TSTP Solution File: GRA002+3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRA002+3 : TPTP v8.1.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 20:47:26 EDT 2022
% Result : Theorem 204.24s 131.11s
% Output : Proof 204.30s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.10 % Problem : GRA002+3 : TPTP v8.1.0. Bugfixed v3.2.0.
% 0.02/0.10 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.30 % Computer : n008.cluster.edu
% 0.10/0.30 % Model : x86_64 x86_64
% 0.10/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.30 % Memory : 8042.1875MB
% 0.10/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Wed Aug 31 13:22:55 EDT 2022
% 0.10/0.31 % CPUTime :
% 0.10/0.31 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.10/0.31 Usage: tptp [options] [-file:]file
% 0.10/0.31 -h, -? prints this message.
% 0.10/0.31 -smt2 print SMT-LIB2 benchmark.
% 0.10/0.31 -m, -model generate model.
% 0.10/0.31 -p, -proof generate proof.
% 0.10/0.31 -c, -core generate unsat core of named formulas.
% 0.10/0.31 -st, -statistics display statistics.
% 0.10/0.31 -t:timeout set timeout (in second).
% 0.10/0.31 -smt2status display status in smt2 format instead of SZS.
% 0.10/0.31 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.10/0.31 -<param>:<value> configuration parameter and value.
% 0.10/0.31 -o:<output-file> file to place output in.
% 204.24/131.11 % SZS status Theorem
% 204.24/131.11 % SZS output start Proof
% 204.24/131.11 tff(triangle_type, type, (
% 204.24/131.11 triangle: ( $i * $i * $i ) > $o)).
% 204.24/131.11 tff(tptp_fun_E3_8_type, type, (
% 204.24/131.11 tptp_fun_E3_8: ( $i * $i ) > $i)).
% 204.24/131.11 tff(tptp_fun_E1_7_type, type, (
% 204.24/131.11 tptp_fun_E1_7: $i > $i)).
% 204.24/131.11 tff(tptp_fun_P_11_type, type, (
% 204.24/131.11 tptp_fun_P_11: $i)).
% 204.24/131.11 tff(tptp_fun_E2_6_type, type, (
% 204.24/131.11 tptp_fun_E2_6: $i > $i)).
% 204.24/131.11 tff(shortest_path_type, type, (
% 204.24/131.11 shortest_path: ( $i * $i * $i ) > $o)).
% 204.24/131.11 tff(tptp_fun_V2_9_type, type, (
% 204.24/131.11 tptp_fun_V2_9: $i)).
% 204.24/131.11 tff(tail_of_type, type, (
% 204.24/131.11 tail_of: $i > $i)).
% 204.24/131.11 tff(tptp_fun_E_1_type, type, (
% 204.24/131.11 tptp_fun_E_1: ( $i * $i * $i ) > $i)).
% 204.24/131.11 tff(tptp_fun_V1_10_type, type, (
% 204.24/131.11 tptp_fun_V1_10: $i)).
% 204.24/131.11 tff(path_cons_type, type, (
% 204.24/131.11 path_cons: ( $i * $i ) > $i)).
% 204.24/131.11 tff(path_type, type, (
% 204.24/131.11 path: ( $i * $i * $i ) > $o)).
% 204.24/131.11 tff(head_of_type, type, (
% 204.24/131.11 head_of: $i > $i)).
% 204.24/131.11 tff(empty_type, type, (
% 204.24/131.11 empty: $i)).
% 204.24/131.11 tff(tptp_fun_TP_2_type, type, (
% 204.24/131.11 tptp_fun_TP_2: ( $i * $i * $i ) > $i)).
% 204.24/131.11 tff(edge_type, type, (
% 204.24/131.11 edge: $i > $o)).
% 204.24/131.11 tff(vertex_type, type, (
% 204.24/131.11 vertex: $i > $o)).
% 204.24/131.11 tff(less_or_equal_type, type, (
% 204.24/131.11 less_or_equal: ( $i * $i ) > $o)).
% 204.24/131.11 tff(length_of_type, type, (
% 204.24/131.11 length_of: $i > $i)).
% 204.24/131.11 tff(tptp_fun_P_5_type, type, (
% 204.24/131.11 tptp_fun_P_5: ( $i * $i * $i ) > $i)).
% 204.24/131.11 tff(number_of_in_type, type, (
% 204.24/131.11 number_of_in: ( $i * $i ) > $i)).
