%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRA007+2 : TPTP v9.0.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.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 05:06:37 AM UTC 2025
% Result : Theorem 18.52s 18.75s
% Output : Assurance 0s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : GRA007+2 : TPTP v9.0.0. Bugfixed v3.2.0.
% 0.10/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 15:22:20 EDT 2025
% 0.12/0.33 % CPUTime :
% 18.52/18.75 % SZS status Theorem
% 18.52/18.75 % SZS output start Proof
% 18.52/18.75 tff(tail_of_type, type, (
% 18.52/18.75 tail_of: $i > $i)).
% 18.52/18.75 tff(tptp_fun_E1_10_type, type, (
% 18.52/18.75 tptp_fun_E1_10: $i)).
% 18.52/18.75 tff(head_of_type, type, (
% 18.52/18.75 head_of: $i > $i)).
% 18.52/18.75 tff(tptp_fun_E_0_type, type, (
% 18.52/18.75 tptp_fun_E_0: ( $i * $i ) > $i)).
% 18.52/18.75 tff(tptp_fun_E_3_type, type, (
% 18.52/18.75 tptp_fun_E_3: ( $i * $i ) > $i)).
% 18.52/18.75 tff(tptp_fun_P_8_type, type, (
% 18.52/18.75 tptp_fun_P_8: $i)).
% 18.52/18.75 tff(tptp_fun_E2_9_type, type, (
% 18.52/18.75 tptp_fun_E2_9: $i)).
% 18.52/18.75 tff(edge_type, type, (
% 18.52/18.75 edge: $i > $o)).
% 18.52/18.75 tff(precedes_type, type, (
% 18.52/18.75 precedes: ( $i * $i * $i ) > $o)).
% 18.52/18.75 tff(shortest_path_type, type, (
% 18.52/18.75 shortest_path: ( $i * $i * $i ) > $o)).
% 18.52/18.75 tff(tptp_fun_V2_11_type, type, (
% 18.52/18.75 tptp_fun_V2_11: $i)).
% 18.52/18.75 tff(tptp_fun_V1_12_type, type, (
% 18.52/18.75 tptp_fun_V1_12: $i)).
% 18.52/18.75 tff(complete_type, type, (
% 18.52/18.75 complete: $o)).
% 18.52/18.75 tff(vertex_type, type, (
% 18.52/18.75 vertex: $i > $o)).
% 18.52/18.75 tff(in_path_type, type, (
% 18.52/18.75 in_path: ( $i * $i ) > $o)).
% 18.52/18.75 tff(path_type, type, (
% 18.52/18.75 path: ( $i * $i * $i ) > $o)).
% 18.52/18.75 tff(tptp_fun_E_1_type, type, (
% 18.52/18.75 tptp_fun_E_1: ( $i * $i * $i ) > $i)).
% 18.52/18.75 tff(path_cons_type, type, (
% 18.52/18.75 path_cons: ( $i * $i ) > $i)).
% 18.52/18.75 tff(empty_type, type, (
% 18.52/18.75 empty: $i)).
% 18.52/18.75 tff(tptp_fun_TP_2_type, type, (
% 18.52/18.75 tptp_fun_TP_2: ( $i * $i * $i ) > $i)).
% 18.52/18.75 tff(less_or_equal_type, type, (
% 18.52/18.75 less_or_equal: ( $i * $i ) > $o)).
% 18.52/18.75 tff(length_of_type, type, (
% 18.52/18.75 length_of: $i > $i)).
% 18.52/18.75 tff(tptp_fun_P_5_type, type, (
% 18.52/18.75 tptp_fun_P_5: ( $i * $i * $i ) > $i)).
% 18.52/18.75 tff(on_path_type, type, (
% 18.52/18.75 on_path: ( $i * $i ) > $o)).
% 18.52/18.75 tff(sequential_type, type, (
% 18.52/18.75 sequential: ( $i * $i ) > $o)).
% 18.52/18.75 tff(tptp_fun_E3_4_type, type, (
% 18.52/18.75 tptp_fun_E3_4: ( $i * $i * $i ) > $i)).
% 18.52/18.75 tff(1,plain,
% 18.52/18.75 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) = tail_of(tptp_fun_E_3(tail_of(E1!10), P!8))) <=> (tail_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_3(tail_of(E1!10), P!8)))))),
% 18.52/18.75 inference(commutativity,[status(thm)],[])).
% 18.52/18.75 tff(2,plain,
% 18.52/18.75 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8))) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))))),
% 18.52/18.75 inference(commutativity,[status(thm)],[])).
% 18.52/18.75 tff(3,assumption,(tail_of(E1!10) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8))), introduced(assumption)).
% 18.52/18.75 tff(4,plain,
% 18.52/18.75 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) = tail_of(E1!10)) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))),
% 18.52/18.75 inference(monotonicity,[status(thm)],[3])).
% 18.52/18.75 tff(5,plain,
% 18.52/18.75 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) = tail_of(E1!10)) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))))),
% 18.52/18.75 inference(transitivity,[status(thm)],[4, 2])).
% 18.52/18.75 tff(6,plain,
% 18.52/18.75 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8))))) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) = tail_of(E1!10))),
% 18.52/18.75 inference(symmetry,[status(thm)],[5])).
% 18.52/18.75 tff(7,assumption,(~((~(tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) | (~(head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))), introduced(assumption)).
% 18.52/18.75 tff(8,plain,
% 18.52/18.75 (((~(tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) | (~(head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.75 inference(tautology,[status(thm)],[])).
% 18.52/18.75 tff(9,plain,
% 18.52/18.75 (tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))),
% 18.52/18.75 inference(unit_resolution,[status(thm)],[8, 7])).
% 18.52/18.75 tff(10,plain,
% 18.52/18.75 (((~(tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) | (~(head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.75 inference(tautology,[status(thm)],[])).
% 18.52/18.75 tff(11,plain,
% 18.52/18.75 (head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))),
% 18.52/18.75 inference(unit_resolution,[status(thm)],[10, 7])).
% 18.52/18.75 tff(12,plain,
% 18.52/18.75 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8))) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.75 inference(commutativity,[status(thm)],[])).
% 18.52/18.75 tff(13,plain,
% 18.52/18.75 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))),
% 18.52/18.75 inference(monotonicity,[status(thm)],[3])).
