TSTP Solution File: GRA003+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRA003+1 : TPTP v8.1.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n020.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 : Fri Sep 16 20:47:26 EDT 2022
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRA003+1 : TPTP v8.1.0. Bugfixed v3.2.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.34 % Computer : n020.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Wed Aug 31 13:23:30 EDT 2022
% 0.14/0.34 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(precedes_type, type, (
% 0.20/0.40 precedes: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_P_9_type, type, (
% 0.20/0.40 tptp_fun_P_9: $i)).
% 0.20/0.40 tff(tptp_fun_E1_11_type, type, (
% 0.20/0.40 tptp_fun_E1_11: $i)).
% 0.20/0.40 tff(tptp_fun_E2_10_type, type, (
% 0.20/0.40 tptp_fun_E2_10: $i)).
% 0.20/0.40 tff(path_type, type, (
% 0.20/0.40 path: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_V2_12_type, type, (
% 0.20/0.40 tptp_fun_V2_12: $i)).
% 0.20/0.40 tff(tptp_fun_V1_13_type, type, (
% 0.20/0.40 tptp_fun_V1_13: $i)).
% 0.20/0.40 tff(less_or_equal_type, type, (
% 0.20/0.40 less_or_equal: ( $i * $i ) > $o)).
% 0.20/0.40 tff(length_of_type, type, (
% 0.20/0.40 length_of: $i > $i)).
% 0.20/0.40 tff(shortest_path_type, type, (
% 0.20/0.40 shortest_path: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_P_6_type, type, (
% 0.20/0.40 tptp_fun_P_6: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(edge_type, type, (
% 0.20/0.40 edge: $i > $o)).
% 0.20/0.40 tff(vertex_type, type, (
% 0.20/0.40 vertex: $i > $o)).
% 0.20/0.40 tff(in_path_type, type, (
% 0.20/0.40 in_path: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tail_of_type, type, (
% 0.20/0.40 tail_of: $i > $i)).
% 0.20/0.40 tff(head_of_type, type, (
% 0.20/0.40 head_of: $i > $i)).
% 0.20/0.40 tff(tptp_fun_E_2_type, type, (
% 0.20/0.40 tptp_fun_E_2: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(path_cons_type, type, (
% 0.20/0.40 path_cons: ( $i * $i ) > $i)).
% 0.20/0.40 tff(empty_type, type, (
% 0.20/0.40 empty: $i)).
% 0.20/0.40 tff(tptp_fun_TP_3_type, type, (
% 0.20/0.40 tptp_fun_TP_3: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(on_path_type, type, (
% 0.20/0.40 on_path: ( $i * $i ) > $o)).
% 0.20/0.40 tff(sequential_type, type, (
% 0.20/0.40 sequential: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_E3_5_type, type, (
% 0.20/0.40 tptp_fun_E3_5: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(1,plain,
% 0.20/0.40 (^[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)))))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 (![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))))))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 (^[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))))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(4,plain,
% 0.20/0.40 (![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)))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.41 tff(5,plain,
% 0.20/0.41 (^[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))))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(6,plain,
% 0.20/0.41 (![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)))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[5])).
% 0.20/0.41 tff(7,plain,
% 0.20/0.41 (![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)))))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[6, 4])).
% 0.20/0.41 tff(8,plain,
% 0.20/0.41 (^[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)))))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 (![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)))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[8])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 (^[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))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 (![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)))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[10])).
% 0.20/0.41 tff(12,plain,
% 0.20/0.41 (![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)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(13,plain,
% 0.20/0.41 (^[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))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(14,plain,
% 0.20/0.41 (![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)))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.41 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')).
% 0.20/0.41 tff(16,plain,
% 0.20/0.41 (![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)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.20/0.41 tff(17,plain,
% 0.20/0.41 (![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)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.20/0.41 tff(18,plain,(
% 0.20/0.41 ![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))))))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[17])).
% 0.20/0.41 tff(19,plain,
% 0.20/0.41 (![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)))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.20/0.41 tff(20,plain,
% 0.20/0.41 (![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)))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[19, 9])).
% 0.20/0.41 tff(21,plain,
% 0.20/0.41 (![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)))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[20, 7])).
% 0.20/0.41 tff(22,plain,
% 0.20/0.41 (![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))))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[21, 2])).
% 0.20/0.41 tff(23,plain,
% 0.20/0.41 (((~![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)))))))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(24,plain,
% 0.20/0.41 ((~((~((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))))))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(25,plain,
% 0.20/0.41 (((~![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)))))))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[24])).
