%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRA003+1 : TPTP v9.0.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:06:36 AM UTC 2025
% Result : Theorem 0.20s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRA003+1 : TPTP v9.0.0. Bugfixed v3.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n006.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 15:22:05 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.20/0.38 % SZS status Theorem
% 0.20/0.38 % SZS output start Proof
% 0.20/0.38 tff(precedes_type, type, (
% 0.20/0.38 precedes: ( $i * $i * $i ) > $o)).
% 0.20/0.38 tff(tptp_fun_P_9_type, type, (
% 0.20/0.38 tptp_fun_P_9: $i)).
% 0.20/0.38 tff(tptp_fun_E1_11_type, type, (
% 0.20/0.38 tptp_fun_E1_11: $i)).
% 0.20/0.38 tff(tptp_fun_E2_10_type, type, (
% 0.20/0.38 tptp_fun_E2_10: $i)).
% 0.20/0.38 tff(path_type, type, (
% 0.20/0.38 path: ( $i * $i * $i ) > $o)).
% 0.20/0.38 tff(tptp_fun_V2_12_type, type, (
% 0.20/0.38 tptp_fun_V2_12: $i)).
% 0.20/0.38 tff(tptp_fun_V1_13_type, type, (
% 0.20/0.38 tptp_fun_V1_13: $i)).
% 0.20/0.38 tff(less_or_equal_type, type, (
% 0.20/0.38 less_or_equal: ( $i * $i ) > $o)).
% 0.20/0.38 tff(length_of_type, type, (
% 0.20/0.38 length_of: $i > $i)).
% 0.20/0.38 tff(shortest_path_type, type, (
% 0.20/0.38 shortest_path: ( $i * $i * $i ) > $o)).
% 0.20/0.38 tff(tptp_fun_P_6_type, type, (
% 0.20/0.38 tptp_fun_P_6: ( $i * $i * $i ) > $i)).
% 0.20/0.38 tff(edge_type, type, (
% 0.20/0.38 edge: $i > $o)).
% 0.20/0.38 tff(vertex_type, type, (
% 0.20/0.38 vertex: $i > $o)).
% 0.20/0.38 tff(in_path_type, type, (
% 0.20/0.38 in_path: ( $i * $i ) > $o)).
% 0.20/0.38 tff(tail_of_type, type, (
% 0.20/0.38 tail_of: $i > $i)).
% 0.20/0.38 tff(head_of_type, type, (
% 0.20/0.38 head_of: $i > $i)).
% 0.20/0.38 tff(tptp_fun_E_2_type, type, (
% 0.20/0.38 tptp_fun_E_2: ( $i * $i * $i ) > $i)).
% 0.20/0.38 tff(path_cons_type, type, (
% 0.20/0.38 path_cons: ( $i * $i ) > $i)).
% 0.20/0.38 tff(empty_type, type, (
% 0.20/0.38 empty: $i)).
% 0.20/0.38 tff(tptp_fun_TP_3_type, type, (
% 0.20/0.38 tptp_fun_TP_3: ( $i * $i * $i ) > $i)).
% 0.20/0.38 tff(on_path_type, type, (
% 0.20/0.38 on_path: ( $i * $i ) > $o)).
% 0.20/0.38 tff(sequential_type, type, (
% 0.20/0.38 sequential: ( $i * $i ) > $o)).
% 0.20/0.38 tff(tptp_fun_E3_5_type, type, (
% 0.20/0.38 tptp_fun_E3_5: ( $i * $i * $i ) > $i)).
% 0.20/0.38 tff(1,plain,
% 0.20/0.38 (^[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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1))))))))) <=> (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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)))))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(2,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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))))))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.38 tff(3,plain,
% 0.20/0.38 (^[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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1))))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(4,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.38 tff(5,plain,
% 0.20/0.38 (^[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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1))))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(6,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[5])).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[6, 4])).