% 18.52/18.75 tff(14,plain,
% 18.52/18.75 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.75 inference(transitivity,[status(thm)],[13, 12])).
% 18.52/18.75 tff(15,plain,
% 18.52/18.75 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))),
% 18.52/18.75 inference(symmetry,[status(thm)],[14])).
% 18.52/18.75 tff(16,assumption,(tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))), introduced(assumption)).
% 18.52/18.75 tff(17,plain,
% 18.52/18.75 (tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)) = tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(E2!9))),
% 18.52/18.75 inference(monotonicity,[status(thm)],[16])).
% 18.52/18.75 tff(18,plain,
% 18.52/18.75 (tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(E2!9)) = tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))),
% 18.52/18.75 inference(symmetry,[status(thm)],[17])).
% 18.52/18.75 tff(19,plain,
% 18.52/18.75 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(E2!9))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))),
% 18.52/18.75 inference(monotonicity,[status(thm)],[18])).
% 18.52/18.75 tff(20,assumption,(head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))), introduced(assumption)).
% 18.52/18.75 tff(21,plain,
% 18.52/18.75 (tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(E2!9)) = tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.75 inference(monotonicity,[status(thm)],[20])).
% 18.52/18.75 tff(22,plain,
% 18.52/18.75 (tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))) = tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(E2!9))),
% 18.52/18.75 inference(symmetry,[status(thm)],[21])).
% 18.52/18.75 tff(23,plain,
% 18.52/18.75 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), head_of(E2!9)))),
% 18.52/18.75 inference(monotonicity,[status(thm)],[22])).
% 18.52/18.75 tff(24,plain,
% 18.52/18.75 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))),
% 18.52/18.75 inference(transitivity,[status(thm)],[23, 19])).
% 18.52/18.75 tff(25,plain,
% 18.52/18.75 (tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) = tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[24])).
% 18.52/18.76 tff(26,plain,
% 18.52/18.76 (tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))) = tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(symmetry,[status(thm)],[25])).
% 18.52/18.76 tff(27,plain,
% 18.52/18.76 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[26])).
% 18.52/18.76 tff(28,plain,
% 18.52/18.76 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))),
% 18.52/18.76 inference(symmetry,[status(thm)],[27])).
% 18.52/18.76 tff(29,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[28])).
% 18.52/18.76 tff(30,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))),
% 18.52/18.76 inference(transitivity,[status(thm)],[29, 15])).
% 18.52/18.76 tff(31,assumption,(~((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) | (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))))))), introduced(assumption)).
% 18.52/18.76 tff(32,plain,
% 18.52/18.76 (((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) | (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))))) | (~(((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))))))),
% 18.52/18.76 inference(tautology,[status(thm)],[])).
% 18.52/18.76 tff(33,plain,
% 18.52/18.76 (~(((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[32, 31])).
% 18.52/18.76 tff(34,plain,
% 18.52/18.76 (edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))) <=> edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[25])).
% 18.52/18.76 tff(35,plain,
% 18.52/18.76 (((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) | (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))))) | edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.76 inference(tautology,[status(thm)],[])).
% 18.52/18.76 tff(36,plain,
% 18.52/18.76 (edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[35, 31])).
% 18.52/18.76 tff(37,plain,
% 18.52/18.76 (edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[36, 34])).
% 18.52/18.76 tff(38,plain,
% 18.52/18.76 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(commutativity,[status(thm)],[])).
% 18.52/18.76 tff(39,plain,
% 18.52/18.76 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[20])).
% 18.52/18.76 tff(40,plain,
% 18.52/18.76 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(transitivity,[status(thm)],[39, 38])).
% 18.52/18.76 tff(41,plain,
% 18.52/18.76 ((tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))),
% 18.52/18.76 inference(symmetry,[status(thm)],[40])).
% 18.52/18.76 tff(42,plain,
% 18.52/18.76 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[24, 28])).
% 18.52/18.76 tff(43,plain,
% 18.52/18.76 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))),
% 18.52/18.76 inference(transitivity,[status(thm)],[42, 41])).
% 18.52/18.76 tff(44,assumption,(~((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))))), introduced(assumption)).
% 18.52/18.76 tff(45,plain,
% 18.52/18.76 (((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) | (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))),
% 18.52/18.76 inference(tautology,[status(thm)],[])).
% 18.52/18.76 tff(46,plain,
% 18.52/18.76 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[45, 44])).
% 18.52/18.76 tff(47,plain,
% 18.52/18.76 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[46, 43])).
% 18.52/18.76 tff(48,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8))) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(commutativity,[status(thm)],[])).
% 18.52/18.76 tff(49,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[3])).
% 18.52/18.76 tff(50,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(transitivity,[status(thm)],[49, 48])).
% 18.52/18.76 tff(51,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))),
% 18.52/18.76 inference(symmetry,[status(thm)],[50])).
% 18.52/18.76 tff(52,plain,
% 18.52/18.76 (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[26])).
% 18.52/18.76 tff(53,plain,
% 18.52/18.76 (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))),
% 18.52/18.76 inference(symmetry,[status(thm)],[52])).
% 18.52/18.76 tff(54,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.76 inference(monotonicity,[status(thm)],[53])).
% 18.52/18.76 tff(55,plain,
% 18.52/18.76 ((head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))),
% 18.52/18.76 inference(transitivity,[status(thm)],[54, 51])).
% 18.52/18.76 tff(56,plain,
% 18.52/18.76 (((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) | (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))),
% 18.52/18.76 inference(tautology,[status(thm)],[])).
% 18.52/18.76 tff(57,plain,
% 18.52/18.76 (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[56, 44])).
% 18.52/18.76 tff(58,plain,
% 18.52/18.76 (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[57, 55])).
% 18.52/18.76 tff(59,assumption,(edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))), introduced(assumption)).
% 18.52/18.76 tff(60,plain,
% 18.52/18.76 (^[E3: $i] : refl(((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10)))) <=> ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10)))))),
% 18.52/18.76 inference(bind,[status(th)],[])).
% 18.52/18.76 tff(61,plain,
% 18.52/18.76 (![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) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))),
% 18.52/18.76 inference(quant_intro,[status(thm)],[60])).