% 204.24/131.11 tff(graph_type, type, (
% 204.24/131.11 graph: $i)).
% 204.24/131.11 tff(triangles_type, type, (
% 204.24/131.11 triangles: $i)).
% 204.24/131.11 tff(minus_type, type, (
% 204.24/131.11 minus: ( $i * $i ) > $i)).
% 204.24/131.11 tff(n1_type, type, (
% 204.24/131.11 n1: $i)).
% 204.24/131.11 tff(complete_type, type, (
% 204.24/131.11 complete: $o)).
% 204.24/131.11 tff(precedes_type, type, (
% 204.24/131.11 precedes: ( $i * $i * $i ) > $o)).
% 204.24/131.11 tff(sequential_type, type, (
% 204.24/131.11 sequential: ( $i * $i ) > $o)).
% 204.24/131.11 tff(on_path_type, type, (
% 204.24/131.11 on_path: ( $i * $i ) > $o)).
% 204.24/131.11 tff(sequential_pairs_type, type, (
% 204.24/131.11 sequential_pairs: $i)).
% 204.24/131.11 tff(1,plain,
% 204.24/131.11 (^[V1: $i, V2: $i, SP: $i] : rewrite((~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))))))),
% 204.24/131.11 inference(bind,[status(th)],[])).
% 204.24/131.11 tff(2,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))),
% 204.24/131.11 inference(quant_intro,[status(thm)],[1])).
% 204.24/131.11 tff(3,plain,
% 204.24/131.11 (^[V1: $i, V2: $i, SP: $i] : refl((~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))))),
% 204.24/131.11 inference(bind,[status(th)],[])).
% 204.24/131.11 tff(4,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 204.24/131.11 inference(quant_intro,[status(thm)],[3])).
% 204.24/131.11 tff(5,plain,
% 204.24/131.11 (^[V1: $i, V2: $i, SP: $i] : rewrite((~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))))),
% 204.24/131.11 inference(bind,[status(th)],[])).
% 204.24/131.11 tff(6,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 204.24/131.11 inference(quant_intro,[status(thm)],[5])).
% 204.24/131.11 tff(7,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 204.24/131.11 inference(transitivity,[status(thm)],[6, 4])).
% 204.24/131.11 tff(8,plain,
% 204.24/131.11 (^[V1: $i, V2: $i, SP: $i] : trans(monotonicity(rewrite(((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) <=> ((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))), rewrite(((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))) <=> ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))), ((((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> (((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))), rewrite((((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))), ((((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))))),
% 204.24/131.11 inference(bind,[status(th)],[])).
% 204.24/131.11 tff(9,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) <=> ![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 204.24/131.11 inference(quant_intro,[status(thm)],[8])).
% 204.24/131.11 tff(10,plain,
% 204.24/131.11 (^[V1: $i, V2: $i, SP: $i] : rewrite((((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & (shortest_path(V1, V2, SP) | ((~path(V1, V2, SP)) | (~(~(V1 = V2))) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))) <=> (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))),
% 204.24/131.11 inference(bind,[status(th)],[])).
% 204.24/131.11 tff(11,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & (shortest_path(V1, V2, SP) | ((~path(V1, V2, SP)) | (~(~(V1 = V2))) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))) <=> ![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))),
% 204.24/131.11 inference(quant_intro,[status(thm)],[10])).
% 204.24/131.11 tff(12,plain,
% 204.24/131.11 (![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)))))),
% 204.24/131.11 inference(rewrite,[status(thm)],[])).
% 204.24/131.11 tff(13,plain,
% 204.24/131.11 (^[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))))))),
% 204.24/131.11 inference(bind,[status(th)],[])).
% 204.24/131.11 tff(14,plain,
% 204.24/131.11 (![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)))))),
% 204.24/131.11 inference(quant_intro,[status(thm)],[13])).
% 204.24/131.11 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')).
% 204.24/131.11 tff(16,plain,
% 204.24/131.11 (![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)))))),
% 204.24/131.11 inference(modus_ponens,[status(thm)],[15, 14])).
% 204.24/131.11 tff(17,plain,
% 204.24/131.11 (![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)))))),
% 204.24/131.11 inference(modus_ponens,[status(thm)],[16, 12])).
% 204.24/131.11 tff(18,plain,(
% 204.24/131.11 ![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & (shortest_path(V1, V2, SP) | ((~path(V1, V2, SP)) | (~(~(V1 = V2))) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))))),
% 204.24/131.11 inference(skolemize,[status(sab)],[17])).
