%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRA007+1 : TPTP v9.0.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:06:36 AM UTC 2025
% Result : Theorem 10.50s 10.70s
% Output : Assurance 0s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.11 % Problem : GRA007+1 : TPTP v9.0.0. Bugfixed v3.2.0.
% 0.03/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n008.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 15:22:18 EDT 2025
% 0.11/0.32 % CPUTime :
% 10.50/10.70 % SZS status Theorem
% 10.50/10.70 % SZS output start Proof
% 10.50/10.70 tff(head_of_type, type, (
% 10.50/10.70 head_of: $i > $i)).
% 10.50/10.70 tff(tptp_fun_E2_9_type, type, (
% 10.50/10.70 tptp_fun_E2_9: $i)).
% 10.50/10.70 tff(tail_of_type, type, (
% 10.50/10.70 tail_of: $i > $i)).
% 10.50/10.70 tff(tptp_fun_E_0_type, type, (
% 10.50/10.70 tptp_fun_E_0: ( $i * $i ) > $i)).
% 10.50/10.70 tff(tptp_fun_E1_10_type, type, (
% 10.50/10.70 tptp_fun_E1_10: $i)).
% 10.50/10.70 tff(edge_type, type, (
% 10.50/10.70 edge: $i > $o)).
% 10.50/10.70 tff(precedes_type, type, (
% 10.50/10.70 precedes: ( $i * $i * $i ) > $o)).
% 10.50/10.70 tff(tptp_fun_P_8_type, type, (
% 10.50/10.70 tptp_fun_P_8: $i)).
% 10.50/10.70 tff(shortest_path_type, type, (
% 10.50/10.70 shortest_path: ( $i * $i * $i ) > $o)).
% 10.50/10.70 tff(tptp_fun_V2_11_type, type, (
% 10.50/10.70 tptp_fun_V2_11: $i)).
% 10.50/10.70 tff(tptp_fun_V1_12_type, type, (
% 10.50/10.70 tptp_fun_V1_12: $i)).
% 10.50/10.70 tff(complete_type, type, (
% 10.50/10.70 complete: $o)).
% 10.50/10.70 tff(sequential_type, type, (
% 10.50/10.70 sequential: ( $i * $i ) > $o)).
% 10.50/10.70 tff(on_path_type, type, (
% 10.50/10.70 on_path: ( $i * $i ) > $o)).
% 10.50/10.70 tff(tptp_fun_E3_4_type, type, (
% 10.50/10.70 tptp_fun_E3_4: ( $i * $i * $i ) > $i)).
% 10.50/10.70 tff(path_type, type, (
% 10.50/10.70 path: ( $i * $i * $i ) > $o)).
% 10.50/10.70 tff(less_or_equal_type, type, (
% 10.50/10.70 less_or_equal: ( $i * $i ) > $o)).
% 10.50/10.70 tff(length_of_type, type, (
% 10.50/10.70 length_of: $i > $i)).
% 10.50/10.70 tff(tptp_fun_P_5_type, type, (
% 10.50/10.70 tptp_fun_P_5: ( $i * $i * $i ) > $i)).
% 10.50/10.70 tff(in_path_type, type, (
% 10.50/10.70 in_path: ( $i * $i ) > $o)).
% 10.50/10.70 tff(tptp_fun_E_1_type, type, (
% 10.50/10.70 tptp_fun_E_1: ( $i * $i * $i ) > $i)).
% 10.50/10.70 tff(path_cons_type, type, (
% 10.50/10.70 path_cons: ( $i * $i ) > $i)).
% 10.50/10.70 tff(empty_type, type, (
% 10.50/10.70 empty: $i)).
% 10.50/10.70 tff(tptp_fun_TP_2_type, type, (
% 10.50/10.70 tptp_fun_TP_2: ( $i * $i * $i ) > $i)).
% 10.50/10.70 tff(vertex_type, type, (
% 10.50/10.70 vertex: $i > $o)).
% 10.50/10.70 tff(1,plain,
% 10.50/10.70 ((tail_of(E1!10) = head_of(E2!9)) <=> (head_of(E2!9) = tail_of(E1!10))),
% 10.50/10.70 inference(commutativity,[status(thm)],[])).
% 10.50/10.70 tff(2,plain,
% 10.50/10.70 ((head_of(E2!9) = tail_of(E1!10)) <=> (tail_of(E1!10) = head_of(E2!9))),
% 10.50/10.70 inference(symmetry,[status(thm)],[1])).
% 10.50/10.70 tff(3,plain,
% 10.50/10.70 ((~(head_of(E2!9) = tail_of(E1!10))) <=> (~(tail_of(E1!10) = head_of(E2!9)))),
% 10.50/10.70 inference(monotonicity,[status(thm)],[2])).
% 10.50/10.70 tff(4,assumption,(E2!9 = E1!10), introduced(assumption)).
% 10.50/10.70 tff(5,plain,
% 10.50/10.70 (E1!10 = E2!9),
% 10.50/10.70 inference(symmetry,[status(thm)],[4])).
% 10.50/10.70 tff(6,plain,
% 10.50/10.70 (precedes(E2!9, E1!10, P!8) <=> precedes(E1!10, E2!9, P!8)),
% 10.50/10.70 inference(monotonicity,[status(thm)],[4, 5])).
% 10.50/10.70 tff(7,plain,
% 10.50/10.70 (precedes(E1!10, E2!9, P!8) <=> precedes(E2!9, E1!10, P!8)),
% 10.50/10.70 inference(symmetry,[status(thm)],[6])).
% 10.50/10.70 tff(8,plain,
% 10.50/10.70 (((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)))))),
% 10.50/10.70 inference(rewrite,[status(thm)],[])).
% 10.50/10.70 tff(9,plain,
% 10.50/10.70 ((~![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)))))),
% 10.50/10.70 inference(rewrite,[status(thm)],[])).
% 10.50/10.70 tff(10,plain,
% 10.50/10.70 ((~(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))))))),
% 10.50/10.70 inference(rewrite,[status(thm)],[])).
% 10.50/10.70 tff(11,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')).
% 10.50/10.70 tff(12,plain,
% 10.50/10.70 (~((~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)))))),
% 10.50/10.70 inference(modus_ponens,[status(thm)],[11, 10])).