% 18.52/18.76 tff(62,plain,
% 18.52/18.76 (^[E3: $i] : trans(monotonicity(rewrite((edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))) <=> (~((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10)))))), ((~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10)))) <=> (~(~((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10)))))))), rewrite((~(~((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10)))))) <=> ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))), ((~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10)))) <=> ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))))),
% 18.52/18.76 inference(bind,[status(th)],[])).
% 18.52/18.76 tff(63,plain,
% 18.52/18.76 (![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) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))),
% 18.52/18.76 inference(quant_intro,[status(thm)],[62])).
% 18.52/18.76 tff(64,plain,
% 18.52/18.76 (((shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8)) & ![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) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10)))))),
% 18.52/18.76 inference(rewrite,[status(thm)],[])).
% 18.52/18.76 tff(65,plain,
% 18.52/18.76 ((~![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) = 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) = head_of(E2)) & (head_of(E3) = tail_of(E1)))))),
% 18.52/18.76 inference(rewrite,[status(thm)],[])).
% 18.52/18.76 tff(66,plain,
% 18.52/18.76 ((~(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) = 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) = head_of(E2)) & (head_of(E3) = tail_of(E1))))))),
% 18.52/18.76 inference(rewrite,[status(thm)],[])).
% 18.52/18.76 tff(67,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) = head_of(E2))) & (head_of(E3) = tail_of(E1)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','back_edge')).
% 18.52/18.76 tff(68,plain,
% 18.52/18.76 (~((~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) = head_of(E2)) & (head_of(E3) = tail_of(E1)))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[67, 66])).
% 18.52/18.76 tff(69,plain,
% 18.52/18.76 (~![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) = head_of(E2)) & (head_of(E3) = tail_of(E1))))),
% 18.52/18.76 inference(or_elim,[status(thm)],[68])).
% 18.52/18.76 tff(70,plain,
% 18.52/18.76 (~![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) = head_of(E2)) & (head_of(E3) = tail_of(E1))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[69, 65])).
% 18.52/18.76 tff(71,plain,
% 18.52/18.76 (~![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) = head_of(E2)) & (head_of(E3) = tail_of(E1))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[70, 65])).
% 18.52/18.76 tff(72,plain,
% 18.52/18.76 (~![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) = head_of(E2)) & (head_of(E3) = tail_of(E1))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[71, 65])).
% 18.52/18.76 tff(73,plain,
% 18.52/18.76 (shortest_path(V1!12, V2!11, P!8) & precedes(E1!10, E2!9, P!8) & ![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[72, 64])).
% 18.52/18.76 tff(74,plain,
% 18.52/18.76 (![E3: $i] : (~(edge(E3) & (tail_of(E3) = head_of(E2!9)) & (head_of(E3) = tail_of(E1!10))))),
% 18.52/18.76 inference(and_elim,[status(thm)],[73])).
% 18.52/18.76 tff(75,plain,
% 18.52/18.76 (![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[74, 63])).
% 18.52/18.76 tff(76,plain,
% 18.52/18.76 (![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[75, 61])).
% 18.52/18.76 tff(77,plain,
% 18.52/18.76 (((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))) <=> ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))) | (~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))),
% 18.52/18.76 inference(rewrite,[status(thm)],[])).
% 18.52/18.76 tff(78,plain,
% 18.52/18.76 ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))),
% 18.52/18.76 inference(quant_inst,[status(thm)],[])).
% 18.52/18.76 tff(79,plain,
% 18.52/18.76 ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = head_of(E2!9))) | (~(head_of(E3) = tail_of(E1!10))))) | (~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)))),
% 18.52/18.76 inference(modus_ponens,[status(thm)],[78, 77])).
% 18.52/18.76 tff(80,plain,
% 18.52/18.76 ((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[79, 76, 59])).
% 18.52/18.76 tff(81,plain,
% 18.52/18.76 ($false),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[80, 58, 47])).
% 18.52/18.76 tff(82,plain,(((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) | (~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(E1!10) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8)))) | (~(tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) | (~(head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))), inference(lemma,lemma(discharge,[]))).
% 18.52/18.76 tff(83,plain,
% 18.52/18.76 ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[82, 37, 3, 16, 20])).
% 18.52/18.76 tff(84,plain,
% 18.52/18.76 ((((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))))) | (~((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))))) | (~((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))))),
% 18.52/18.76 inference(tautology,[status(thm)],[])).
% 18.52/18.76 tff(85,plain,
% 18.52/18.76 (~((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))))),
% 18.52/18.76 inference(unit_resolution,[status(thm)],[84, 83, 33])).
% 18.52/18.76 tff(86,plain,
% 18.52/18.76 (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) | (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))),
% 18.52/18.77 inference(tautology,[status(thm)],[])).
% 18.52/18.77 tff(87,plain,
% 18.52/18.77 (head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.77 inference(unit_resolution,[status(thm)],[86, 85])).
% 18.52/18.77 tff(88,plain,
% 18.52/18.77 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[87, 30])).
% 18.52/18.77 tff(89,plain,
% 18.52/18.77 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.77 inference(commutativity,[status(thm)],[])).
% 18.52/18.77 tff(90,plain,
% 18.52/18.77 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.77 inference(monotonicity,[status(thm)],[20])).
% 18.52/18.77 tff(91,plain,
% 18.52/18.77 ((head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.77 inference(transitivity,[status(thm)],[90, 89])).
% 18.52/18.77 tff(92,plain,
% 18.52/18.77 ((tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))),
% 18.52/18.77 inference(symmetry,[status(thm)],[91])).
% 18.52/18.77 tff(93,plain,
% 18.52/18.77 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))),
% 18.52/18.77 inference(monotonicity,[status(thm)],[24, 53])).
% 18.52/18.77 tff(94,plain,
% 18.52/18.77 ((tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) <=> (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))),
% 18.52/18.77 inference(transitivity,[status(thm)],[93, 92])).
% 18.52/18.77 tff(95,plain,
% 18.52/18.77 (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) | (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))),
% 18.52/18.77 inference(tautology,[status(thm)],[])).
% 18.52/18.77 tff(96,plain,
% 18.52/18.77 (tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))),
% 18.52/18.77 inference(unit_resolution,[status(thm)],[95, 85])).
% 18.52/18.77 tff(97,plain,
% 18.52/18.77 (head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[96, 94])).