% 204.24/131.11 tff(19,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (((~shortest_path(V1, V2, SP)) | (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) & ((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))),
% 204.24/131.11 inference(modus_ponens,[status(thm)],[18, 11])).
% 204.24/131.11 tff(20,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 204.24/131.11 inference(modus_ponens,[status(thm)],[19, 9])).
% 204.24/131.11 tff(21,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))))) | (~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1)))))))))),
% 204.24/131.11 inference(modus_ponens,[status(thm)],[20, 7])).
% 204.24/131.11 tff(22,plain,
% 204.24/131.11 (![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))),
% 204.24/131.11 inference(modus_ponens,[status(thm)],[21, 2])).
% 204.24/131.11 tff(23,plain,
% 204.24/131.11 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))))) <=> ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))))))),
% 204.24/131.11 inference(rewrite,[status(thm)],[])).
% 204.24/131.11 tff(24,plain,
% 204.24/131.11 ((~((~((V1!10 = V2!9) | shortest_path(V1!10, V2!9, P!11) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))))) <=> (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))))),
% 204.24/131.11 inference(rewrite,[status(thm)],[])).
% 204.24/131.11 tff(25,plain,
% 204.24/131.11 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~((V1!10 = V2!9) | shortest_path(V1!10, V2!9, P!11) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))))) <=> ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))))))),
% 204.24/131.12 inference(monotonicity,[status(thm)],[24])).
% 204.24/131.12 tff(26,plain,
% 204.24/131.12 (((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~((V1!10 = V2!9) | shortest_path(V1!10, V2!9, P!11) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))))) <=> ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))))))),
% 204.24/131.12 inference(transitivity,[status(thm)],[25, 23])).
% 204.24/131.12 tff(27,plain,
% 204.24/131.12 ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~((V1!10 = V2!9) | shortest_path(V1!10, V2!9, P!11) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))))),
% 204.24/131.12 inference(quant_inst,[status(thm)],[])).
% 204.24/131.12 tff(28,plain,
% 204.24/131.12 ((~![V1: $i, V2: $i, SP: $i] : (~((~((V1 = V2) | shortest_path(V1, V2, SP) | (~path(V1, V2, SP)) | (~((~path(V1, V2, tptp_fun_P_5(SP, V2, V1))) | less_or_equal(length_of(SP), length_of(tptp_fun_P_5(SP, V2, V1))))))) | (~((~shortest_path(V1, V2, SP)) | (~((V1 = V2) | (~path(V1, V2, SP)) | (~![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))))))) | (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[27, 26])).
% 204.24/131.12 tff(29,plain,
% 204.24/131.12 (~((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))))),
% 204.24/131.12 inference(unit_resolution,[status(thm)],[28, 22])).
% 204.24/131.12 tff(30,plain,
% 204.24/131.12 (((~(shortest_path(V1!10, V2!9, P!11) | (V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~((~path(V1!10, V2!9, tptp_fun_P_5(P!11, V2!9, V1!10))) | less_or_equal(length_of(P!11), length_of(tptp_fun_P_5(P!11, V2!9, V1!10))))))) | (~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))))) | ((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))),
% 204.24/131.12 inference(tautology,[status(thm)],[])).
% 204.24/131.12 tff(31,plain,
% 204.24/131.12 ((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))),
% 204.24/131.12 inference(unit_resolution,[status(thm)],[30, 29])).
% 204.24/131.12 tff(32,plain,
% 204.24/131.12 ((~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))) <=> (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph))))),
% 204.24/131.12 inference(rewrite,[status(thm)],[])).
% 204.24/131.12 tff(33,plain,
% 204.24/131.12 ((~(complete => ![P: $i, V1: $i, V2: $i] : (shortest_path(V1, V2, P) => less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph))))) <=> (~((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))))),
% 204.24/131.12 inference(rewrite,[status(thm)],[])).
% 204.24/131.12 tff(34,axiom,(~(complete => ![P: $i, V1: $i, V2: $i] : (shortest_path(V1, V2, P) => less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','maximal_path_length')).
% 204.24/131.12 tff(35,plain,
% 204.24/131.12 (~((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[34, 33])).
% 204.24/131.12 tff(36,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(or_elim,[status(thm)],[35])).
% 204.24/131.12 tff(37,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[36, 32])).
