~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))))),
% 21.26/21.47 inference(rewrite,[status(thm)],[])).
% 21.26/21.47 tff(171,plain,
% 21.26/21.47 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | ((~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))))),
% 21.26/21.47 inference(quant_inst,[status(thm)],[])).
% 21.26/21.47 tff(172,plain,
% 21.26/21.47 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | (~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8))))),
% 21.26/21.47 inference(modus_ponens,[status(thm)],[171, 170])).
% 21.26/21.47 tff(173,plain,
% 21.26/21.47 ((~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8))))),
% 21.26/21.47 inference(unit_resolution,[status(thm)],[172, 169, 156])).
% 21.26/21.47 tff(174,plain,
% 21.26/21.47 (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))),
% 21.26/21.47 inference(unit_resolution,[status(thm)],[173, 120])).
% 21.26/21.47 tff(175,plain,
% 21.26/21.47 (((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8))) | edge(E1!10)),
% 21.32/21.47 inference(tautology,[status(thm)],[])).
% 21.32/21.47 tff(176,plain,
% 21.32/21.47 (edge(E1!10)),
% 21.32/21.47 inference(unit_resolution,[status(thm)],[175, 174])).
% 21.32/21.47 tff(177,plain,
% 21.32/21.47 (^[E: $i] : refl(((~edge(E)) | (~(head_of(E) = tail_of(E)))) <=> ((~edge(E)) | (~(head_of(E) = tail_of(E)))))),
% 21.32/21.47 inference(bind,[status(th)],[])).
% 21.32/21.47 tff(178,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(quant_intro,[status(thm)],[177])).
% 21.32/21.47 tff(179,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(180,plain,
% 21.32/21.47 (^[E: $i] : rewrite((edge(E) => (~(head_of(E) = tail_of(E)))) <=> ((~edge(E)) | (~(head_of(E) = tail_of(E)))))),
% 21.32/21.47 inference(bind,[status(th)],[])).
% 21.32/21.47 tff(181,plain,
% 21.32/21.47 (![E: $i] : (edge(E) => (~(head_of(E) = tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(quant_intro,[status(thm)],[180])).
% 21.32/21.47 tff(182,axiom,(![E: $i] : (edge(E) => (~(head_of(E) = tail_of(E))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','no_loops')).
% 21.32/21.47 tff(183,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[182, 181])).
% 21.32/21.47 tff(184,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[183, 179])).
% 21.32/21.47 tff(185,plain,(
% 21.32/21.47 ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(skolemize,[status(sab)],[184])).
% 21.32/21.47 tff(186,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[185, 178])).
% 21.32/21.47 tff(187,plain,
% 21.32/21.47 (((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | ((~edge(E1!10)) | (~(head_of(E1!10) = tail_of(E1!10))))) <=> ((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | (~edge(E1!10)) | (~(head_of(E1!10) = tail_of(E1!10))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(188,plain,
% 21.32/21.47 ((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | ((~edge(E1!10)) | (~(head_of(E1!10) = tail_of(E1!10))))),
% 21.32/21.47 inference(quant_inst,[status(thm)],[])).
% 21.32/21.47 tff(189,plain,
% 21.32/21.47 ((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | (~edge(E1!10)) | (~(head_of(E1!10) = tail_of(E1!10)))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[188, 187])).
% 21.32/21.47 tff(190,plain,
% 21.32/21.47 (~(head_of(E1!10) = tail_of(E1!10))),
% 21.32/21.47 inference(unit_resolution,[status(thm)],[189, 186, 176])).
% 21.32/21.47 tff(191,plain,
% 21.32/21.47 (~(tail_of(E1!10) = head_of(E1!10))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[190, 41])).
% 21.32/21.47 tff(192,plain,
% 21.32/21.47 (^[E: $i] : refl(((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))) <=> ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))))),
% 21.32/21.47 inference(bind,[status(th)],[])).
% 21.32/21.47 tff(193,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))) <=> ![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 21.32/21.47 inference(quant_intro,[status(thm)],[192])).
% 21.32/21.47 tff(194,plain,
% 21.32/21.47 (^[E: $i] : rewrite(((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))))),
% 21.32/21.47 inference(bind,[status(th)],[])).
% 21.32/21.47 tff(195,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 21.32/21.47 inference(quant_intro,[status(thm)],[194])).