% 18.52/18.77 tff(98,plain,
% 18.52/18.77 (precedes(E1!10, E2!9, P!8)),
% 18.52/18.77 inference(and_elim,[status(thm)],[73])).
% 18.52/18.77 tff(99,plain,
% 18.52/18.77 (shortest_path(V1!12, V2!11, P!8)),
% 18.52/18.77 inference(and_elim,[status(thm)],[73])).
% 18.52/18.77 tff(100,plain,
% 18.52/18.77 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : refl(((~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))))) <=> ((~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))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(101,plain,
% 18.52/18.77 (![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))))) <=> ![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)))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[100])).
% 18.52/18.77 tff(102,plain,
% 18.52/18.77 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : rewrite(((~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))))) <=> ((~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))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(103,plain,
% 18.52/18.77 (![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))))) <=> ![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)))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[102])).
% 18.52/18.77 tff(104,plain,
% 18.52/18.77 (![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))))) <=> ![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)))))),
% 18.52/18.77 inference(transitivity,[status(thm)],[103, 101])).
% 18.52/18.77 tff(105,plain,
% 18.52/18.77 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : trans(monotonicity(trans(monotonicity(rewrite((shortest_path(V1, V2, P) & precedes(E1, E2, P)) <=> (~((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P))))), ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) <=> (~(~((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P))))))), rewrite((~(~((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P))))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)))), ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P))))), quant_intro(proof_bind(^[E3: $i] : trans(monotonicity(rewrite((edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> (~((~edge(E3)) | (~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))), ((~(edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> (~(~((~edge(E3)) | (~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))))), rewrite((~(~((~edge(E3)) | (~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))) <=> ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))), ((~(edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))))), (![E3: $i] : (~(edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> ![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))), (((~(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))))) <=> (((~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))))))), rewrite((((~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))))) <=> ((~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)))))), (((~(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))))) <=> ((~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)))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(106,plain,
% 18.52/18.77 (![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))))) <=> ![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)))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[105])).
% 18.52/18.77 tff(107,plain,
% 18.52/18.77 (![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))))) <=> ![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)))))),
% 18.52/18.77 inference(rewrite,[status(thm)],[])).
% 18.52/18.77 tff(108,plain,
% 18.52/18.77 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : trans(monotonicity(rewrite((~?[E3: $i] : ((edge(E3) & (tail_of(E3) = tail_of(E1))) & (head_of(E3) = head_of(E2)))) <=> (~?[E3: $i] : (edge(E3) & (tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))))), (((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))))) <=> ((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))))))), rewrite(((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))))) <=> ((~(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)))))), (((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))))) <=> ((~(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)))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(109,plain,
% 18.52/18.77 (![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))))) <=> ![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)))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[108])).
% 18.52/18.77 tff(110,axiom,(![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)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','no_short_cut_edge')).
% 18.52/18.77 tff(111,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[110, 109])).
% 18.52/18.77 tff(112,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[111, 107])).
% 18.52/18.77 tff(113,plain,(
% 18.52/18.77 ![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)))))),
% 18.52/18.77 inference(skolemize,[status(sab)],[112])).
% 18.52/18.77 tff(114,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[113, 106])).
% 18.52/18.77 tff(115,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[114, 104])).
% 18.52/18.77 tff(116,plain,
% 18.52/18.77 (((~![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) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))) | (~precedes(E1!10, E2!9, P!8)) | (~shortest_path(V1!12, V2!11, P!8)))) <=> ((~![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) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))) | (~precedes(E1!10, E2!9, P!8)) | (~shortest_path(V1!12, V2!11, P!8)))),
% 18.52/18.77 inference(rewrite,[status(thm)],[])).
% 18.52/18.77 tff(117,plain,
% 18.52/18.77 (((~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))))) <=> (![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))) | (~precedes(E1!10, E2!9, P!8)) | (~shortest_path(V1!12, V2!11, P!8)))),
% 18.52/18.77 inference(rewrite,[status(thm)],[])).
% 18.52/18.77 tff(118,plain,
% 18.52/18.77 (((~![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)))))) | ((~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)))))) <=> ((~![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) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))) | (~precedes(E1!10, E2!9, P!8)) | (~shortest_path(V1!12, V2!11, P!8))))),
% 18.52/18.77 inference(monotonicity,[status(thm)],[117])).
% 18.52/18.77 tff(119,plain,
% 18.52/18.77 (((~![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)))))) | ((~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)))))) <=> ((~![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) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))) | (~precedes(E1!10, E2!9, P!8)) | (~shortest_path(V1!12, V2!11, P!8)))),
% 18.52/18.77 inference(transitivity,[status(thm)],[118, 116])).
% 18.52/18.77 tff(120,plain,
% 18.52/18.77 ((~![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)))))) | ((~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)))))),
% 18.52/18.77 inference(quant_inst,[status(thm)],[])).
% 18.52/18.77 tff(121,plain,
% 18.52/18.77 ((~![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) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))) | (~precedes(E1!10, E2!9, P!8)) | (~shortest_path(V1!12, V2!11, P!8))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[120, 119])).
% 18.52/18.77 tff(122,plain,
% 18.52/18.77 (![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))),
% 18.52/18.77 inference(unit_resolution,[status(thm)],[121, 115, 99, 98])).
% 18.52/18.77 tff(123,plain,
% 18.52/18.77 (((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))) <=> ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | (~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))),
% 18.52/18.77 inference(rewrite,[status(thm)],[])).
% 18.52/18.77 tff(124,plain,
% 18.52/18.77 (((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9)))) <=> ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))),
% 18.52/18.77 inference(rewrite,[status(thm)],[])).
% 18.52/18.77 tff(125,plain,
% 18.52/18.77 (((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))))) <=> ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)))))),
% 18.52/18.77 inference(monotonicity,[status(thm)],[124])).
% 18.52/18.77 tff(126,plain,
% 18.52/18.77 (((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))))) <=> ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | (~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))))),
% 18.52/18.77 inference(transitivity,[status(thm)],[125, 123])).
% 18.52/18.77 tff(127,plain,
% 18.52/18.77 ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))))),
% 18.52/18.77 inference(quant_inst,[status(thm)],[])).
% 18.52/18.77 tff(128,plain,
% 18.52/18.77 ((~![E3: $i] : ((~edge(E3)) | (~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | (~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))) | (~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[127, 126])).