% 10.50/10.70 tff(13,plain,
% 10.50/10.70 (~![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))))),
% 10.50/10.70 inference(or_elim,[status(thm)],[12])).
% 10.50/10.70 tff(14,plain,
% 10.50/10.70 (~![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))))),
% 10.50/10.70 inference(modus_ponens,[status(thm)],[13, 9])).
% 10.50/10.70 tff(15,plain,
% 10.50/10.70 (~![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))))),
% 10.50/10.70 inference(modus_ponens,[status(thm)],[14, 9])).
% 10.50/10.70 tff(16,plain,
% 10.50/10.70 (~![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))))),
% 10.50/10.70 inference(modus_ponens,[status(thm)],[15, 9])).
% 10.50/10.70 tff(17,plain,
% 10.50/10.70 (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))))),
% 10.50/10.70 inference(modus_ponens,[status(thm)],[16, 8])).
% 10.50/10.70 tff(18,plain,
% 10.50/10.70 (precedes(E1!10, E2!9, P!8)),
% 10.50/10.70 inference(and_elim,[status(thm)],[17])).
% 10.50/10.70 tff(19,plain,
% 10.50/10.70 (precedes(E2!9, E1!10, P!8)),
% 10.50/10.70 inference(modus_ponens,[status(thm)],[18, 7])).
% 10.50/10.70 tff(20,plain,
% 10.50/10.70 (shortest_path(V1!12, V2!11, P!8)),
% 10.50/10.70 inference(and_elim,[status(thm)],[17])).
% 10.50/10.70 tff(21,plain,
% 10.50/10.70 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : refl(((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))))),
% 10.50/10.70 inference(bind,[status(th)],[])).
% 10.50/10.70 tff(22,plain,
% 10.50/10.70 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))),
% 10.50/10.70 inference(quant_intro,[status(thm)],[21])).
% 10.50/10.70 tff(23,plain,
% 10.50/10.70 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : rewrite(((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))))),
% 10.50/10.70 inference(bind,[status(th)],[])).
% 10.50/10.70 tff(24,plain,
% 10.50/10.70 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))),
% 10.50/10.70 inference(quant_intro,[status(thm)],[23])).
% 10.50/10.70 tff(25,plain,
% 10.50/10.70 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))),
% 10.50/10.70 inference(transitivity,[status(thm)],[24, 22])).
% 10.50/10.70 tff(26,plain,
% 10.50/10.70 (^[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))))), trans(monotonicity(quant_intro(proof_bind(^[E3: $i] : trans(monotonicity(rewrite(((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))) <=> (~((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))), ((~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> (~(~((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))))), rewrite((~(~((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))) <=> ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))), ((~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))))), (![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> ![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))))), ((![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))) <=> (![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))), rewrite((![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))) <=> (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))), ((![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))) <=> (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))), (((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> (((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P))) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))))), rewrite((((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P))) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P)))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))), (((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))))),
% 10.50/10.70 inference(bind,[status(th)],[])).
% 10.50/10.70 tff(27,plain,
% 10.50/10.70 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))),
% 10.50/10.70 inference(quant_intro,[status(thm)],[26])).
% 10.50/10.70 tff(28,plain,
% 10.50/10.70 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))),
% 10.50/10.70 inference(rewrite,[status(thm)],[])).
% 10.50/10.70 tff(29,plain,
% 10.50/10.70 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : trans(monotonicity(monotonicity(rewrite((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) <=> (~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2))))), (((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))) <=> ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))), (((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))))), rewrite(((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))), (((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(30,plain,
% 10.50/10.71 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P)))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[29])).
% 10.50/10.71 tff(31,axiom,(![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','shortest_path_properties')).
% 10.50/10.71 tff(32,plain,
% 10.50/10.71 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[31, 30])).
% 10.50/10.71 tff(33,plain,
% 10.50/10.71 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | ((~?[E3: $i] : ((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[32, 28])).
% 10.50/10.71 tff(34,plain,(
% 10.50/10.71 ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (![E3: $i] : (~((tail_of(E3) = tail_of(E1)) & (head_of(E3) = head_of(E2)))) & (~precedes(E2, E1, P))))),
% 10.50/10.71 inference(skolemize,[status(sab)],[33])).
% 10.50/10.71 tff(35,plain,
% 10.50/10.71 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[34, 27])).
% 10.50/10.71 tff(36,plain,
% 10.50/10.71 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[35, 25])).
% 10.50/10.71 tff(37,plain,
% 10.50/10.71 (((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | ((~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8)))) <=> ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | (~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8)))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(38,plain,
% 10.50/10.71 (((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8)))) <=> ((~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8)))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(39,plain,
% 10.50/10.71 ((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) <=> (~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9)))))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(40,plain,
% 10.50/10.71 (((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8)) <=> ((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))),
% 10.50/10.71 inference(monotonicity,[status(thm)],[39])).
% 10.50/10.71 tff(41,plain,
% 10.50/10.71 ((~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) <=> (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8)))),
% 10.50/10.71 inference(monotonicity,[status(thm)],[40])).
% 10.50/10.71 tff(42,plain,
% 10.50/10.71 (((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8)))) <=> ((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))))),
% 10.50/10.71 inference(monotonicity,[status(thm)],[41])).
% 10.50/10.71 tff(43,plain,
% 10.50/10.71 (((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8)))) <=> ((~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8)))),
% 10.50/10.71 inference(transitivity,[status(thm)],[42, 38])).
% 10.50/10.71 tff(44,plain,
% 10.50/10.71 (((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | ((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))))) <=> ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | ((~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8))))),
% 10.50/10.71 inference(monotonicity,[status(thm)],[43])).
% 10.50/10.71 tff(45,plain,
% 10.50/10.71 (((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | ((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))))) <=> ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | (~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8)))),
% 10.50/10.71 inference(transitivity,[status(thm)],[44, 37])).
% 10.50/10.71 tff(46,plain,
% 10.50/10.71 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | ((~shortest_path(V1!12, V2!11, P!8)) | (~precedes(E1!10, E2!9, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))))),
% 10.50/10.71 inference(quant_inst,[status(thm)],[])).
% 10.50/10.71 tff(47,plain,
% 10.50/10.71 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1))) | (~(head_of(E3) = head_of(E2))))) | precedes(E2, E1, P))))) | (~shortest_path(V1!12, V2!11, P!8)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))) | (~precedes(E1!10, E2!9, P!8))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[46, 45])).