% 0.20/0.42 tff(26,plain,
% 0.20/0.42 (((~![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)))))))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[25, 23])).
% 0.20/0.42 tff(27,plain,
% 0.20/0.42 ((~![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))))))))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(28,plain,
% 0.20/0.42 ((~![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))))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.42 tff(29,plain,
% 0.20/0.42 (~((~(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)))))))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[28, 22])).
% 0.20/0.42 tff(30,plain,
% 0.20/0.42 (((~(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)))))))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(31,plain,
% 0.20/0.42 ((~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))))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.20/0.42 tff(32,plain,
% 0.20/0.42 ((~![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(33,plain,
% 0.20/0.42 ((~![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 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')).
% 0.20/0.42 tff(35,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.20/0.42 tff(36,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[35, 32])).
% 0.20/0.42 tff(37,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.20/0.42 tff(38,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[37, 32])).
% 0.20/0.42 tff(39,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[38, 32])).
% 0.20/0.42 tff(40,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.20/0.42 tff(41,plain,
% 0.20/0.42 (~![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[40, 32])).
% 0.20/0.42 tff(42,plain,(
% 0.20/0.42 ~((~(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)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[41])).
% 0.20/0.42 tff(43,plain,
% 0.20/0.42 (shortest_path(V1!13, V2!12, P!9) & precedes(E1!11, E2!10, P!9)),
% 0.20/0.42 inference(or_elim,[status(thm)],[42])).
% 0.20/0.42 tff(44,plain,
% 0.20/0.42 (shortest_path(V1!13, V2!12, P!9)),
% 0.20/0.42 inference(and_elim,[status(thm)],[43])).
% 0.20/0.42 tff(45,plain,
% 0.20/0.42 ((~((~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))))))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(46,plain,
% 0.20/0.42 ((~((~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))))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[45, 44])).
% 0.20/0.42 tff(47,plain,
% 0.20/0.42 (~((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)))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[46, 31])).
% 0.20/0.42 tff(48,plain,
% 0.20/0.42 (((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)),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(49,plain,
% 0.20/0.42 (path(V1!13, V2!12, P!9)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.20/0.42 tff(50,plain,
% 0.20/0.42 (^[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))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(51,plain,
% 0.20/0.42 (![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)))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[50])).
% 0.20/0.42 tff(52,plain,
% 0.20/0.42 (^[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))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(53,plain,
% 0.20/0.42 (![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)))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[52])).
% 0.20/0.42 tff(54,plain,
% 0.20/0.42 (^[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))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(55,plain,
% 0.20/0.43 (![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)))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.43 tff(56,plain,
% 0.20/0.43 (![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)))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(57,plain,
% 0.20/0.43 (^[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)))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(58,plain,
% 0.20/0.43 (![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)))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.43 tff(59,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')).
% 0.20/0.43 tff(60,plain,
% 0.20/0.43 (![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)))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.43 tff(61,plain,
% 0.20/0.43 (![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)))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.20/0.43 tff(62,plain,(
% 0.20/0.43 ![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)))))))))),
% 0.20/0.43 inference(skolemize,[status(sab)],[61])).
% 0.20/0.43 tff(63,plain,
% 0.20/0.43 (![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)))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.20/0.43 tff(64,plain,
% 0.20/0.43 (![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)))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[63, 53])).
% 0.20/0.43 tff(65,plain,
% 0.20/0.43 (![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)))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[64, 51])).
% 0.20/0.43 tff(66,plain,
% 0.20/0.43 (((~![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)))) | (~((~((~(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)))))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(67,plain,
% 0.20/0.43 ((~![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)))))))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(68,plain,
% 0.20/0.43 ((~![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))))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.20/0.44 tff(69,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)))) | (~((~((~(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)))))))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[68, 65, 49])).
% 0.20/0.44 tff(70,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)))) | (~((~((~(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!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)))),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(71,plain,
% 0.20/0.44 (V1!13 = tail_of(tptp_fun_E_2(P!9, V2!12, V1!13))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.20/0.44 tff(72,plain,
% 0.20/0.44 (tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)) = V1!13),
% 0.20/0.44 inference(symmetry,[status(thm)],[71])).
% 0.20/0.44 tff(73,plain,
% 0.20/0.44 (path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9) <=> path(V1!13, V2!12, P!9)),
% 0.20/0.44 inference(monotonicity,[status(thm)],[72])).