% 18.52/18.77 tff(129,plain,
% 18.52/18.77 ((~(head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(E2!9))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(E1!10)))),
% 18.52/18.77 inference(unit_resolution,[status(thm)],[128, 122, 37])).
% 18.52/18.77 tff(130,plain,
% 18.52/18.77 ($false),
% 18.52/18.77 inference(unit_resolution,[status(thm)],[129, 97, 88])).
% 18.52/18.77 tff(131,plain,(((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) | (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))))) | (~(head_of(E2!9) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) | (~(tail_of(E2!9) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) | (~(tail_of(E1!10) = head_of(tptp_fun_E_3(tail_of(E1!10), P!8))))), inference(lemma,lemma(discharge,[]))).
% 18.52/18.77 tff(132,plain,
% 18.52/18.77 ((~edge(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))) | (((~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))) <=> ((~(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)) = head_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))))))) | (~(tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) = tail_of(tptp_fun_E_0(head_of(tptp_fun_E_3(tail_of(E1!10), P!8)), tail_of(tptp_fun_E_0(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))), tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))))))))),
% 18.52/18.77 inference(unit_resolution,[status(thm)],[131, 11, 9, 3])).
% 18.52/18.77 tff(133,plain,
% 18.52/18.77 ((tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))) <=> (head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.77 inference(commutativity,[status(thm)],[])).
% 18.52/18.77 tff(134,plain,
% 18.52/18.77 ((head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))) <=> (tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))),
% 18.52/18.77 inference(symmetry,[status(thm)],[133])).
% 18.52/18.77 tff(135,plain,
% 18.52/18.77 ((~(head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))))) <=> (~(tail_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9))) = head_of(tptp_fun_E_0(tail_of(E2!9), head_of(E2!9)))))),
% 18.52/18.77 inference(monotonicity,[status(thm)],[134])).
% 18.52/18.77 tff(136,plain,
% 18.52/18.77 (^[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)))))))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(137,plain,
% 18.52/18.77 (![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))))))))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[136])).
% 18.52/18.77 tff(138,plain,
% 18.52/18.77 (^[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))))))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(139,plain,
% 18.52/18.77 (![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)))))))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[138])).
% 18.52/18.77 tff(140,plain,
% 18.52/18.77 (^[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))))))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(141,plain,
% 18.52/18.77 (![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)))))))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[140])).
% 18.52/18.77 tff(142,plain,
% 18.52/18.77 (![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)))))))))),
% 18.52/18.77 inference(transitivity,[status(thm)],[141, 139])).
% 18.52/18.77 tff(143,plain,
% 18.52/18.77 (^[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)))))))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(144,plain,
% 18.52/18.77 (![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)))))))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[143])).
% 18.52/18.77 tff(145,plain,
% 18.52/18.77 (^[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))))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(146,plain,
% 18.52/18.77 (![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)))))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[145])).
% 18.52/18.77 tff(147,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(rewrite,[status(thm)],[])).
% 18.52/18.77 tff(148,plain,
% 18.52/18.77 (^[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))))))),
% 18.52/18.77 inference(bind,[status(th)],[])).
% 18.52/18.77 tff(149,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(quant_intro,[status(thm)],[148])).
% 18.52/18.77 tff(150,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')).
% 18.52/18.77 tff(151,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[150, 149])).
% 18.52/18.77 tff(152,plain,
% 18.52/18.77 (![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)))))),
% 18.52/18.77 inference(modus_ponens,[status(thm)],[151, 147])).
% 18.52/18.77 tff(153,plain,(
% 18.52/18.77 ![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))))))))),
% 18.61/18.78 inference(skolemize,[status(sab)],[152])).
% 18.61/18.78 tff(154,plain,
% 18.61/18.78 (![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)))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[153, 146])).
% 18.61/18.78 tff(155,plain,
% 18.61/18.78 (![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)))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[154, 144])).
% 18.61/18.78 tff(156,plain,
% 18.61/18.78 (![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)))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[155, 142])).
% 18.61/18.78 tff(157,plain,
% 18.61/18.78 (![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))))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[156, 137])).
% 18.61/18.78 tff(158,plain,
% 18.61/18.78 (((~![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)))))))))),
% 18.61/18.78 inference(rewrite,[status(thm)],[])).
% 18.61/18.78 tff(159,plain,
% 18.61/18.78 ((~((~((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))))))))),
% 18.61/18.78 inference(rewrite,[status(thm)],[])).
% 18.61/18.78 tff(160,plain,
% 18.61/18.78 (((~![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)))))))))),
% 18.61/18.78 inference(monotonicity,[status(thm)],[159])).
% 18.61/18.78 tff(161,plain,
% 18.61/18.78 (((~![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)))))))))),
% 18.61/18.78 inference(transitivity,[status(thm)],[160, 158])).
% 18.61/18.78 tff(162,plain,
% 18.61/18.78 ((~![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))))))))))),
% 18.61/18.78 inference(quant_inst,[status(thm)],[])).
% 18.61/18.78 tff(163,plain,
% 18.61/18.78 ((~![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))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[162, 161])).
% 18.61/18.78 tff(164,plain,
% 18.61/18.78 (~((~(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)))))))),
% 18.61/18.78 inference(unit_resolution,[status(thm)],[163, 157])).
% 18.61/18.78 tff(165,plain,
% 18.61/18.78 (((~(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)))))),
% 18.61/18.78 inference(tautology,[status(thm)],[])).
% 18.61/18.78 tff(166,plain,
% 18.61/18.78 ((~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))))),
% 18.61/18.78 inference(unit_resolution,[status(thm)],[165, 164])).
% 18.61/18.78 tff(167,plain,
% 18.61/18.78 ((~((~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))))),
% 18.61/18.78 inference(tautology,[status(thm)],[])).
% 18.61/18.78 tff(168,plain,
% 18.61/18.78 ((~((~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))))),
% 18.61/18.78 inference(unit_resolution,[status(thm)],[167, 99])).
% 18.61/18.78 tff(169,plain,
% 18.61/18.78 (~((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)))),
% 18.61/18.78 inference(unit_resolution,[status(thm)],[168, 166])).
% 18.61/18.78 tff(170,plain,
% 18.61/18.78 (((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)),
% 18.61/18.78 inference(tautology,[status(thm)],[])).