% 0.20/0.38 tff(8,plain,
% 0.20/0.38 (^[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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))) <=> ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(9,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[8])).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (^[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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1))))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(11,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (![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)))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(13,plain,
% 0.20/0.38 (^[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))))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (![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)))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.38 tff(15,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')).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (![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)))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.20/0.38 tff(17,plain,
% 0.20/0.38 (![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)))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.20/0.38 tff(18,plain,(
% 0.20/0.38 ![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1))))))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[17])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.20/0.38 tff(20,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[19, 9])).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 (![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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(SP, V2, V1)))))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[20, 7])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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))))))))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[21, 2])).
% 0.20/0.39 tff(23,plain,
% 0.20/0.39 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), 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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(24,plain,
% 0.20/0.39 ((~((~((V1!13 = V2!12) | shortest_path(V1!13, V2!12, P!9) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))))) <=> (~((~(shortest_path(V1!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(25,plain,
% 0.20/0.39 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13 = V2!12) | shortest_path(V1!13, V2!12, P!9) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), 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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[24])).
% 0.20/0.39 tff(26,plain,
% 0.20/0.39 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13 = V2!12) | shortest_path(V1!13, V2!12, P!9) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), 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_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[25, 23])).
% 0.20/0.39 tff(27,plain,
% 0.20/0.39 ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13 = V2!12) | shortest_path(V1!13, V2!12, P!9) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(28,plain,
% 0.20/0.39 ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_6(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_6(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!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.39 tff(29,plain,
% 0.20/0.39 (~((~(shortest_path(V1!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[28, 22])).
% 0.20/0.39 tff(30,plain,
% 0.20/0.39 (((~(shortest_path(V1!13, V2!12, P!9) | (V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~((~path(V1!13, V2!12, tptp_fun_P_6(P!9, V2!12, V1!13))) | less_or_equal(length_of(P!9), length_of(tptp_fun_P_6(P!9, V2!12, V1!13))))))) | (~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))))) | ((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(31,plain,
% 0.20/0.39 ((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.20/0.39 tff(32,plain,
% 0.20/0.39 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P)))) <=> (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(33,plain,
% 0.20/0.39 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((((((vertex(V1) & vertex(V2)) & (~(V1 = V2))) & edge(E1)) & edge(E2)) & (~(E1 = E2))) & path(V1, V2, P)))) <=> (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(34,axiom,(~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((shortest_path(V1, V2, P) & precedes(E1, E2, P)) => ((((((vertex(V1) & vertex(V2)) & (~(V1 = V2))) & edge(E1)) & edge(E2)) & (~(E1 = E2))) & path(V1, V2, P)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','vertices_and_edges')).
% 0.20/0.39 tff(35,plain,
% 0.20/0.39 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.20/0.39 tff(36,plain,
% 0.20/0.39 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[35, 32])).
% 0.20/0.39 tff(37,plain,
% 0.20/0.39 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.20/0.39 tff(38,plain,
% 0.20/0.39 (~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P))) | (vertex(V1) & vertex(V2) & (~(V1 = V2)) & edge(E1) & edge(E2) & (~(E1 = E2)) & path(V1, V2, P)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[37, 32])).
% 0.20/0.39 tff(39,plain,(
% 0.20/0.39 ~((~(shortest_path(V1!13, V2!12, P!9) & precedes(E1!11, E2!10, P!9))) | (vertex(V1!13) & vertex(V2!12) & (~(V1!13 = V2!12)) & edge(E1!11) & edge(E2!10) & (~(E1!11 = E2!10)) & path(V1!13, V2!12, P!9)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[38])).
% 0.20/0.39 tff(40,plain,
% 0.20/0.39 (shortest_path(V1!13, V2!12, P!9) & precedes(E1!11, E2!10, P!9)),
% 0.20/0.39 inference(or_elim,[status(thm)],[39])).
% 0.20/0.39 tff(41,plain,
% 0.20/0.39 (shortest_path(V1!13, V2!12, P!9)),
% 0.20/0.39 inference(and_elim,[status(thm)],[40])).
% 0.20/0.39 tff(42,plain,
% 0.20/0.39 ((~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))) | (~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(43,plain,
% 0.20/0.39 ((~((~shortest_path(V1!13, V2!12, P!9)) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))))) | (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.20/0.39 tff(44,plain,
% 0.20/0.39 (~((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P)))))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[43, 31])).