% 0.20/0.44 tff(74,plain,
% 0.20/0.44 (path(V1!13, V2!12, P!9) <=> path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)),
% 0.20/0.44 inference(symmetry,[status(thm)],[73])).
% 0.20/0.44 tff(75,plain,
% 0.20/0.44 (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)],[49, 74])).
% 0.20/0.44 tff(76,plain,
% 0.20/0.44 (^[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))))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(77,plain,
% 0.20/0.44 (![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)))))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[76])).
% 0.20/0.44 tff(78,plain,
% 0.20/0.44 (^[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))))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(79,plain,
% 0.20/0.44 (![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)))))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[78])).
% 0.20/0.44 tff(80,plain,
% 0.20/0.44 (![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)))))))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[79, 77])).
% 0.20/0.44 tff(81,plain,
% 0.20/0.44 (^[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))))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(82,plain,
% 0.20/0.44 (![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)))))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[81])).
% 0.20/0.44 tff(83,plain,
% 0.20/0.44 (^[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))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(84,plain,
% 0.20/0.44 (![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)))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[83])).
% 0.20/0.44 tff(85,plain,
% 0.20/0.44 (![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))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(86,plain,
% 0.20/0.44 (^[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))))))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(87,plain,
% 0.20/0.44 (![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))))))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[86])).
% 0.20/0.44 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')).
% 0.20/0.44 tff(89,plain,
% 0.20/0.44 (![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))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.44 tff(90,plain,
% 0.20/0.44 (![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))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.20/0.44 tff(91,plain,(
% 0.20/0.44 ![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))))))))),
% 0.20/0.44 inference(skolemize,[status(sab)],[90])).
% 0.20/0.44 tff(92,plain,
% 0.20/0.44 (![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)))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[91, 84])).
% 0.20/0.45 tff(93,plain,
% 0.20/0.45 (![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)))))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[92, 82])).
% 0.20/0.45 tff(94,plain,
% 0.20/0.45 (![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)))))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[93, 80])).
% 0.20/0.45 tff(95,plain,
% 0.20/0.45 (((~![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)) | (~((~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)))))))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(96,plain,
% 0.20/0.45 ((~![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)))))))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(97,plain,
% 0.20/0.45 ((~![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))))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.20/0.45 tff(98,plain,
% 0.20/0.45 (![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))))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[97, 94, 49])).
% 0.20/0.45 tff(99,plain,
% 0.20/0.45 (precedes(E1!11, E2!10, P!9)),
% 0.20/0.45 inference(and_elim,[status(thm)],[43])).
% 0.20/0.45 tff(100,plain,
% 0.20/0.45 (((~![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))))))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, 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))))))))) | (~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(101,plain,
% 0.20/0.45 ((~![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))))))))) | ((~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(102,plain,
% 0.20/0.45 ((~![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))))))))) | (~precedes(E1!11, E2!10, P!9)) | (~((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9)))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.20/0.45 tff(103,plain,
% 0.20/0.45 (~((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[102, 99, 98])).
% 0.20/0.45 tff(104,plain,
% 0.20/0.45 (((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9)))))) | on_path(E2!10, P!9)),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(105,plain,
% 0.20/0.45 (on_path(E2!10, P!9)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[104, 103])).
% 0.20/0.45 tff(106,plain,
% 0.20/0.45 (^[V1: $i, V2: $i, P: $i, E: $i] : refl(((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P))) <=> ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(107,plain,
% 0.20/0.45 (![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P))) <=> ![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[106])).
% 0.20/0.45 tff(108,plain,
% 0.20/0.45 (^[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))))) <=> ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))), (((~(path(V1, V2, P) & on_path(E, P))) | (edge(E) & in_path(head_of(E), P) & in_path(tail_of(E), P))) <=> ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(109,plain,
% 0.20/0.45 (![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)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[108])).
% 0.20/0.45 tff(110,plain,
% 0.20/0.45 (![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)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(111,plain,
% 0.20/0.45 (^[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)))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(112,plain,
% 0.20/0.45 (![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)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[111])).
% 0.20/0.46 tff(113,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')).
% 0.20/0.46 tff(114,plain,
% 0.20/0.46 (![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)))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.20/0.46 tff(115,plain,
% 0.20/0.46 (![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)))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[114, 110])).
% 0.20/0.46 tff(116,plain,(
% 0.20/0.46 ![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)))),
% 0.20/0.46 inference(skolemize,[status(sab)],[115])).