% 18.61/18.78 tff(171,plain,
% 18.61/18.78 (path(V1!12, V2!11, P!8)),
% 18.61/18.78 inference(unit_resolution,[status(thm)],[170, 169])).
% 18.61/18.78 tff(172,plain,
% 18.61/18.78 (^[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))))))))))),
% 18.61/18.78 inference(bind,[status(th)],[])).
% 18.61/18.78 tff(173,plain,
% 18.61/18.78 (![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)))))))))),
% 18.61/18.78 inference(quant_intro,[status(thm)],[172])).
% 18.61/18.78 tff(174,plain,
% 18.61/18.78 (^[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))))))))))),
% 18.61/18.78 inference(bind,[status(th)],[])).
% 18.61/18.78 tff(175,plain,
% 18.61/18.78 (![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)))))))))),
% 18.61/18.78 inference(quant_intro,[status(thm)],[174])).
% 18.61/18.78 tff(176,plain,
% 18.61/18.78 (^[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))))))))),
% 18.61/18.78 inference(bind,[status(th)],[])).
% 18.61/18.78 tff(177,plain,
% 18.61/18.78 (![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)))))))),
% 18.61/18.78 inference(quant_intro,[status(thm)],[176])).
% 18.61/18.78 tff(178,plain,
% 18.61/18.78 (![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))))))))),
% 18.61/18.78 inference(rewrite,[status(thm)],[])).
% 18.61/18.78 tff(179,plain,
% 18.61/18.78 (^[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))))))))))),
% 18.61/18.78 inference(bind,[status(th)],[])).
% 18.61/18.78 tff(180,plain,
% 18.61/18.78 (![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))))))))),
% 18.61/18.78 inference(quant_intro,[status(thm)],[179])).
% 18.61/18.78 tff(181,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')).
% 18.61/18.78 tff(182,plain,
% 18.61/18.78 (![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))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[181, 180])).
% 18.61/18.78 tff(183,plain,
% 18.61/18.78 (![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))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[182, 178])).
% 18.61/18.78 tff(184,plain,(
% 18.61/18.78 ![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)))))))))),
% 18.61/18.78 inference(skolemize,[status(sab)],[183])).
% 18.61/18.78 tff(185,plain,
% 18.61/18.78 (![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)))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[184, 177])).
% 18.61/18.78 tff(186,plain,
% 18.61/18.78 (![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)))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[185, 175])).
% 18.61/18.78 tff(187,plain,
% 18.61/18.78 (![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)))))))))),
% 18.61/18.78 inference(modus_ponens,[status(thm)],[186, 173])).
% 18.61/18.78 tff(188,plain,
% 18.61/18.78 (((~![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)))))))))),
% 18.61/18.79 inference(rewrite,[status(thm)],[])).
% 18.61/18.79 tff(189,plain,
% 18.61/18.79 (((~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)))))))))),
% 18.61/18.79 inference(rewrite,[status(thm)],[])).
% 18.61/18.79 tff(190,plain,
% 18.61/18.79 (((~![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))))))))))),
% 18.61/18.79 inference(monotonicity,[status(thm)],[189])).
% 18.61/18.79 tff(191,plain,
% 18.61/18.79 (((~![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)))))))))),
% 18.61/18.79 inference(transitivity,[status(thm)],[190, 188])).
% 18.61/18.79 tff(192,plain,
% 18.61/18.79 ((~![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)))))))))),
% 18.61/18.79 inference(quant_inst,[status(thm)],[])).
% 18.61/18.79 tff(193,plain,
% 18.61/18.79 ((~![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))))))))),
% 18.61/18.79 inference(modus_ponens,[status(thm)],[192, 191])).
% 18.61/18.79 tff(194,plain,
% 18.61/18.79 (~((~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)))))))),
% 18.61/18.79 inference(unit_resolution,[status(thm)],[193, 187, 171])).
% 18.61/18.79 tff(195,plain,
% 18.61/18.79 (((~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)))),
% 18.61/18.79 inference(tautology,[status(thm)],[])).
% 18.61/18.79 tff(196,plain,
% 18.61/18.79 (V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12))),
% 18.61/18.79 inference(unit_resolution,[status(thm)],[195, 194])).
% 18.61/18.79 tff(197,plain,
% 18.61/18.79 (tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)) = V1!12),
% 18.61/18.79 inference(symmetry,[status(thm)],[196])).
% 18.61/18.79 tff(198,plain,
% 18.61/18.79 (path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8) <=> path(V1!12, V2!11, P!8)),
% 18.61/18.79 inference(monotonicity,[status(thm)],[197])).
% 18.61/18.79 tff(199,plain,
% 18.61/18.79 (path(V1!12, V2!11, P!8) <=> path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)),
% 18.61/18.79 inference(symmetry,[status(thm)],[198])).
% 18.61/18.79 tff(200,plain,
% 18.61/18.79 (path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)),
% 18.61/18.79 inference(modus_ponens,[status(thm)],[171, 199])).
% 18.61/18.79 tff(201,plain,
% 18.61/18.79 (^[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))))))))))),
% 18.61/18.79 inference(bind,[status(th)],[])).
% 18.61/18.79 tff(202,plain,
% 18.61/18.79 (![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)))))))))),
% 18.61/18.79 inference(quant_intro,[status(thm)],[201])).
% 18.61/18.79 tff(203,plain,
% 18.61/18.79 (^[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))))))))))),
% 18.61/18.79 inference(bind,[status(th)],[])).
% 18.61/18.79 tff(204,plain,
% 18.61/18.79 (![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)))))))))),
% 18.61/18.79 inference(quant_intro,[status(thm)],[203])).
% 18.61/18.79 tff(205,plain,
% 18.61/18.79 (![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)))))))))),
% 18.61/18.79 inference(transitivity,[status(thm)],[204, 202])).
% 18.61/18.79 tff(206,plain,
% 18.61/18.79 (^[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))))))))))),
% 18.61/18.79 inference(bind,[status(th)],[])).
% 18.61/18.79 tff(207,plain,
% 18.61/18.79 (![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)))))))))),
% 18.61/18.79 inference(quant_intro,[status(thm)],[206])).
% 18.61/18.79 tff(208,plain,
% 18.61/18.79 (^[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))))))))),
% 18.61/18.79 inference(bind,[status(th)],[])).