% 0.20/0.39 tff(45,plain,
% 0.20/0.39 (((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))) | path(V1!13, V2!12, P!9)),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(46,plain,
% 0.20/0.39 (path(V1!13, V2!12, P!9)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[45, 44])).
% 0.20/0.39 tff(47,plain,
% 0.20/0.39 (^[V1: $i, V2: $i, P: $i] : refl(((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))) <=> ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(48,plain,
% 0.20/0.39 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[47])).
% 0.20/0.39 tff(49,plain,
% 0.20/0.39 (^[V1: $i, V2: $i, P: $i] : rewrite(((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))) <=> ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(50,plain,
% 0.20/0.40 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[49])).
% 0.20/0.40 tff(51,plain,
% 0.20/0.40 (^[V1: $i, V2: $i, P: $i] : rewrite(((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & ((((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))) <=> ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(52,plain,
% 0.20/0.40 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & ((((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP))))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 (![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))))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 (^[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))))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(55,plain,
% 0.20/0.40 (![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))))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.40 tff(56,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')).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 (![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))))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.20/0.40 tff(58,plain,
% 0.20/0.40 (![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))))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.20/0.40 tff(59,plain,(
% 0.20/0.40 ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & ((((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[58])).
% 0.20/0.40 tff(60,plain,
% 0.20/0.40 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_2(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_2(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1)) & (P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_2(P, V2, V1))) & (P = path_cons(tptp_fun_E_2(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.20/0.40 tff(61,plain,
% 0.20/0.40 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[60, 50])).
% 0.20/0.40 tff(62,plain,
% 0.20/0.40 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[61, 48])).
% 0.20/0.40 tff(63,plain,
% 0.20/0.40 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | ((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | (~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))))) | (~((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))))))) | (~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP))))))))) <=> ((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | ((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))))) | (~((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))))))) | (~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | ((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13))))))))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[64])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | ((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))))) | (~((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))))))) | (~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | (~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[65, 63])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | ((~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))))) | (~((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))))))) | (~((~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_2(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_2(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, tptp_fun_TP_3(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), tptp_fun_TP_3(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_2(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_2(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_2(P, V2, V1), TP)))))))))) | (~path(V1!13, V2!12, P!9)) | (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 (~((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[68, 62, 46])).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 (((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13))))))) | (V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(71,plain,
% 0.20/0.42 (V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.20/0.42 tff(72,plain,
% 0.20/0.42 (tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)) = V1!13),
% 0.20/0.42 inference(symmetry,[status(thm)],[71])).
% 0.20/0.42 tff(73,plain,
% 0.20/0.42 (path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9) <=> path(V1!13, V2!12, P!9)),
% 0.20/0.42 inference(monotonicity,[status(thm)],[72])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 (path(V1!13, V2!12, P!9) <=> path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)),
% 0.20/0.42 inference(symmetry,[status(thm)],[73])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 (path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[46, 74])).
% 0.20/0.42 tff(76,plain,
% 0.20/0.42 (^[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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 (![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[76])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (^[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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(79,plain,
% 0.20/0.42 (![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[78])).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 (![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[79, 77])).
% 0.20/0.42 tff(81,plain,
% 0.20/0.42 (^[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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(82,plain,
% 0.20/0.42 (![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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[81])).
% 0.20/0.42 tff(83,plain,
% 0.20/0.42 (^[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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(84,plain,
% 0.20/0.42 (![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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[83])).
% 0.20/0.42 tff(85,plain,
% 0.20/0.42 (![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)))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(86,plain,
% 0.20/0.42 (^[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)))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 (![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)))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[86])).
% 0.20/0.42 tff(88,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')).
% 0.20/0.42 tff(89,plain,
% 0.20/0.42 (![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)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.42 tff(90,plain,
% 0.20/0.42 (![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)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.20/0.42 tff(91,plain,(
% 0.20/0.42 ![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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[90])).
% 0.20/0.42 tff(92,plain,
% 0.20/0.42 (![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_5(E2, E1, P)) & precedes(tptp_fun_E3_5(E2, E1, P), E2, P))) & ((~sequential(E1, E2)) | ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[91, 84])).
% 0.20/0.42 tff(93,plain,
% 0.20/0.42 (![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[92, 82])).