% 10.50/10.71 tff(48,plain,
% 10.50/10.71 (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8))),
% 10.50/10.71 inference(unit_resolution,[status(thm)],[47, 36, 20, 18])).
% 10.50/10.71 tff(49,plain,
% 10.50/10.71 (((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!10))) | (~(head_of(E3) = head_of(E2!9))))) | precedes(E2!9, E1!10, P!8)) | (~precedes(E2!9, E1!10, P!8))),
% 10.50/10.71 inference(tautology,[status(thm)],[])).
% 10.50/10.71 tff(50,plain,
% 10.50/10.71 (~precedes(E2!9, E1!10, P!8)),
% 10.50/10.71 inference(unit_resolution,[status(thm)],[49, 48])).
% 10.50/10.71 tff(51,plain,
% 10.50/10.71 ($false),
% 10.50/10.71 inference(unit_resolution,[status(thm)],[50, 19])).
% 10.50/10.71 tff(52,plain,(~(E2!9 = E1!10)), inference(lemma,lemma(discharge,[]))).
% 10.50/10.71 tff(53,plain,
% 10.50/10.71 (^[E1: $i, E2: $i] : refl((~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2))) <=> (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(54,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2))) <=> ![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[53])).
% 10.50/10.71 tff(55,plain,
% 10.50/10.71 (^[E1: $i, E2: $i] : trans(trans(monotonicity(rewrite((edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2))) <=> (~((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))))), ((sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2)))) <=> (sequential(E1, E2) <=> (~((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))))))), rewrite((sequential(E1, E2) <=> (~((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))))) <=> (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))), ((sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2)))) <=> (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2))))), rewrite((~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2))) <=> (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))), ((sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2)))) <=> (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(56,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2)))) <=> ![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[55])).
% 10.50/10.71 tff(57,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2)))) <=> ![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2))))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(58,plain,
% 10.50/10.71 (^[E1: $i, E2: $i] : rewrite((sequential(E1, E2) <=> (((edge(E1) & edge(E2)) & (~(E1 = E2))) & (head_of(E1) = tail_of(E2)))) <=> (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2)))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(59,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (sequential(E1, E2) <=> (((edge(E1) & edge(E2)) & (~(E1 = E2))) & (head_of(E1) = tail_of(E2)))) <=> ![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[58])).
% 10.50/10.71 tff(60,axiom,(![E1: $i, E2: $i] : (sequential(E1, E2) <=> (((edge(E1) & edge(E2)) & (~(E1 = E2))) & (head_of(E1) = tail_of(E2))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','sequential_defn')).
% 10.50/10.71 tff(61,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[60, 59])).
% 10.50/10.71 tff(62,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[61, 57])).
% 10.50/10.71 tff(63,plain,(
% 10.50/10.71 ![E1: $i, E2: $i] : (sequential(E1, E2) <=> (edge(E1) & edge(E2) & (~(E1 = E2)) & (head_of(E1) = tail_of(E2))))),
% 10.50/10.71 inference(skolemize,[status(sab)],[62])).
% 10.50/10.71 tff(64,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[63, 56])).
% 10.50/10.71 tff(65,plain,
% 10.50/10.71 (![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[64, 54])).
% 10.50/10.71 tff(66,plain,
% 10.50/10.71 (((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))) <=> ((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(67,plain,
% 10.50/10.71 ((~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))) <=> (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(68,plain,
% 10.50/10.71 ((((E2!9 = E1!10) | (~edge(E1!10)) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)) <=> (((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(69,plain,
% 10.50/10.71 ((~(((E2!9 = E1!10) | (~edge(E1!10)) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))) <=> (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))),
% 10.50/10.71 inference(monotonicity,[status(thm)],[68])).
% 10.50/10.71 tff(70,plain,
% 10.50/10.71 ((~(((E2!9 = E1!10) | (~edge(E1!10)) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))) <=> (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))),
% 10.50/10.71 inference(transitivity,[status(thm)],[69, 67])).
% 10.50/10.71 tff(71,plain,
% 10.50/10.71 (((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((E2!9 = E1!10) | (~edge(E1!10)) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))) <=> ((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))))),
% 10.50/10.71 inference(monotonicity,[status(thm)],[70])).
% 10.50/10.71 tff(72,plain,
% 10.50/10.71 (((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((E2!9 = E1!10) | (~edge(E1!10)) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))) <=> ((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))))),
% 10.50/10.71 inference(transitivity,[status(thm)],[71, 66])).
% 10.50/10.71 tff(73,plain,
% 10.50/10.71 ((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((E2!9 = E1!10) | (~edge(E1!10)) | (~edge(E2!9)) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))),
% 10.50/10.71 inference(quant_inst,[status(thm)],[])).
% 10.50/10.71 tff(74,plain,
% 10.50/10.71 ((~![E1: $i, E2: $i] : (~(((E1 = E2) | (~edge(E2)) | (~edge(E1)) | (~(head_of(E1) = tail_of(E2)))) <=> sequential(E1, E2)))) | (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[73, 72])).
% 10.50/10.71 tff(75,plain,
% 10.50/10.71 (~(((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10))),
% 10.50/10.71 inference(unit_resolution,[status(thm)],[74, 65])).
% 10.50/10.71 tff(76,plain,
% 10.50/10.71 (^[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)))))))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(77,plain,
% 10.50/10.71 (![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))))))))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[76])).
% 10.50/10.71 tff(78,plain,
% 10.50/10.71 (^[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))))))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(79,plain,
% 10.50/10.71 (![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)))))))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[78])).
% 10.50/10.71 tff(80,plain,
% 10.50/10.71 (^[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))))))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(81,plain,
% 10.50/10.71 (![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)))))))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[80])).
% 10.50/10.71 tff(82,plain,
% 10.50/10.71 (![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)))))))))),
% 10.50/10.71 inference(transitivity,[status(thm)],[81, 79])).
% 10.50/10.71 tff(83,plain,
% 10.50/10.71 (^[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)))))))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(84,plain,
% 10.50/10.71 (![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)))))))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[83])).
% 10.50/10.71 tff(85,plain,
% 10.50/10.71 (^[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))))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(86,plain,
% 10.50/10.71 (![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)))))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[85])).