% 21.32/21.47 tff(196,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(197,plain,
% 21.32/21.47 (^[E: $i] : rewrite((edge(E) => (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))))),
% 21.32/21.47 inference(bind,[status(th)],[])).
% 21.32/21.47 tff(198,plain,
% 21.32/21.47 (![E: $i] : (edge(E) => (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 21.32/21.47 inference(quant_intro,[status(thm)],[197])).
% 21.32/21.47 tff(199,axiom,(![E: $i] : (edge(E) => (vertex(head_of(E)) & vertex(tail_of(E))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','edge_ends_are_vertices')).
% 21.32/21.47 tff(200,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[199, 198])).
% 21.32/21.47 tff(201,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[200, 196])).
% 21.32/21.47 tff(202,plain,(
% 21.32/21.47 ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 21.32/21.47 inference(skolemize,[status(sab)],[201])).
% 21.32/21.47 tff(203,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[202, 195])).
% 21.32/21.47 tff(204,plain,
% 21.32/21.47 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[203, 193])).
% 21.32/21.47 tff(205,plain,
% 21.32/21.47 (((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | ((~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))))) <=> ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | (~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(206,plain,
% 21.32/21.47 ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | ((~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))))),
% 21.32/21.47 inference(quant_inst,[status(thm)],[])).
% 21.32/21.47 tff(207,plain,
% 21.32/21.47 ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | (~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10)))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[206, 205])).
% 21.32/21.47 tff(208,plain,
% 21.32/21.47 (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))),
% 21.32/21.47 inference(unit_resolution,[status(thm)],[207, 204, 176])).
% 21.32/21.47 tff(209,plain,
% 21.32/21.47 (((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10)))) | vertex(tail_of(E1!10))),
% 21.32/21.47 inference(tautology,[status(thm)],[])).
% 21.32/21.47 tff(210,plain,
% 21.32/21.47 (vertex(tail_of(E1!10))),
% 21.32/21.47 inference(unit_resolution,[status(thm)],[209, 208])).
% 21.32/21.47 tff(211,plain,
% 21.32/21.47 (((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10)))) | vertex(head_of(E1!10))),
% 21.32/21.47 inference(tautology,[status(thm)],[])).
% 21.32/21.47 tff(212,plain,
% 21.32/21.47 (vertex(head_of(E1!10))),
% 21.32/21.47 inference(unit_resolution,[status(thm)],[211, 208])).
% 21.32/21.47 tff(213,plain,
% 21.32/21.47 (complete <=> complete),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(214,plain,
% 21.32/21.47 (complete),
% 21.32/21.47 inference(or_elim,[status(thm)],[80])).
% 21.32/21.47 tff(215,plain,
% 21.32/21.47 (complete),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[214, 213])).
% 21.32/21.47 tff(216,plain,
% 21.32/21.47 (((~complete) | ![V1: $i, V2: $i] : ((V1 = V2) | (~vertex(V1)) | (~vertex(V2)) | (~((~edge(tptp_fun_E_0(V2, V1))) | (((~(V1 = head_of(tptp_fun_E_0(V2, V1)))) | (~(V2 = tail_of(tptp_fun_E_0(V2, V1))))) <=> ((~(V2 = head_of(tptp_fun_E_0(V2, V1)))) | (~(V1 = tail_of(tptp_fun_E_0(V2, V1)))))))))) <=> ((~complete) | ![V1: $i, V2: $i] : ((V1 = V2) | (~vertex(V1)) | (~vertex(V2)) | (~((~edge(tptp_fun_E_0(V2, V1))) | (((~(V1 = head_of(tptp_fun_E_0(V2, V1)))) | (~(V2 = tail_of(tptp_fun_E_0(V2, V1))))) <=> ((~(V2 = head_of(tptp_fun_E_0(V2, V1)))) | (~(V1 = tail_of(tptp_fun_E_0(V2, V1))))))))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(217,plain,