% 0.20/0.42 tff(94,plain,
% 0.20/0.42 (![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[93, 80])).
% 0.20/0.42 tff(95,plain,
% 0.20/0.42 (((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9))))))) <=> ((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | (~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(96,plain,
% 0.20/0.42 (((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~on_path(E1, P!9)) | (~on_path(E2, P!9)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))))))) <=> ((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(97,plain,
% 0.20/0.42 (((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~on_path(E1, P!9)) | (~on_path(E2, P!9)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9)))))))))) <=> ((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[96])).
% 0.20/0.42 tff(98,plain,
% 0.20/0.42 (((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~on_path(E1, P!9)) | (~on_path(E2, P!9)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9)))))))))) <=> ((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | (~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9))))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[97, 95])).
% 0.20/0.43 tff(99,plain,
% 0.20/0.43 ((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | ((~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~on_path(E1, P!9)) | (~on_path(E2, P!9)) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9)))))))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(100,plain,
% 0.20/0.43 ((~![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_5(E2, E1, P))) | (~precedes(tptp_fun_E3_5(E2, E1, P), E2, P)))))) | (~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))) | (~path(V1!13, V2!12, P!9)) | ![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.20/0.43 tff(101,plain,
% 0.20/0.43 (![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[100, 94, 46])).
% 0.20/0.43 tff(102,plain,
% 0.20/0.43 (precedes(E1!11, E2!10, P!9)),
% 0.20/0.43 inference(and_elim,[status(thm)],[40])).
% 0.20/0.43 tff(103,plain,
% 0.20/0.43 (((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10))))))) <=> ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | (~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(104,plain,
% 0.20/0.43 (((~precedes(E1!11, E2!10, P!9)) | (~((~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~(sequential(E1!11, E2!10) | (~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))))) | (~on_path(E2!10, P!9)) | (~on_path(E1!11, P!9))))) <=> ((~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(105,plain,
% 0.20/0.43 (((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~(sequential(E1!11, E2!10) | (~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))))) | (~on_path(E2!10, P!9)) | (~on_path(E1!11, P!9)))))) <=> ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10)))))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[104])).
% 0.20/0.43 tff(106,plain,
% 0.20/0.43 (((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~(sequential(E1!11, E2!10) | (~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))))) | (~on_path(E2!10, P!9)) | (~on_path(E1!11, P!9)))))) <=> ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | (~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10))))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[105, 103])).
% 0.20/0.43 tff(107,plain,
% 0.20/0.43 ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~(sequential(E1!11, E2!10) | (~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))))) | (~on_path(E2!10, P!9)) | (~on_path(E1!11, P!9)))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(108,plain,
% 0.20/0.43 ((~![E1: $i, E2: $i] : ((~precedes(E1, E2, P!9)) | (~((~((~sequential(E1, E2)) | ![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!9))))) | (~(sequential(E1, E2) | (~((~sequential(E1, tptp_fun_E3_5(E2, E1, P!9))) | (~precedes(tptp_fun_E3_5(E2, E1, P!9), E2, P!9)))))) | (~on_path(E2, P!9)) | (~on_path(E1, P!9)))))) | (~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10)))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.20/0.43 tff(109,plain,
% 0.20/0.43 (~((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[108, 102, 101])).
% 0.20/0.43 tff(110,plain,
% 0.20/0.43 (((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10)))) | on_path(E2!10, P!9)),
% 0.20/0.43 inference(tautology,[status(thm)],[])).
% 0.20/0.43 tff(111,plain,
% 0.20/0.43 (on_path(E2!10, P!9)),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.20/0.43 tff(112,plain,
% 0.20/0.43 (^[V1: $i, V2: $i, P: $i, E: $i] : refl(((~((~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))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(113,plain,
% 0.20/0.43 (![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P))) <=> ![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[112])).