% 0.20/0.46 tff(117,plain,
% 0.20/0.46 (![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[116, 109])).
% 0.20/0.46 tff(118,plain,
% 0.20/0.46 (![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[117, 107])).
% 0.20/0.46 tff(119,plain,
% 0.20/0.46 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~on_path(E2!10, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | (~on_path(E2!10, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(120,plain,
% 0.20/0.46 (((~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~on_path(E2!10, P!9))) <=> ((~on_path(E2!10, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(121,plain,
% 0.20/0.46 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~on_path(E2!10, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~on_path(E2!10, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9))))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[120])).
% 0.20/0.46 tff(122,plain,
% 0.20/0.46 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~on_path(E2!10, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | (~on_path(E2!10, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[121, 119])).
% 0.20/0.46 tff(123,plain,
% 0.20/0.46 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))) | (~on_path(E2!10, P!9)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(124,plain,
% 0.20/0.46 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | (~on_path(E2!10, P!9)) | (~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.20/0.46 tff(125,plain,
% 0.20/0.46 ((~path(tail_of(tptp_fun_E_2(P!9, V2!12, V1!13)), V2!12, P!9)) | (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9))))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[124, 118, 105])).
% 0.20/0.46 tff(126,plain,
% 0.20/0.46 (~((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9)))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[125, 75])).
% 0.20/0.46 tff(127,plain,
% 0.20/0.46 (((~edge(E2!10)) | (~in_path(head_of(E2!10), P!9)) | (~in_path(tail_of(E2!10), P!9))) | edge(E2!10)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(128,plain,
% 0.20/0.46 (edge(E2!10)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[127, 126])).
% 0.20/0.46 tff(129,plain,
% 0.20/0.46 (((~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))))))) | vertex(V1!13)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(130,plain,
% 0.20/0.46 (vertex(V1!13)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[129, 69])).
% 0.20/0.46 tff(131,plain,
% 0.20/0.46 (((~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))))))) | vertex(V2!12)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(132,plain,
% 0.20/0.46 (vertex(V2!12)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[131, 69])).
% 0.20/0.46 tff(133,plain,
% 0.20/0.46 (((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))),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(134,plain,
% 0.20/0.46 (~(V1!13 = V2!12)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[133, 47])).
% 0.20/0.46 tff(135,plain,
% 0.20/0.46 (((~on_path(E1!11, P!9)) | (~on_path(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)))))) | (~((~sequential(E1!11, E2!10)) | ![E3: $i] : ((~sequential(E1!11, E3)) | (~precedes(E3, E2!10, P!9)))))) | on_path(E1!11, P!9)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(136,plain,
% 0.20/0.46 (on_path(E1!11, P!9)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[135, 103])).
% 0.20/0.46 tff(137,plain,
% 0.20/0.46 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), 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] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), 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)))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(138,plain,
% 0.20/0.46 (((~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)))) | (~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)))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(139,plain,
% 0.20/0.46 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~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)))) | (~on_path(E1!11, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), 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))))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[138])).
% 0.20/0.46 tff(140,plain,
% 0.20/0.46 (((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~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)))) | (~on_path(E1!11, P!9)))) <=> ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), 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)))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[139, 137])).
% 0.20/0.46 tff(141,plain,
% 0.20/0.46 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), P)))) | (~on_path(E, P)))) | ((~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)))) | (~on_path(E1!11, P!9)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(142,plain,
% 0.20/0.46 ((~![V1: $i, V2: $i, P: $i, E: $i] : ((~path(V1, V2, P)) | (~((~edge(E)) | (~in_path(head_of(E), P)) | (~in_path(tail_of(E), 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))))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[141, 140])).
% 0.20/0.46 tff(143,plain,
% 0.20/0.46 ((~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))))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[142, 118, 136])).
% 0.20/0.46 tff(144,plain,
% 0.20/0.46 (~((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9)))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[143, 75])).
% 0.20/0.46 tff(145,plain,
% 0.20/0.46 (((~edge(E1!11)) | (~in_path(head_of(E1!11), P!9)) | (~in_path(tail_of(E1!11), P!9))) | edge(E1!11)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(146,plain,
% 0.20/0.46 (edge(E1!11)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[145, 144])).
% 0.20/0.46 tff(147,plain,
% 0.20/0.46 ((~(~((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)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(148,plain,
% 0.20/0.46 ((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))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(149,plain,
% 0.20/0.46 ((~(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)))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[148])).