% 10.50/10.71 tff(87,plain,
% 10.50/10.71 (![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)))))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(88,plain,
% 10.50/10.71 (^[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))))))),
% 10.50/10.71 inference(bind,[status(th)],[])).
% 10.50/10.71 tff(89,plain,
% 10.50/10.71 (![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)))))),
% 10.50/10.71 inference(quant_intro,[status(thm)],[88])).
% 10.50/10.71 tff(90,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')).
% 10.50/10.71 tff(91,plain,
% 10.50/10.71 (![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)))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[90, 89])).
% 10.50/10.71 tff(92,plain,
% 10.50/10.71 (![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)))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[91, 87])).
% 10.50/10.71 tff(93,plain,(
% 10.50/10.71 ![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))))))))),
% 10.50/10.71 inference(skolemize,[status(sab)],[92])).
% 10.50/10.71 tff(94,plain,
% 10.50/10.71 (![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)))))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[93, 86])).
% 10.50/10.71 tff(95,plain,
% 10.50/10.71 (![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)))))))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[94, 84])).
% 10.50/10.71 tff(96,plain,
% 10.50/10.71 (![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)))))))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[95, 82])).
% 10.50/10.71 tff(97,plain,
% 10.50/10.71 (![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))))))))))),
% 10.50/10.71 inference(modus_ponens,[status(thm)],[96, 77])).
% 10.50/10.71 tff(98,plain,
% 10.50/10.71 (((~![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)))))))))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(99,plain,
% 10.50/10.71 ((~((~((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))))))))),
% 10.50/10.71 inference(rewrite,[status(thm)],[])).
% 10.50/10.71 tff(100,plain,
% 10.50/10.71 (((~![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)))))))))),
% 10.50/10.72 inference(monotonicity,[status(thm)],[99])).
% 10.50/10.72 tff(101,plain,
% 10.50/10.72 (((~![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)))))))))),
% 10.50/10.72 inference(transitivity,[status(thm)],[100, 98])).
% 10.50/10.72 tff(102,plain,
% 10.50/10.72 ((~![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))))))))))),
% 10.50/10.72 inference(quant_inst,[status(thm)],[])).
% 10.50/10.72 tff(103,plain,
% 10.50/10.72 ((~![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))))))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[102, 101])).
% 10.50/10.72 tff(104,plain,
% 10.50/10.72 (~((~(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)))))))),
% 10.50/10.72 inference(unit_resolution,[status(thm)],[103, 97])).
% 10.50/10.72 tff(105,plain,
% 10.50/10.72 (((~(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)))))),
% 10.50/10.72 inference(tautology,[status(thm)],[])).
% 10.50/10.72 tff(106,plain,
% 10.50/10.72 ((~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))))),
% 10.50/10.72 inference(unit_resolution,[status(thm)],[105, 104])).
% 10.50/10.72 tff(107,plain,
% 10.50/10.72 ((~((~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))))),
% 10.50/10.72 inference(tautology,[status(thm)],[])).
% 10.50/10.72 tff(108,plain,
% 10.50/10.72 ((~((~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))))),
% 10.50/10.72 inference(unit_resolution,[status(thm)],[107, 20])).
% 10.50/10.72 tff(109,plain,
% 10.50/10.72 (~((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)))),
% 10.50/10.72 inference(unit_resolution,[status(thm)],[108, 106])).
% 10.50/10.72 tff(110,plain,
% 10.50/10.72 (((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)),
% 10.50/10.72 inference(tautology,[status(thm)],[])).
% 10.50/10.72 tff(111,plain,
% 10.50/10.72 (path(V1!12, V2!11, P!8)),
% 10.50/10.72 inference(unit_resolution,[status(thm)],[110, 109])).
% 10.50/10.72 tff(112,plain,
% 10.50/10.72 (^[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))))))))))),
% 10.50/10.72 inference(bind,[status(th)],[])).
% 10.50/10.72 tff(113,plain,
% 10.50/10.72 (![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)))))))))),
% 10.50/10.72 inference(quant_intro,[status(thm)],[112])).
% 10.50/10.72 tff(114,plain,
% 10.50/10.72 (^[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))))))))))),
% 10.50/10.72 inference(bind,[status(th)],[])).
% 10.50/10.72 tff(115,plain,
% 10.50/10.72 (![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)))))))))),
% 10.50/10.72 inference(quant_intro,[status(thm)],[114])).
% 10.50/10.72 tff(116,plain,
% 10.50/10.72 (![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)))))))))),
% 10.50/10.72 inference(transitivity,[status(thm)],[115, 113])).
% 10.50/10.72 tff(117,plain,
% 10.50/10.72 (^[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))))))))))),
% 10.50/10.72 inference(bind,[status(th)],[])).
% 10.50/10.72 tff(118,plain,
% 10.50/10.72 (![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)))))))))),
% 10.50/10.72 inference(quant_intro,[status(thm)],[117])).
% 10.50/10.72 tff(119,plain,
% 10.50/10.72 (^[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))))))))),
% 10.50/10.72 inference(bind,[status(th)],[])).
% 10.50/10.72 tff(120,plain,
% 10.50/10.72 (![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)))))))),
% 10.50/10.72 inference(quant_intro,[status(thm)],[119])).
% 10.50/10.72 tff(121,plain,
% 10.50/10.72 (![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)))))))),
% 10.50/10.72 inference(rewrite,[status(thm)],[])).
% 10.50/10.72 tff(122,plain,
% 10.50/10.72 (^[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)))))))))),
% 10.50/10.72 inference(bind,[status(th)],[])).
% 10.50/10.72 tff(123,plain,
% 10.50/10.72 (![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)))))))),
% 10.50/10.72 inference(quant_intro,[status(thm)],[122])).
% 10.50/10.72 tff(124,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')).
% 10.50/10.72 tff(125,plain,
% 10.50/10.72 (![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)))))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[124, 123])).
% 10.50/10.72 tff(126,plain,
% 10.50/10.72 (![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)))))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[125, 121])).
% 10.50/10.72 tff(127,plain,(
% 10.50/10.72 ![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))))))))),
% 10.50/10.72 inference(skolemize,[status(sab)],[126])).
% 10.50/10.72 tff(128,plain,
% 10.50/10.72 (![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)))))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[127, 120])).