% 0.20/0.43 tff(114,plain,
% 0.20/0.43 (^[V1: $i, V2: $i, P: $i, E: $i] : trans(monotonicity(trans(monotonicity(rewrite((path(V1, V2, P) & on_path(E, P)) <=> (~((~path(V1, V2, P)) | (~on_path(E, P))))), ((~(path(V1, V2, P) & on_path(E, P))) <=> (~(~((~path(V1, V2, P)) | (~on_path(E, P))))))), rewrite((~(~((~path(V1, V2, P)) | (~on_path(E, P))))) <=> ((~path(V1, V2, P)) | (~on_path(E, P)))), ((~(path(V1, V2, P) & on_path(E, P))) <=> ((~path(V1, V2, P)) | (~on_path(E, 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))) <=> (((~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))))) <=> ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))), (((~(path(V1, V2, P) & on_path(E, P))) | (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)))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(115,plain,
% 0.20/0.43 (![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] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[114])).
% 0.20/0.44 tff(116,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(117,plain,
% 0.20/0.44 (^[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)))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(118,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[117])).
% 0.20/0.44 tff(119,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')).
% 0.20/0.44 tff(120,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.20/0.44 tff(121,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[120, 116])).
% 0.20/0.44 tff(122,plain,(
% 0.20/0.44 ![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)))),
% 0.20/0.44 inference(skolemize,[status(sab)],[121])).
% 0.20/0.44 tff(123,plain,
% 0.20/0.44 (![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[122, 115])).
% 0.20/0.44 tff(124,plain,
% 0.20/0.44 (![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[123, 113])).
% 0.20/0.44 tff(125,plain,
% 0.20/0.44 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~on_path(E2!10, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | (~on_path(E2!10, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(126,plain,
% 0.20/0.44 (((~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E2!10, P!9))) <=> ((~on_path(E2!10, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(127,plain,
% 0.20/0.44 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E2!10, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~on_path(E2!10, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[126])).
% 0.20/0.44 tff(128,plain,
% 0.20/0.44 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E2!10, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | (~on_path(E2!10, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)))),
% 0.20/0.44 inference(transitivity,[status(thm)],[127, 125])).
% 0.20/0.44 tff(129,plain,
% 0.20/0.44 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E2!10, P!9)))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(130,plain,
% 0.20/0.44 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | (~on_path(E2!10, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[129, 128])).
% 0.20/0.44 tff(131,plain,
% 0.20/0.44 ((~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[130, 124, 111])).
% 0.20/0.44 tff(132,plain,
% 0.20/0.44 (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[131, 75])).
% 0.20/0.44 tff(133,plain,
% 0.20/0.44 (((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9))) | edge(E2!10)),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(134,plain,
% 0.20/0.44 (edge(E2!10)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[133, 132])).
% 0.20/0.44 tff(135,plain,
% 0.20/0.44 (((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13))))))) | vertex(V1!13)),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(136,plain,
% 0.20/0.44 (vertex(V1!13)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[135, 69])).
% 0.20/0.44 tff(137,plain,
% 0.20/0.44 (((~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(tptp_fun_E_2(P!9, V2!12, V1!13))) | (~(V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))) | (~((~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13)))))) | (~((~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), tptp_fun_TP_3(P!9, V2!12, V1!13)))) | (~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, tptp_fun_TP_3(P!9, V2!12, V1!13))))))) | (~(![TP: $i] : ((~path(head_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, TP)) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), TP)))) | (~(P!9 = path_cons(tptp_fun_E_2(P!9, V2!12, V1!13), empty))) | (~(V2!12 = head_of(tptp_fun_E_2(P!9, V2!12, V1!13))))))) | vertex(V2!12)),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(138,plain,
% 0.20/0.44 (vertex(V2!12)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[137, 69])).
% 0.20/0.44 tff(139,plain,
% 0.20/0.44 (((V1!13 = V2!12) | (~path(V1!13, V2!12, P!9)) | (~![P: $i] : ((~path(V1!13, V2!12, P)) | less_or_equal(length_of(P!9), length_of(P))))) | (~(V1!13 = V2!12))),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(140,plain,
% 0.20/0.44 (~(V1!13 = V2!12)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[139, 44])).
% 0.20/0.44 tff(141,plain,
% 0.20/0.44 (((~on_path(E1!11, P!9)) | (~on_path(E2!10, P!9)) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))) | (~((~((~sequential(E1!11, tptp_fun_E3_5(E2!10, E1!11, P!9))) | (~precedes(tptp_fun_E3_5(E2!10, E1!11, P!9), E2!10, P!9)))) | sequential(E1!11, E2!10)))) | on_path(E1!11, P!9)),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(142,plain,
% 0.20/0.44 (on_path(E1!11, P!9)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[141, 109])).