% 204.24/131.12 tff(38,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[37, 32])).
% 204.24/131.12 tff(39,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[38, 32])).
% 204.24/131.12 tff(40,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[39, 32])).
% 204.24/131.12 tff(41,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[40, 32])).
% 204.24/131.12 tff(42,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[41, 32])).
% 204.24/131.12 tff(43,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[42, 32])).
% 204.24/131.12 tff(44,plain,
% 204.24/131.12 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[43, 32])).
% 204.24/131.12 tff(45,plain,(
% 204.24/131.12 ~((~shortest_path(V1!10, V2!9, P!11)) | less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph)))),
% 204.24/131.12 inference(skolemize,[status(sab)],[44])).
% 204.24/131.12 tff(46,plain,
% 204.24/131.12 (shortest_path(V1!10, V2!9, P!11)),
% 204.24/131.12 inference(or_elim,[status(thm)],[45])).
% 204.24/131.12 tff(47,plain,
% 204.24/131.12 ((~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))) | (~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))),
% 204.24/131.12 inference(tautology,[status(thm)],[])).
% 204.24/131.12 tff(48,plain,
% 204.24/131.12 ((~((~shortest_path(V1!10, V2!9, P!11)) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))))) | (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))))),
% 204.24/131.12 inference(unit_resolution,[status(thm)],[47, 46])).
% 204.24/131.12 tff(49,plain,
% 204.24/131.12 (~((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P)))))),
% 204.24/131.12 inference(unit_resolution,[status(thm)],[48, 31])).
% 204.24/131.12 tff(50,plain,
% 204.24/131.12 (((V1!10 = V2!9) | (~path(V1!10, V2!9, P!11)) | (~![P: $i] : ((~path(V1!10, V2!9, P)) | less_or_equal(length_of(P!11), length_of(P))))) | path(V1!10, V2!9, P!11)),
% 204.24/131.12 inference(tautology,[status(thm)],[])).
% 204.24/131.12 tff(51,plain,
% 204.24/131.12 (path(V1!10, V2!9, P!11)),
% 204.24/131.12 inference(unit_resolution,[status(thm)],[50, 49])).
% 204.24/131.12 tff(52,plain,
% 204.24/131.12 (^[V1: $i, V2: $i, P: $i] : refl(((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))))),
% 204.24/131.12 inference(bind,[status(th)],[])).
% 204.24/131.12 tff(53,plain,
% 204.24/131.12 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 204.24/131.12 inference(quant_intro,[status(thm)],[52])).
% 204.24/131.12 tff(54,plain,
% 204.24/131.12 (^[V1: $i, V2: $i, P: $i] : rewrite(((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))) <=> ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))))),
% 204.24/131.12 inference(bind,[status(th)],[])).
% 204.24/131.12 tff(55,plain,
% 204.24/131.12 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 204.24/131.12 inference(quant_intro,[status(thm)],[54])).
% 204.24/131.12 tff(56,plain,
% 204.24/131.12 (^[V1: $i, V2: $i, P: $i] : rewrite(((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((~(~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))),
% 204.24/131.12 inference(bind,[status(th)],[])).
% 204.24/131.12 tff(57,plain,
% 204.24/131.12 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((~(~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP))))))))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))),
% 204.24/131.12 inference(quant_intro,[status(thm)],[56])).
% 204.24/131.12 tff(58,plain,
% 204.24/131.12 (![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)))))))),
% 204.24/131.12 inference(rewrite,[status(thm)],[])).
% 204.24/131.12 tff(59,plain,
% 204.24/131.12 (^[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)))))))))),
% 204.24/131.12 inference(bind,[status(th)],[])).
% 204.24/131.12 tff(60,plain,
% 204.24/131.12 (![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)))))))),
% 204.24/131.12 inference(quant_intro,[status(thm)],[59])).
% 204.24/131.12 tff(61,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')).
% 204.24/131.12 tff(62,plain,
% 204.24/131.12 (![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)))))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[61, 60])).
% 204.24/131.12 tff(63,plain,
% 204.24/131.12 (![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)))))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[62, 58])).
% 204.24/131.12 tff(64,plain,(
% 204.24/131.12 ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & (edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((~(~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 204.24/131.12 inference(skolemize,[status(sab)],[63])).