% 10.50/10.72 tff(129,plain,
% 10.50/10.72 (![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)))))))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[128, 118])).
% 10.50/10.72 tff(130,plain,
% 10.50/10.72 (![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)))))))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[129, 116])).
% 10.50/10.72 tff(131,plain,
% 10.50/10.72 (((~![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))))))),
% 10.50/10.72 inference(rewrite,[status(thm)],[])).
% 10.50/10.72 tff(132,plain,
% 10.50/10.72 (((~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))))))),
% 10.50/10.72 inference(rewrite,[status(thm)],[])).
% 10.50/10.72 tff(133,plain,
% 10.50/10.72 (((~![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)))))))),
% 10.50/10.72 inference(monotonicity,[status(thm)],[132])).
% 10.50/10.72 tff(134,plain,
% 10.50/10.72 (((~![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))))))),
% 10.50/10.72 inference(transitivity,[status(thm)],[133, 131])).
% 10.50/10.72 tff(135,plain,
% 10.50/10.72 ((~![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)))))))))),
% 10.50/10.72 inference(quant_inst,[status(thm)],[])).
% 10.50/10.72 tff(136,plain,
% 10.50/10.72 ((~![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)))))),
% 10.50/10.72 inference(modus_ponens,[status(thm)],[135, 134])).
% 10.50/10.72 tff(137,plain,
% 10.50/10.72 (![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)))))),
% 10.50/10.72 inference(unit_resolution,[status(thm)],[136, 130, 111])).
% 10.50/10.72 tff(138,plain,
% 10.50/10.72 (((~![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))))))),
% 10.50/10.72 inference(rewrite,[status(thm)],[])).
% 10.50/10.72 tff(139,plain,
% 10.50/10.72 (((~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))))))),
% 10.50/10.72 inference(rewrite,[status(thm)],[])).
% 10.50/10.72 tff(140,plain,
% 10.50/10.72 (((~![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)))))))),
% 10.50/10.72 inference(monotonicity,[status(thm)],[139])).
% 10.50/10.72 tff(141,plain,
% 10.50/10.72 (((~![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))))))),
% 10.50/10.73 inference(transitivity,[status(thm)],[140, 138])).
% 10.50/10.73 tff(142,plain,
% 10.50/10.73 ((~![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)))))),
% 10.50/10.73 inference(quant_inst,[status(thm)],[])).
% 10.50/10.73 tff(143,plain,
% 10.50/10.73 ((~![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)))))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[142, 141])).
% 10.50/10.73 tff(144,plain,
% 10.50/10.73 (~((~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))))),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[143, 18, 137])).
% 10.50/10.73 tff(145,plain,
% 10.50/10.73 (((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~sequential(E1!10, E2!9)) | ![E3: $i] : ((~sequential(E1!10, E3)) | (~precedes(E3, E2!9, P!8))))) | (~((~((~sequential(E1!10, tptp_fun_E3_4(E2!9, E1!10, P!8))) | (~precedes(tptp_fun_E3_4(E2!9, E1!10, P!8), E2!9, P!8)))) | sequential(E1!10, E2!9)))) | on_path(E1!10, P!8)),
% 10.50/10.73 inference(tautology,[status(thm)],[])).
% 10.50/10.73 tff(146,plain,
% 10.50/10.73 (on_path(E1!10, P!8)),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[145, 144])).
% 10.50/10.73 tff(147,plain,
% 10.50/10.73 (((~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)),
% 10.50/10.73 inference(tautology,[status(thm)],[])).
% 10.50/10.73 tff(148,plain,
% 10.50/10.73 (on_path(E2!9, P!8)),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[147, 144])).
% 10.50/10.73 tff(149,plain,
% 10.50/10.73 (^[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) | (~![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)))))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(150,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[149])).
% 10.50/10.73 tff(151,plain,
% 10.50/10.73 (^[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) | (~![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)))))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(152,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[151])).
% 10.50/10.73 tff(153,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(transitivity,[status(thm)],[152, 150])).
% 10.50/10.73 tff(154,plain,
% 10.50/10.73 (^[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)) & ![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)))))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(155,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[154])).
% 10.50/10.73 tff(156,plain,
% 10.50/10.73 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (((~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, 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)))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(157,plain,
% 10.50/10.73 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (((~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, 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))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[156])).
% 10.50/10.73 tff(158,plain,
% 10.50/10.73 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(159,plain,
% 10.50/10.73 (^[P: $i, V1: $i, V2: $i] : trans(monotonicity(quant_intro(proof_bind(^[E1: $i, E2: $i] : 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)))), (((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)))))), ((((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)))), rewrite(((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))))) | precedes(E1, E2, P))))), (![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)) <=> ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))), ((path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P))) <=> (path(V1, V2, P) => ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))))), rewrite((path(V1, V2, P) => ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))), ((path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(160,plain,
% 10.50/10.73 (![P: $i, V1: $i, V2: $i] : (path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[159])).
% 10.50/10.73 tff(161,axiom,(![P: $i, V1: $i, V2: $i] : (path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','precedes_defn')).
% 10.50/10.73 tff(162,plain,
% 10.50/10.73 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[161, 160])).
% 10.50/10.73 tff(163,plain,
% 10.50/10.73 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[162, 158])).
% 10.50/10.73 tff(164,plain,(
% 10.50/10.73 ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (((~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 10.50/10.73 inference(skolemize,[status(sab)],[163])).
% 10.50/10.73 tff(165,plain,
% 10.50/10.73 (![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))))))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[164, 157])).
% 10.50/10.73 tff(166,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[165, 155])).
% 10.50/10.73 tff(167,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[166, 153])).
% 10.50/10.73 tff(168,plain,
% 10.50/10.73 (((~![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))))))))) | ((~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, 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) | (~![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(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))))))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(169,plain,
% 10.50/10.73 (((~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) | (~![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) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))))))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(170,plain,
% 10.50/10.73 (((~![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))))))))) | ((~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) | (~![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) | (~![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(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))))),
% 10.50/10.73 inference(monotonicity,[status(thm)],[169])).
% 10.50/10.73 tff(171,plain,
% 10.50/10.73 (((~![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))))))))) | ((~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) | (~![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) | (~![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(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))))))),
% 10.50/10.73 inference(transitivity,[status(thm)],[170, 168])).