% 0.20/0.46 tff(150,plain,
% 0.20/0.46 ((~(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)))),
% 0.20/0.46 inference(transitivity,[status(thm)],[149, 147])).
% 0.20/0.46 tff(151,plain,
% 0.20/0.46 (~(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))),
% 0.20/0.46 inference(or_elim,[status(thm)],[42])).
% 0.20/0.46 tff(152,plain,
% 0.20/0.46 ((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))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[151, 150])).
% 0.20/0.46 tff(153,plain,
% 0.20/0.46 (E1!11 = E2!10),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[152, 146, 134, 132, 130, 128, 49])).
% 0.20/0.46 tff(154,plain,
% 0.20/0.46 (E2!10 = E1!11),
% 0.20/0.46 inference(symmetry,[status(thm)],[153])).
% 0.20/0.46 tff(155,plain,
% 0.20/0.46 (precedes(E2!10, E1!11, P!9) <=> precedes(E1!11, E2!10, P!9)),
% 0.20/0.46 inference(monotonicity,[status(thm)],[154, 153])).
% 0.20/0.46 tff(156,plain,
% 0.20/0.46 (precedes(E1!11, E2!10, P!9) <=> precedes(E2!10, E1!11, P!9)),
% 0.20/0.46 inference(symmetry,[status(thm)],[155])).
% 0.20/0.46 tff(157,plain,
% 0.20/0.46 (precedes(E2!10, E1!11, P!9)),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[99, 156])).
% 0.20/0.46 tff(158,plain,
% 0.20/0.46 (^[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)))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(159,plain,
% 0.20/0.46 (![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))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[158])).
% 0.20/0.46 tff(160,plain,
% 0.20/0.46 (^[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)))))),
% 0.20/0.46 inference(bind,[status(th)],[])).
% 0.20/0.46 tff(161,plain,
% 0.20/0.46 (![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))))),
% 0.20/0.46 inference(quant_intro,[status(thm)],[160])).
% 0.20/0.47 tff(162,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(transitivity,[status(thm)],[161, 159])).
% 0.20/0.47 tff(163,plain,
% 0.20/0.47 (^[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))))))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(164,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(quant_intro,[status(thm)],[163])).
% 0.20/0.47 tff(165,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(166,plain,
% 0.20/0.47 (^[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))))))),
% 0.20/0.47 inference(bind,[status(th)],[])).
% 0.20/0.47 tff(167,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(quant_intro,[status(thm)],[166])).
% 0.20/0.47 tff(168,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')).
% 0.20/0.47 tff(169,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[168, 167])).
% 0.20/0.47 tff(170,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[169, 165])).
% 0.20/0.47 tff(171,plain,(
% 0.20/0.47 ![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))))),
% 0.20/0.47 inference(skolemize,[status(sab)],[170])).
% 0.20/0.47 tff(172,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[171, 164])).
% 0.20/0.47 tff(173,plain,
% 0.20/0.47 (![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))))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[172, 162])).
% 0.20/0.47 tff(174,plain,
% 0.20/0.47 (((~![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)) | (~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))))),
% 0.20/0.47 inference(rewrite,[status(thm)],[])).
% 0.20/0.47 tff(175,plain,
% 0.20/0.47 ((~![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))))),
% 0.20/0.47 inference(quant_inst,[status(thm)],[])).
% 0.20/0.47 tff(176,plain,
% 0.20/0.47 ((~![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)))),
% 0.20/0.47 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.20/0.47 tff(177,plain,
% 0.20/0.47 (~((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9))),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[176, 173, 44, 99])).
% 0.20/0.47 tff(178,plain,
% 0.20/0.47 (((~![E3: $i] : ((~(tail_of(E3) = tail_of(E1!11))) | (~(head_of(E3) = head_of(E2!10))))) | precedes(E2!10, E1!11, P!9)) | (~precedes(E2!10, E1!11, P!9))),
% 0.20/0.47 inference(tautology,[status(thm)],[])).
% 0.20/0.47 tff(179,plain,
% 0.20/0.47 (~precedes(E2!10, E1!11, P!9)),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[178, 177])).
% 0.20/0.47 tff(180,plain,
% 0.20/0.47 ($false),
% 0.20/0.47 inference(unit_resolution,[status(thm)],[179, 157])).
% 0.20/0.47 % SZS output end Proof
%------------------------------------------------------------------------------