% 21.32/21.47 (((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | (edge(tptp_fun_E_0(V2, V1)) & (~(((V1 = head_of(tptp_fun_E_0(V2, V1))) & (V2 = tail_of(tptp_fun_E_0(V2, V1)))) <=> ((V2 = head_of(tptp_fun_E_0(V2, V1))) & (V1 = tail_of(tptp_fun_E_0(V2, V1))))))))) <=> ((~complete) | ![V1: $i, V2: $i] : ((V1 = V2) | (~vertex(V1)) | (~vertex(V2)) | (~((~edge(tptp_fun_E_0(V2, V1))) | (((~(V1 = head_of(tptp_fun_E_0(V2, V1)))) | (~(V2 = tail_of(tptp_fun_E_0(V2, V1))))) <=> ((~(V2 = head_of(tptp_fun_E_0(V2, V1)))) | (~(V1 = tail_of(tptp_fun_E_0(V2, V1))))))))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(218,plain,
% 21.32/21.47 (((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | (edge(tptp_fun_E_0(V2, V1)) & (~(((V1 = head_of(tptp_fun_E_0(V2, V1))) & (V2 = tail_of(tptp_fun_E_0(V2, V1)))) <=> ((V2 = head_of(tptp_fun_E_0(V2, V1))) & (V1 = tail_of(tptp_fun_E_0(V2, V1))))))))) <=> ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | (edge(tptp_fun_E_0(V2, V1)) & (~(((V1 = head_of(tptp_fun_E_0(V2, V1))) & (V2 = tail_of(tptp_fun_E_0(V2, V1)))) <=> ((V2 = head_of(tptp_fun_E_0(V2, V1))) & (V1 = tail_of(tptp_fun_E_0(V2, V1)))))))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(219,plain,
% 21.32/21.47 (((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))) <=> ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(220,plain,
% 21.32/21.47 ((complete => ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))) <=> ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))))),
% 21.32/21.47 inference(rewrite,[status(thm)],[])).
% 21.32/21.47 tff(221,plain,
% 21.32/21.47 (^[V1: $i, V2: $i] : trans(monotonicity(rewrite(((vertex(V1) & vertex(V2)) & (~(V1 = V2))) <=> (vertex(V1) & vertex(V2) & (~(V1 = V2)))), ((((vertex(V1) & vertex(V2)) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))) <=> ((vertex(V1) & vertex(V2) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))))), rewrite(((vertex(V1) & vertex(V2) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))) <=> ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))), ((((vertex(V1) & vertex(V2)) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))) <=> ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))))),
% 21.32/21.47 inference(bind,[status(th)],[])).
% 21.32/21.47 tff(222,plain,
% 21.32/21.47 (![V1: $i, V2: $i] : (((vertex(V1) & vertex(V2)) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))) <=> ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))),
% 21.32/21.47 inference(quant_intro,[status(thm)],[221])).
% 21.32/21.47 tff(223,plain,
% 21.32/21.47 ((complete => ![V1: $i, V2: $i] : (((vertex(V1) & vertex(V2)) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))) <=> (complete => ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))))),
% 21.32/21.47 inference(monotonicity,[status(thm)],[222])).
% 21.32/21.47 tff(224,plain,
% 21.32/21.47 ((complete => ![V1: $i, V2: $i] : (((vertex(V1) & vertex(V2)) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))) <=> ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E))))))))),
% 21.32/21.47 inference(transitivity,[status(thm)],[223, 220])).
% 21.32/21.47 tff(225,axiom,(complete => ![V1: $i, V2: $i] : (((vertex(V1) & vertex(V2)) & (~(V1 = V2))) => ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','complete_properties')).
% 21.32/21.47 tff(226,plain,
% 21.32/21.47 ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[225, 224])).
% 21.32/21.47 tff(227,plain,
% 21.32/21.47 ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[226, 219])).
% 21.32/21.47 tff(228,plain,
% 21.32/21.47 ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[227, 219])).
% 21.32/21.47 tff(229,plain,
% 21.32/21.47 ((~complete) | ![V1: $i, V2: $i] : ((~(vertex(V1) & vertex(V2) & (~(V1 = V2)))) | ?[E: $i] : (edge(E) & (~(((V1 = head_of(E)) & (V2 = tail_of(E))) <=> ((V2 = head_of(E)) & (V1 = tail_of(E)))))))),
% 21.32/21.47 inference(modus_ponens,[status(thm)],[228, 219])).
% 21.32/21.47 Proof display could not be completed: monotonicity rule is not handled
% 21.32/21.50 % E exiting
%------------------------------------------------------------------------------