% 10.50/10.73 tff(172,plain,
% 10.50/10.73 ((~![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))))))))) | ((~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) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8))))))))),
% 10.50/10.73 inference(quant_inst,[status(thm)],[])).
% 10.50/10.73 tff(173,plain,
% 10.50/10.73 ((~![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))))))))) | (~path(V1!12, V2!11, P!8)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[172, 171])).
% 10.50/10.73 tff(174,plain,
% 10.50/10.73 (![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[173, 167, 111])).
% 10.50/10.73 tff(175,plain,
% 10.50/10.73 (((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))))) <=> ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(176,plain,
% 10.50/10.73 ((precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10)))) <=> (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(177,plain,
% 10.50/10.73 ((~(sequential(E2!9, E1!10) | (~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))))) <=> (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10)))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(178,plain,
% 10.50/10.73 ((precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~(sequential(E2!9, E1!10) | (~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8))))))) <=> (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))))),
% 10.50/10.73 inference(monotonicity,[status(thm)],[177])).
% 10.50/10.73 tff(179,plain,
% 10.50/10.73 ((precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~(sequential(E2!9, E1!10) | (~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8))))))) <=> (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))))),
% 10.50/10.73 inference(transitivity,[status(thm)],[178, 176])).
% 10.50/10.73 tff(180,plain,
% 10.50/10.73 (((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~(sequential(E2!9, E1!10) | (~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))))))) <=> ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10)))))),
% 10.50/10.73 inference(monotonicity,[status(thm)],[179])).
% 10.50/10.73 tff(181,plain,
% 10.50/10.73 (((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~(sequential(E2!9, E1!10) | (~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))))))) <=> ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))))),
% 10.50/10.73 inference(transitivity,[status(thm)],[180, 175])).
% 10.50/10.73 tff(182,plain,
% 10.50/10.73 ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | (precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~(sequential(E2!9, E1!10) | (~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))))))),
% 10.50/10.73 inference(quant_inst,[status(thm)],[])).
% 10.50/10.73 tff(183,plain,
% 10.50/10.73 ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!8) | (~on_path(E2, P!8)) | (~on_path(E1, P!8)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!8)))))))) | precedes(E2!9, E1!10, P!8) | (~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10)))),
% 10.50/10.73 inference(modus_ponens,[status(thm)],[182, 181])).
% 10.50/10.73 tff(184,plain,
% 10.50/10.73 ((~on_path(E1!10, P!8)) | (~on_path(E2!9, P!8)) | (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10)))),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[183, 50, 174])).
% 10.50/10.73 tff(185,plain,
% 10.50/10.73 (~((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10))),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[184, 148, 146])).
% 10.50/10.73 tff(186,plain,
% 10.50/10.73 (((~![E3: $i] : ((~sequential(E2!9, E3)) | (~precedes(E3, E1!10, P!8)))) | sequential(E2!9, E1!10)) | (~sequential(E2!9, E1!10))),
% 10.50/10.73 inference(tautology,[status(thm)],[])).
% 10.50/10.73 tff(187,plain,
% 10.50/10.73 (~sequential(E2!9, E1!10)),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[186, 185])).
% 10.50/10.73 tff(188,plain,
% 10.50/10.73 ((((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)) | ((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) | sequential(E2!9, E1!10)),
% 10.50/10.73 inference(tautology,[status(thm)],[])).
% 10.50/10.73 tff(189,plain,
% 10.50/10.73 ((((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))) <=> sequential(E2!9, E1!10)) | ((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10))))),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[188, 187])).
% 10.50/10.73 tff(190,plain,
% 10.50/10.73 ((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))),
% 10.50/10.73 inference(unit_resolution,[status(thm)],[189, 75])).
% 10.50/10.73 tff(191,plain,
% 10.50/10.73 (^[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))))))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(192,plain,
% 10.50/10.73 (![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)))))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[191])).
% 10.50/10.73 tff(193,plain,
% 10.50/10.73 (^[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))))))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(194,plain,
% 10.50/10.73 (![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)))))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[193])).
% 10.50/10.73 tff(195,plain,
% 10.50/10.73 (^[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))))))))),
% 10.50/10.73 inference(bind,[status(th)],[])).
% 10.50/10.73 tff(196,plain,
% 10.50/10.73 (![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)))))))),
% 10.50/10.73 inference(quant_intro,[status(thm)],[195])).
% 10.50/10.73 tff(197,plain,
% 10.50/10.73 (![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))))))))),
% 10.50/10.73 inference(rewrite,[status(thm)],[])).
% 10.50/10.73 tff(198,plain,
% 10.50/10.73 (^[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))))))))))),
% 10.50/10.74 inference(bind,[status(th)],[])).
% 10.50/10.74 tff(199,plain,
% 10.50/10.74 (![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))))))))),
% 10.50/10.74 inference(quant_intro,[status(thm)],[198])).
% 10.50/10.74 tff(200,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')).
% 10.50/10.74 tff(201,plain,
% 10.50/10.74 (![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))))))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[200, 199])).
% 10.50/10.74 tff(202,plain,
% 10.50/10.74 (![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))))))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[201, 197])).
% 10.50/10.74 tff(203,plain,(
% 10.50/10.74 ![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)))))))))),
% 10.50/10.74 inference(skolemize,[status(sab)],[202])).
% 10.50/10.74 tff(204,plain,
% 10.50/10.74 (![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)))))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[203, 196])).
% 10.50/10.74 tff(205,plain,
% 10.50/10.74 (![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)))))))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[204, 194])).
% 10.50/10.74 tff(206,plain,
% 10.50/10.74 (![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)))))))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[205, 192])).
% 10.50/10.74 tff(207,plain,
% 10.50/10.74 (((~![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)))))))))),
% 10.50/10.74 inference(rewrite,[status(thm)],[])).
% 10.50/10.74 tff(208,plain,
% 10.50/10.74 (((~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)))))))))),
% 10.50/10.74 inference(rewrite,[status(thm)],[])).
% 10.50/10.74 tff(209,plain,
% 10.50/10.74 (((~![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))))))))))),
% 10.50/10.74 inference(monotonicity,[status(thm)],[208])).
% 10.50/10.74 tff(210,plain,
% 10.50/10.74 (((~![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)))))))))),
% 10.50/10.74 inference(transitivity,[status(thm)],[209, 207])).