% 0.20/0.44 tff(143,plain,
% 0.20/0.44 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~on_path(E1!11, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | (~on_path(E1!11, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(144,plain,
% 0.20/0.44 (((~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E1!11, P!9))) <=> ((~on_path(E1!11, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(145,plain,
% 0.20/0.44 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E1!11, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~on_path(E1!11, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9))))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[144])).
% 0.20/0.44 tff(146,plain,
% 0.20/0.44 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E1!11, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | (~on_path(E1!11, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[145, 143])).
% 0.20/0.44 tff(147,plain,
% 0.20/0.44 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | ((~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~on_path(E1!11, P!9)))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(148,plain,
% 0.20/0.45 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~path(V1, V2, P)) | (~on_path(E, P)))) | (~on_path(E1!11, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[147, 146])).
% 0.20/0.45 tff(149,plain,
% 0.20/0.45 ((~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[148, 124, 142])).
% 0.20/0.45 tff(150,plain,
% 0.20/0.45 (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[149, 75])).
% 0.20/0.45 tff(151,plain,
% 0.20/0.45 (((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9))) | edge(E1!11)),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(152,plain,
% 0.20/0.45 (edge(E1!11)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[151, 150])).
% 0.20/0.45 tff(153,plain,
% 0.20/0.45 ((~(~((V1!13 = V2!12) | (E1!11 = E2!10) | (~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(E1!11)) | (~edge(E2!10)) | (~path(V1!13, V2!12, P!9))))) <=> ((V1!13 = V2!12) | (E1!11 = E2!10) | (~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(E1!11)) | (~edge(E2!10)) | (~path(V1!13, V2!12, P!9)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(154,plain,
% 0.20/0.45 ((vertex(V1!13) & vertex(V2!12) & (~(V1!13 = V2!12)) & edge(E1!11) & edge(E2!10) & (~(E1!11 = E2!10)) & path(V1!13, V2!12, P!9)) <=> (~((V1!13 = V2!12) | (E1!11 = E2!10) | (~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(E1!11)) | (~edge(E2!10)) | (~path(V1!13, V2!12, P!9))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(155,plain,
% 0.20/0.45 ((~(vertex(V1!13) & vertex(V2!12) & (~(V1!13 = V2!12)) & edge(E1!11) & edge(E2!10) & (~(E1!11 = E2!10)) & path(V1!13, V2!12, P!9))) <=> (~(~((V1!13 = V2!12) | (E1!11 = E2!10) | (~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(E1!11)) | (~edge(E2!10)) | (~path(V1!13, V2!12, P!9)))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[154])).
% 0.20/0.45 tff(156,plain,
% 0.20/0.45 ((~(vertex(V1!13) & vertex(V2!12) & (~(V1!13 = V2!12)) & edge(E1!11) & edge(E2!10) & (~(E1!11 = E2!10)) & path(V1!13, V2!12, P!9))) <=> ((V1!13 = V2!12) | (E1!11 = E2!10) | (~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(E1!11)) | (~edge(E2!10)) | (~path(V1!13, V2!12, P!9)))),
% 0.20/0.45 inference(transitivity,[status(thm)],[155, 153])).
% 0.20/0.45 tff(157,plain,
% 0.20/0.45 (~(vertex(V1!13) & vertex(V2!12) & (~(V1!13 = V2!12)) & edge(E1!11) & edge(E2!10) & (~(E1!11 = E2!10)) & path(V1!13, V2!12, P!9))),
% 0.20/0.45 inference(or_elim,[status(thm)],[39])).
% 0.20/0.45 tff(158,plain,
% 0.20/0.45 ((V1!13 = V2!12) | (E1!11 = E2!10) | (~vertex(V1!13)) | (~vertex(V2!12)) | (~edge(E1!11)) | (~edge(E2!10)) | (~path(V1!13, V2!12, P!9))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[157, 156])).