% 204.24/131.12 tff(65,plain,
% 204.24/131.12 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (vertex(V1) & vertex(V2) & edge(tptp_fun_E_1(P, V2, V1)) & (V1 = tail_of(tptp_fun_E_1(P, V2, V1))) & (((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | (path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1)) & (P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1))))) & ((~((V2 = head_of(tptp_fun_E_1(P, V2, V1))) & (P = path_cons(tptp_fun_E_1(P, V2, V1), empty)))) | ![TP: $i] : (~(path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP) & (P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[64, 57])).
% 204.24/131.12 tff(66,plain,
% 204.24/131.12 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[65, 55])).
% 204.24/131.12 tff(67,plain,
% 204.24/131.12 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))),
% 204.24/131.12 inference(modus_ponens,[status(thm)],[66, 53])).
% 204.24/131.12 tff(68,plain,
% 204.24/131.12 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!10, V2!9, P!11)) | (~((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP)))))))))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | (~path(V1!10, V2!9, P!11)) | (~((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP)))))))))),
% 204.24/131.12 inference(rewrite,[status(thm)],[])).
% 204.24/131.12 tff(69,plain,
% 204.24/131.12 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | ((~path(V1!10, V2!9, P!11)) | (~((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP)))))))))),
% 204.24/131.12 inference(quant_inst,[status(thm)],[])).
% 204.24/131.12 tff(70,plain,
% 204.24/131.12 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (~((~vertex(V1)) | (~vertex(V2)) | (~edge(tptp_fun_E_1(P, V2, V1))) | (~(V1 = tail_of(tptp_fun_E_1(P, V2, V1)))) | (~((~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))))) | (~((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, tptp_fun_TP_2(P, V2, V1))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), tptp_fun_TP_2(P, V2, V1)))))))) | (~((~(V2 = head_of(tptp_fun_E_1(P, V2, V1)))) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P, V2, V1)), V2, TP)) | (~(P = path_cons(tptp_fun_E_1(P, V2, V1), TP)))))))))) | (~path(V1!10, V2!9, P!11)) | (~((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP))))))))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[69, 68])).
% 204.24/131.13 tff(71,plain,
% 204.24/131.13 (~((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP)))))))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[70, 67, 51])).
% 204.24/131.13 tff(72,plain,
% 204.24/131.13 (((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP))))))) | (V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))),
% 204.24/131.13 inference(tautology,[status(thm)],[])).
% 204.24/131.13 tff(73,plain,
% 204.24/131.13 (V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[72, 71])).
% 204.24/131.13 tff(74,plain,
% 204.24/131.13 (tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)) = V1!10),
% 204.24/131.13 inference(symmetry,[status(thm)],[73])).
% 204.24/131.13 tff(75,plain,
% 204.24/131.13 (shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11) <=> shortest_path(V1!10, V2!9, P!11)),
% 204.24/131.13 inference(monotonicity,[status(thm)],[74])).
% 204.24/131.13 tff(76,plain,
% 204.24/131.13 (shortest_path(V1!10, V2!9, P!11) <=> shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)),
% 204.24/131.13 inference(symmetry,[status(thm)],[75])).
% 204.24/131.13 tff(77,plain,
% 204.24/131.13 (shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[46, 76])).
% 204.24/131.13 tff(78,plain,
% 204.24/131.13 (^[V1: $i, V2: $i, P: $i] : refl(((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))) <=> ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(79,plain,
% 204.24/131.13 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[78])).
% 204.24/131.13 tff(80,plain,
% 204.24/131.13 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(81,plain,
% 204.24/131.13 (^[V1: $i, V2: $i, P: $i] : rewrite((path(V1, V2, P) => (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))) <=> ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(82,plain,
% 204.24/131.13 (![V1: $i, V2: $i, P: $i] : (path(V1, V2, P) => (number_of_in(sequential_pairs, P) = minus(length_of(P), n1))) <=> ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[81])).
% 204.24/131.13 tff(83,axiom,(![V1: $i, V2: $i, P: $i] : (path(V1, V2, P) => (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','path_length_sequential_pairs')).
% 204.24/131.13 tff(84,plain,
% 204.24/131.13 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[83, 82])).
% 204.24/131.13 tff(85,plain,
% 204.24/131.13 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[84, 80])).
% 204.24/131.13 tff(86,plain,(
% 204.24/131.13 ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(skolemize,[status(sab)],[85])).
% 204.24/131.13 tff(87,plain,
% 204.24/131.13 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[86, 79])).