% 10.50/10.74 tff(211,plain,
% 10.50/10.74 ((~![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)))))))))),
% 10.50/10.74 inference(quant_inst,[status(thm)],[])).
% 10.50/10.74 tff(212,plain,
% 10.50/10.74 ((~![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))))))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[211, 210])).
% 10.50/10.74 tff(213,plain,
% 10.50/10.74 (~((~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)))))))),
% 10.50/10.74 inference(unit_resolution,[status(thm)],[212, 206, 111])).
% 10.50/10.74 tff(214,plain,
% 10.50/10.74 (((~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)))),
% 10.50/10.74 inference(tautology,[status(thm)],[])).
% 10.50/10.74 tff(215,plain,
% 10.50/10.74 (V1!12 = tail_of(tptp_fun_E_1(P!8, V2!11, V1!12))),
% 10.50/10.74 inference(unit_resolution,[status(thm)],[214, 213])).
% 10.50/10.74 tff(216,plain,
% 10.50/10.74 (tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)) = V1!12),
% 10.50/10.74 inference(symmetry,[status(thm)],[215])).
% 10.50/10.74 tff(217,plain,
% 10.50/10.74 (path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8) <=> path(V1!12, V2!11, P!8)),
% 10.50/10.74 inference(monotonicity,[status(thm)],[216])).
% 10.50/10.74 tff(218,plain,
% 10.50/10.74 (path(V1!12, V2!11, P!8) <=> path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)),
% 10.50/10.74 inference(symmetry,[status(thm)],[217])).
% 10.50/10.74 tff(219,plain,
% 10.50/10.74 (path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[111, 218])).
% 10.50/10.74 tff(220,plain,
% 10.50/10.74 (^[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))))))),
% 10.50/10.74 inference(bind,[status(th)],[])).
% 10.50/10.74 tff(221,plain,
% 10.50/10.74 (![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)))))),
% 10.50/10.74 inference(quant_intro,[status(thm)],[220])).
% 10.50/10.74 tff(222,plain,
% 10.50/10.74 (^[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)))))))),
% 10.50/10.74 inference(bind,[status(th)],[])).
% 10.50/10.74 tff(223,plain,
% 10.50/10.74 (![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)))))),
% 10.50/10.74 inference(quant_intro,[status(thm)],[222])).
% 10.50/10.74 tff(224,plain,
% 10.50/10.74 (![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)))),
% 10.50/10.74 inference(rewrite,[status(thm)],[])).
% 10.50/10.74 tff(225,plain,
% 10.50/10.74 (^[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)))))),
% 10.50/10.74 inference(bind,[status(th)],[])).
% 10.50/10.74 tff(226,plain,
% 10.50/10.74 (![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)))),
% 10.50/10.74 inference(quant_intro,[status(thm)],[225])).
% 10.50/10.74 tff(227,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')).
% 10.50/10.74 tff(228,plain,
% 10.50/10.74 (![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)))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[227, 226])).
% 10.50/10.74 tff(229,plain,
% 10.50/10.74 (![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)))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[228, 224])).
% 10.50/10.74 tff(230,plain,(
% 10.50/10.74 ![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)))),
% 10.50/10.74 inference(skolemize,[status(sab)],[229])).
% 10.50/10.74 tff(231,plain,
% 10.50/10.74 (![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)))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[230, 223])).
% 10.50/10.74 tff(232,plain,
% 10.50/10.74 (![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)))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[231, 221])).
% 10.50/10.74 tff(233,plain,
% 10.50/10.74 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | ((~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | (~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))))),
% 10.50/10.74 inference(rewrite,[status(thm)],[])).
% 10.50/10.74 tff(234,plain,
% 10.50/10.74 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | ((~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))))),
% 10.50/10.74 inference(quant_inst,[status(thm)],[])).
% 10.50/10.74 tff(235,plain,
% 10.50/10.74 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~on_path(E, P)) | (~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))))) | (~on_path(E1!10, P!8)) | (~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8))))),
% 10.50/10.74 inference(modus_ponens,[status(thm)],[234, 233])).
% 10.50/10.74 tff(236,plain,
% 10.50/10.74 ((~path(tail_of(tptp_fun_E_1(P!8, V2!11, V1!12)), V2!11, P!8)) | (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8))))),
% 10.50/10.74 inference(unit_resolution,[status(thm)],[235, 232, 146])).
% 10.50/10.74 tff(237,plain,
% 10.50/10.74 (~((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8)))),
% 10.50/10.74 inference(unit_resolution,[status(thm)],[236, 219])).
% 10.50/10.74 tff(238,plain,
% 10.50/10.74 (((~edge(E1!10)) | (~in_path(head_of(E1!10), P!8)) | (~in_path(tail_of(E1!10), P!8))) | edge(E1!10)),
% 10.50/10.74 inference(tautology,[status(thm)],[])).
% 10.50/10.74 tff(239,plain,
% 10.50/10.74 (edge(E1!10)),
% 10.50/10.74 inference(unit_resolution,[status(thm)],[238, 237])).
% 10.50/10.74 tff(240,plain,
% 10.50/10.74 (((~![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)))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(241,plain,
% 10.50/10.75 ((~![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)))))),
% 10.50/10.75 inference(quant_inst,[status(thm)],[])).
% 10.50/10.75 tff(242,plain,
% 10.50/10.75 ((~![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))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[241, 240])).
% 10.50/10.75 tff(243,plain,
% 10.50/10.75 ((~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))))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[242, 232, 148])).
% 10.50/10.75 tff(244,plain,
% 10.50/10.75 (~((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8)))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[243, 219])).
% 10.50/10.75 tff(245,plain,
% 10.50/10.75 (((~edge(E2!9)) | (~in_path(head_of(E2!9), P!8)) | (~in_path(tail_of(E2!9), P!8))) | edge(E2!9)),
% 10.50/10.75 inference(tautology,[status(thm)],[])).
% 10.50/10.75 tff(246,plain,
% 10.50/10.75 (edge(E2!9)),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[245, 244])).
% 10.50/10.75 tff(247,plain,
% 10.50/10.75 ((~((~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10))))) | (~edge(E2!9)) | (~edge(E1!10)) | (E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))),
% 10.50/10.75 inference(tautology,[status(thm)],[])).