% 0.20/0.45 tff(159,plain,
% 0.20/0.45 (E1!11 = E2!10),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[158, 152, 140, 138, 136, 134, 46])).
% 0.20/0.45 tff(160,plain,
% 0.20/0.45 (E2!10 = E1!11),
% 0.20/0.45 inference(symmetry,[status(thm)],[159])).
% 0.20/0.45 tff(161,plain,
% 0.20/0.45 (precedes(E2!10, E1!11, P!9) <=> precedes(E1!11, E2!10, P!9)),
% 0.20/0.45 inference(monotonicity,[status(thm)],[160, 159])).
% 0.20/0.45 tff(162,plain,
% 0.20/0.45 (precedes(E1!11, E2!10, P!9) <=> precedes(E2!10, E1!11, P!9)),
% 0.20/0.45 inference(symmetry,[status(thm)],[161])).
% 0.20/0.45 tff(163,plain,
% 0.20/0.45 (precedes(E2!10, E1!11, P!9)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[102, 162])).
% 0.20/0.45 tff(164,plain,
% 0.20/0.45 (^[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)))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(165,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[164])).
% 0.20/0.45 tff(166,plain,
% 0.20/0.45 (^[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)))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(167,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[166])).
% 0.20/0.45 tff(168,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[167, 165])).
% 0.20/0.45 tff(169,plain,
% 0.20/0.45 (^[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))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(170,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[169])).
% 0.20/0.45 tff(171,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(172,plain,
% 0.20/0.45 (^[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))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(173,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[172])).
% 0.20/0.45 tff(174,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')).
% 0.20/0.45 tff(175,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[174, 173])).
% 0.20/0.45 tff(176,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[175, 171])).
% 0.20/0.45 tff(177,plain,(
% 0.20/0.45 ![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))))),
% 0.20/0.45 inference(skolemize,[status(sab)],[176])).
% 0.20/0.45 tff(178,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[177, 170])).
% 0.20/0.45 tff(179,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[178, 168])).
% 0.20/0.45 tff(180,plain,
% 0.20/0.45 (((~![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!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9)))) <=> ((~![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!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(181,plain,
% 0.20/0.45 (((~shortest_path(V1!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)))) <=> ((~shortest_path(V1!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(182,plain,
% 0.20/0.45 ((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) <=> (~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10)))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(183,plain,
% 0.20/0.45 (((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)) <=> ((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[182])).
% 0.20/0.45 tff(184,plain,
% 0.20/0.45 ((~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) <=> (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[183])).
% 0.20/0.45 tff(185,plain,
% 0.20/0.45 (((~shortest_path(V1!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)))) <=> ((~shortest_path(V1!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[184])).
% 0.20/0.45 tff(186,plain,
% 0.20/0.45 (((~shortest_path(V1!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)))) <=> ((~shortest_path(V1!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9)))),
% 0.20/0.45 inference(transitivity,[status(thm)],[185, 181])).
% 0.20/0.45 tff(187,plain,
% 0.20/0.45 (((~![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!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))))) <=> ((~![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!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[186])).
% 0.20/0.45 tff(188,plain,
% 0.20/0.45 (((~![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!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))))) <=> ((~![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!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9)))),
% 0.20/0.45 inference(transitivity,[status(thm)],[187, 180])).
% 0.20/0.45 tff(189,plain,
% 0.20/0.45 ((~![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!13, V2!12, P!9)) | (~precedes(E1!11, E2!10, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(190,plain,
% 0.20/0.45 ((~![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!13, V2!12, P!9)) | (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))) | (~precedes(E1!11, E2!10, P!9))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[189, 188])).
% 0.20/0.45 tff(191,plain,
% 0.20/0.45 (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[190, 179, 41, 102])).
% 0.20/0.45 tff(192,plain,
% 0.20/0.45 (((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)) | (~precedes(E2!10, E1!11, P!9))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(193,plain,
% 0.20/0.45 (~precedes(E2!10, E1!11, P!9)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[192, 191])).
% 0.20/0.45 tff(194,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[193, 163])).
% 0.20/0.45 % SZS output end Proof
% 0.20/0.46 % E exiting
%------------------------------------------------------------------------------