% 18.61/18.79 tff(209,plain,
% 18.61/18.79 (![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)))))))),
% 18.61/18.79 inference(quant_intro,[status(thm)],[208])).
% 18.61/18.79 tff(210,plain,
% 18.61/18.79 (![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)))))))),
% 18.61/18.79 inference(rewrite,[status(thm)],[])).
% 18.61/18.79 tff(211,plain,
% 18.61/18.79 (^[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)))))))))),
% 18.63/18.79 inference(bind,[status(th)],[])).
% 18.63/18.79 tff(212,plain,
% 18.63/18.79 (![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)))))))),
% 18.63/18.79 inference(quant_intro,[status(thm)],[211])).
% 18.63/18.79 tff(213,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')).
% 18.63/18.79 tff(214,plain,
% 18.63/18.79 (![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)))))))),
% 18.63/18.79 inference(modus_ponens,[status(thm)],[213, 212])).
% 18.63/18.79 tff(215,plain,
% 18.63/18.79 (![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)))))))),
% 18.63/18.79 inference(modus_ponens,[status(thm)],[214, 210])).
% 18.63/18.79 tff(216,plain,(
% 18.63/18.79 ![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))))))))),
% 18.63/18.79 inference(skolemize,[status(sab)],[215])).
% 18.63/18.79 tff(217,plain,
% 18.63/18.79 (![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)))))))),
% 18.63/18.79 inference(modus_ponens,[status(thm)],[216, 209])).
% 18.63/18.79 tff(218,plain,
% 18.63/18.79 (![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)))))))))),
% 18.63/18.79 inference(modus_ponens,[status(thm)],[217, 207])).
% 18.63/18.79 tff(219,plain,
% 18.63/18.79 (![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)))))))))),
% 18.63/18.79 inference(modus_ponens,[status(thm)],[218, 205])).
% 18.63/18.79 tff(220,plain,
% 18.63/18.79 (((~![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))))))),
% 18.63/18.79 inference(rewrite,[status(thm)],[])).
% 18.63/18.79 tff(221,plain,
% 18.63/18.79 (((~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))))))),
% 18.63/18.79 inference(rewrite,[status(thm)],[])).
% 18.63/18.79 tff(222,plain,
% 18.63/18.79 (((~![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)))))))),
% 18.63/18.79 inference(monotonicity,[status(thm)],[221])).
% 18.63/18.79 tff(223,plain,
% 18.63/18.79 (((~![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))))))),
% 18.63/18.79 inference(transitivity,[status(thm)],[222, 220])).
% 18.63/18.79 tff(224,plain,
% 18.63/18.79 ((~![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)))))))))),
% 18.63/18.80 inference(quant_inst,[status(thm)],[])).
% 18.63/18.80 tff(225,plain,
% 18.63/18.80 ((~![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)))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[224, 223])).
% 18.63/18.80 tff(226,plain,
% 18.63/18.80 (![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)))))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[225, 219, 171])).
% 18.63/18.80 tff(227,plain,
% 18.63/18.80 (((~![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))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(228,plain,
% 18.63/18.80 (((~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))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(229,plain,
% 18.63/18.80 (((~![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)))))))),
% 18.63/18.80 inference(monotonicity,[status(thm)],[228])).
% 18.63/18.80 tff(230,plain,
% 18.63/18.80 (((~![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))))))),
% 18.63/18.80 inference(transitivity,[status(thm)],[229, 227])).
% 18.63/18.80 tff(231,plain,
% 18.63/18.80 ((~![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)))))),
% 18.63/18.80 inference(quant_inst,[status(thm)],[])).
% 18.63/18.80 tff(232,plain,
% 18.63/18.80 ((~![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)))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[231, 230])).
% 18.63/18.80 tff(233,plain,
% 18.63/18.80 (~((~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))))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[232, 98, 226])).
% 18.63/18.80 tff(234,plain,
% 18.63/18.80 (((~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(E2!9, P!8)),
% 18.63/18.80 inference(tautology,[status(thm)],[])).
% 18.63/18.80 tff(235,plain,
% 18.63/18.80 (on_path(E2!9, P!8)),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[234, 233])).
% 18.63/18.80 tff(236,plain,
% 18.63/18.80 (^[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))))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(237,plain,
% 18.63/18.80 (![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)))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[236])).
% 18.63/18.80 tff(238,plain,
% 18.63/18.80 (^[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)))))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(239,plain,
% 18.63/18.80 (![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)))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[238])).
% 18.63/18.80 tff(240,plain,
% 18.63/18.80 (![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)))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(241,plain,
% 18.63/18.80 (^[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)))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(242,plain,
% 18.63/18.80 (![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)))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[241])).
% 18.63/18.80 tff(243,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')).
% 18.63/18.80 tff(244,plain,
% 18.63/18.80 (![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)))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[243, 242])).
% 18.63/18.80 tff(245,plain,
% 18.63/18.80 (![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)))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[244, 240])).
% 18.63/18.80 tff(246,plain,(
% 18.63/18.80 ![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)))),
% 18.63/18.80 inference(skolemize,[status(sab)],[245])).
% 18.63/18.80 tff(247,plain,
% 18.63/18.80 (![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)))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[246, 239])).
% 18.63/18.80 tff(248,plain,
% 18.63/18.80 (![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)))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[247, 237])).
% 18.63/18.80 tff(249,plain,
% 18.63/18.80 (((~![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(E2!9, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), 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(E2!9, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8)))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(250,plain,
% 18.63/18.80 ((~![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(E2!9, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8)))))),
% 18.63/18.80 inference(quant_inst,[status(thm)],[])).
% 18.63/18.80 tff(251,plain,
% 18.63/18.80 ((~![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(E2!9, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[250, 249])).
% 18.63/18.80 tff(252,plain,
% 18.63/18.80 ((~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8))))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[251, 248, 235])).
% 18.63/18.80 tff(253,plain,
% 18.63/18.80 (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8)))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[252, 200])).
% 18.63/18.80 tff(254,plain,
% 18.63/18.80 (((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8))) | edge(E2!9)),
% 18.63/18.80 inference(tautology,[status(thm)],[])).
% 18.63/18.80 tff(255,plain,
% 18.63/18.80 (edge(E2!9)),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[254, 253])).