% 10.50/10.75 tff(248,plain,
% 10.50/10.75 ((E2!9 = E1!10) | (~(head_of(E2!9) = tail_of(E1!10)))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[247, 246, 239, 190])).
% 10.50/10.75 tff(249,plain,
% 10.50/10.75 (~(head_of(E2!9) = tail_of(E1!10))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[248, 52])).
% 10.50/10.75 tff(250,plain,
% 10.50/10.75 (~(tail_of(E1!10) = head_of(E2!9))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[249, 3])).
% 10.50/10.75 tff(251,plain,
% 10.50/10.75 (^[E: $i] : refl(((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))) <=> ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))))),
% 10.50/10.75 inference(bind,[status(th)],[])).
% 10.50/10.75 tff(252,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))) <=> ![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 10.50/10.75 inference(quant_intro,[status(thm)],[251])).
% 10.50/10.75 tff(253,plain,
% 10.50/10.75 (^[E: $i] : rewrite(((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E)))))))),
% 10.50/10.75 inference(bind,[status(th)],[])).
% 10.50/10.75 tff(254,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 10.50/10.75 inference(quant_intro,[status(thm)],[253])).
% 10.50/10.75 tff(255,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(256,plain,
% 10.50/10.75 (^[E: $i] : rewrite((edge(E) => (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E)))))),
% 10.50/10.75 inference(bind,[status(th)],[])).
% 10.50/10.75 tff(257,plain,
% 10.50/10.75 (![E: $i] : (edge(E) => (vertex(head_of(E)) & vertex(tail_of(E)))) <=> ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 10.50/10.75 inference(quant_intro,[status(thm)],[256])).
% 10.50/10.75 tff(258,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')).
% 10.50/10.75 tff(259,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[258, 257])).
% 10.50/10.75 tff(260,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[259, 255])).
% 10.50/10.75 tff(261,plain,(
% 10.50/10.75 ![E: $i] : ((~edge(E)) | (vertex(head_of(E)) & vertex(tail_of(E))))),
% 10.50/10.75 inference(skolemize,[status(sab)],[260])).
% 10.50/10.75 tff(262,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[261, 254])).
% 10.50/10.75 tff(263,plain,
% 10.50/10.75 (![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[262, 252])).
% 10.50/10.75 tff(264,plain,
% 10.50/10.75 (((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | ((~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))))) <=> ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | (~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(265,plain,
% 10.50/10.75 ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | ((~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))))),
% 10.50/10.75 inference(quant_inst,[status(thm)],[])).
% 10.50/10.75 tff(266,plain,
% 10.50/10.75 ((~![E: $i] : ((~edge(E)) | (~((~vertex(head_of(E))) | (~vertex(tail_of(E))))))) | (~edge(E1!10)) | (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10)))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[265, 264])).
% 10.50/10.75 tff(267,plain,
% 10.50/10.75 (~((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10))))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[266, 263, 239])).
% 10.50/10.75 tff(268,plain,
% 10.50/10.75 (((~vertex(head_of(E1!10))) | (~vertex(tail_of(E1!10)))) | vertex(tail_of(E1!10))),
% 10.50/10.75 inference(tautology,[status(thm)],[])).
% 10.50/10.75 tff(269,plain,
% 10.50/10.75 (vertex(tail_of(E1!10))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[268, 267])).
% 10.50/10.75 tff(270,plain,
% 10.50/10.75 (((~![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))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(271,plain,
% 10.50/10.75 ((~![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))))))),
% 10.50/10.75 inference(quant_inst,[status(thm)],[])).
% 10.50/10.75 tff(272,plain,
% 10.50/10.75 ((~![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)))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[271, 270])).
% 10.50/10.75 tff(273,plain,
% 10.50/10.75 (~((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9))))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[272, 263, 246])).
% 10.50/10.75 tff(274,plain,
% 10.50/10.75 (((~vertex(head_of(E2!9))) | (~vertex(tail_of(E2!9)))) | vertex(head_of(E2!9))),
% 10.50/10.75 inference(tautology,[status(thm)],[])).
% 10.50/10.75 tff(275,plain,
% 10.50/10.75 (vertex(head_of(E2!9))),
% 10.50/10.75 inference(unit_resolution,[status(thm)],[274, 273])).
% 10.50/10.75 tff(276,plain,
% 10.50/10.75 (complete <=> complete),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(277,plain,
% 10.50/10.75 (complete),
% 10.50/10.75 inference(or_elim,[status(thm)],[12])).
% 10.50/10.75 tff(278,plain,
% 10.50/10.75 (complete),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[277, 276])).
% 10.50/10.75 tff(279,plain,
% 10.50/10.75 (((~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))))))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(280,plain,
% 10.50/10.75 (((~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))))))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(281,plain,
% 10.50/10.75 (((~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)))))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(282,plain,
% 10.50/10.75 (((~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))))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(283,plain,
% 10.50/10.75 ((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))))))))),
% 10.50/10.75 inference(rewrite,[status(thm)],[])).
% 10.50/10.75 tff(284,plain,
% 10.50/10.75 (^[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)))))))))),
% 10.50/10.75 inference(bind,[status(th)],[])).
% 10.50/10.75 tff(285,plain,
% 10.50/10.75 (![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)))))))),
% 10.50/10.75 inference(quant_intro,[status(thm)],[284])).
% 10.50/10.75 tff(286,plain,
% 10.50/10.75 ((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))))))))),
% 10.50/10.75 inference(monotonicity,[status(thm)],[285])).
% 10.50/10.75 tff(287,plain,
% 10.50/10.75 ((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))))))))),
% 10.50/10.75 inference(transitivity,[status(thm)],[286, 283])).
% 10.50/10.75 tff(288,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')).
% 10.50/10.75 tff(289,plain,
% 10.50/10.75 ((~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)))))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[288, 287])).
% 10.50/10.75 tff(290,plain,
% 10.50/10.75 ((~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)))))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[289, 282])).
% 10.50/10.75 tff(291,plain,
% 10.50/10.75 ((~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)))))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[290, 282])).
% 10.50/10.75 tff(292,plain,
% 10.50/10.75 ((~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)))))))),
% 10.50/10.75 inference(modus_ponens,[status(thm)],[291, 282])).
% 10.50/10.75 Proof display could not be completed: monotonicity rule is not handled
% 10.50/10.78 % E exiting
%------------------------------------------------------------------------------