% 18.63/18.80 tff(256,plain,
% 18.63/18.80 (^[E: $i] : refl(((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))) <=> ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(257,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))) <=> ![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[256])).
% 18.63/18.80 tff(258,plain,
% 18.63/18.80 (^[E: $i] : rewrite(((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(259,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[258])).
% 18.63/18.80 tff(260,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(261,plain,
% 18.63/18.80 (^[E: $i] : rewrite((edge(E) => (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(262,plain,
% 18.63/18.80 (![E: $i] : (edge(E) => (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[261])).
% 18.63/18.80 tff(263,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')).
% 18.63/18.80 tff(264,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[263, 262])).
% 18.63/18.80 tff(265,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[264, 260])).
% 18.63/18.80 tff(266,plain,(
% 18.63/18.80 ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 18.63/18.80 inference(skolemize,[status(sab)],[265])).
% 18.63/18.80 tff(267,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[266, 259])).
% 18.63/18.80 tff(268,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[267, 257])).
% 18.63/18.80 tff(269,plain,
% 18.63/18.80 (((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | ((~edge(E2!9)) | (~((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9))))))) <=> ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | (~edge(E2!9)) | (~((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(270,plain,
% 18.63/18.80 ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | ((~edge(E2!9)) | (~((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9))))))),
% 18.63/18.80 inference(quant_inst,[status(thm)],[])).
% 18.63/18.80 tff(271,plain,
% 18.63/18.80 ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | (~edge(E2!9)) | (~((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9)))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[270, 269])).
% 18.63/18.80 tff(272,plain,
% 18.63/18.80 (~((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9))))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[271, 268, 255])).
% 18.63/18.80 tff(273,plain,
% 18.63/18.80 (((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9)))) | vertex(tail_of(E2!9))),
% 18.63/18.80 inference(tautology,[status(thm)],[])).
% 18.63/18.80 tff(274,plain,
% 18.63/18.80 (vertex(tail_of(E2!9))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[273, 272])).
% 18.63/18.80 tff(275,plain,
% 18.63/18.80 (((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9)))) | vertex(head_of(E2!9))),
% 18.63/18.80 inference(tautology,[status(thm)],[])).
% 18.63/18.80 tff(276,plain,
% 18.63/18.80 (vertex(head_of(E2!9))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[275, 272])).
% 18.63/18.80 tff(277,plain,
% 18.63/18.80 (^[E: $i] : refl(((~edge(E)) | (~(head_of(E) = tail_of(E)))) <=> ((~edge(E)) | (~(head_of(E) = tail_of(E)))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(278,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[277])).
% 18.63/18.80 tff(279,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(280,plain,
% 18.63/18.80 (^[E: $i] : rewrite((edge(E) => (~(head_of(E) = tail_of(E)))) <=> ((~edge(E)) | (~(head_of(E) = tail_of(E)))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(281,plain,
% 18.63/18.80 (![E: $i] : (edge(E) => (~(head_of(E) = tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[280])).
% 18.63/18.80 tff(282,axiom,(![E: $i] : (edge(E) => (~(head_of(E) = tail_of(E))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','no_loops')).
% 18.63/18.80 tff(283,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[282, 281])).
% 18.63/18.80 tff(284,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[283, 279])).
% 18.63/18.80 tff(285,plain,(
% 18.63/18.80 ![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(skolemize,[status(sab)],[284])).
% 18.63/18.80 tff(286,plain,
% 18.63/18.80 (![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[285, 278])).
% 18.63/18.80 tff(287,plain,
% 18.63/18.80 (((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | ((~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E2!9))))) <=> ((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E2!9))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(288,plain,
% 18.63/18.80 ((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | ((~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E2!9))))),
% 18.63/18.80 inference(quant_inst,[status(thm)],[])).
% 18.63/18.80 tff(289,plain,
% 18.63/18.80 ((~![E: $i] : ((~edge(E)) | (~(head_of(E) = tail_of(E))))) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E2!9)))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[288, 287])).
% 18.63/18.80 tff(290,plain,
% 18.63/18.80 (~(head_of(E2!9) = tail_of(E2!9))),
% 18.63/18.80 inference(unit_resolution,[status(thm)],[289, 286, 255])).
% 18.63/18.80 tff(291,plain,
% 18.63/18.80 (complete <=> complete),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(292,plain,
% 18.63/18.80 (complete),
% 18.63/18.80 inference(or_elim,[status(thm)],[68])).
% 18.63/18.80 tff(293,plain,
% 18.63/18.80 (complete),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[292, 291])).
% 18.63/18.80 tff(294,plain,
% 18.63/18.80 (((~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))))))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(295,plain,
% 18.63/18.80 (((~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))))))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(296,plain,
% 18.63/18.80 (((~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)))))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(297,plain,
% 18.63/18.80 (((~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))))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(298,plain,
% 18.63/18.80 ((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))))))))),
% 18.63/18.80 inference(rewrite,[status(thm)],[])).
% 18.63/18.80 tff(299,plain,
% 18.63/18.80 (^[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)))))))))),
% 18.63/18.80 inference(bind,[status(th)],[])).
% 18.63/18.80 tff(300,plain,
% 18.63/18.80 (![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)))))))),
% 18.63/18.80 inference(quant_intro,[status(thm)],[299])).
% 18.63/18.80 tff(301,plain,
% 18.63/18.80 ((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))))))))),
% 18.63/18.80 inference(monotonicity,[status(thm)],[300])).
% 18.63/18.80 tff(302,plain,
% 18.63/18.80 ((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))))))))),
% 18.63/18.80 inference(transitivity,[status(thm)],[301, 298])).
% 18.63/18.80 tff(303,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')).
% 18.63/18.80 tff(304,plain,
% 18.63/18.80 ((~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)))))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[303, 302])).
% 18.63/18.80 tff(305,plain,
% 18.63/18.80 ((~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)))))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[304, 297])).
% 18.63/18.80 tff(306,plain,
% 18.63/18.80 ((~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)))))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[305, 297])).
% 18.63/18.80 tff(307,plain,
% 18.63/18.80 ((~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)))))))),
% 18.63/18.80 inference(modus_ponens,[status(thm)],[306, 297])).
% 18.63/18.80 Proof display could not be completed: monotonicity rule is not handled
% 18.66/18.84 % E exiting
%------------------------------------------------------------------------------