% 204.24/131.13 tff(88,plain,
% 204.24/131.13 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | ((~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = minus(length_of(P!11), n1)))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | (~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = minus(length_of(P!11), n1)))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(89,plain,
% 204.24/131.13 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | ((~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = minus(length_of(P!11), n1)))),
% 204.24/131.13 inference(quant_inst,[status(thm)],[])).
% 204.24/131.13 tff(90,plain,
% 204.24/131.13 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | (~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = minus(length_of(P!11), n1))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[89, 88])).
% 204.24/131.13 tff(91,plain,
% 204.24/131.13 (number_of_in(sequential_pairs, P!11) = minus(length_of(P!11), n1)),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[90, 87, 51])).
% 204.24/131.13 tff(92,assumption,(number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)), introduced(assumption)).
% 204.24/131.13 tff(93,plain,
% 204.24/131.13 (number_of_in(triangles, P!11) = number_of_in(sequential_pairs, P!11)),
% 204.24/131.13 inference(symmetry,[status(thm)],[92])).
% 204.24/131.13 tff(94,assumption,(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), introduced(assumption)).
% 204.24/131.13 tff(95,plain,
% 204.24/131.13 (path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)) = P!11),
% 204.24/131.13 inference(symmetry,[status(thm)],[94])).
% 204.24/131.13 tff(96,plain,
% 204.24/131.13 (number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))) = number_of_in(triangles, P!11)),
% 204.24/131.13 inference(monotonicity,[status(thm)],[95])).
% 204.24/131.13 tff(97,plain,
% 204.24/131.13 (number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))) = minus(length_of(P!11), n1)),
% 204.24/131.13 inference(transitivity,[status(thm)],[96, 93, 91])).
% 204.24/131.13 tff(98,plain,
% 204.24/131.13 (less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), number_of_in(triangles, graph)) <=> less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph))),
% 204.24/131.13 inference(monotonicity,[status(thm)],[97])).
% 204.24/131.13 tff(99,plain,
% 204.24/131.13 (less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph)) <=> less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), number_of_in(triangles, graph))),
% 204.24/131.13 inference(symmetry,[status(thm)],[98])).
% 204.24/131.13 tff(100,plain,
% 204.24/131.13 ((~less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph))) <=> (~less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), number_of_in(triangles, graph)))),
% 204.24/131.13 inference(monotonicity,[status(thm)],[99])).
% 204.24/131.13 tff(101,plain,
% 204.24/131.13 (~less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph))),
% 204.24/131.13 inference(or_elim,[status(thm)],[45])).
% 204.24/131.13 tff(102,plain,
% 204.24/131.13 (~less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), number_of_in(triangles, graph))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[101, 100])).
% 204.24/131.13 tff(103,plain,
% 204.24/131.13 (^[Things: $i, InThese: $i] : refl(less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph)) <=> less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph)))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(104,plain,
% 204.24/131.13 (![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph)) <=> ![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[103])).
% 204.24/131.13 tff(105,plain,
% 204.24/131.13 (![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph)) <=> ![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(106,axiom,(![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','graph_has_them_all')).
% 204.24/131.13 tff(107,plain,
% 204.24/131.13 (![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[106, 105])).
% 204.24/131.13 tff(108,plain,(
% 204.24/131.13 ![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 204.24/131.13 inference(skolemize,[status(sab)],[107])).
% 204.24/131.13 tff(109,plain,
% 204.24/131.13 (![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[108, 104])).
% 204.24/131.13 tff(110,plain,
% 204.24/131.13 ((~![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))) | less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), number_of_in(triangles, graph))),
% 204.24/131.13 inference(quant_inst,[status(thm)],[])).
% 204.24/131.13 tff(111,plain,
% 204.24/131.13 (less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))), number_of_in(triangles, graph))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[110, 109])).
% 204.24/131.13 tff(112,plain,
% 204.24/131.13 ($false),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[111, 102])).
% 204.24/131.13 tff(113,plain,((~(number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))))), inference(lemma,lemma(discharge,[]))).
% 204.24/131.13 tff(114,plain,
% 204.24/131.13 (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[113, 92])).
% 204.24/131.13 tff(115,plain,
% 204.24/131.13 (((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))))) | (P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))),
% 204.24/131.13 inference(tautology,[status(thm)],[])).
% 204.24/131.13 tff(116,plain,
% 204.24/131.13 ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[115, 114])).
% 204.24/131.13 tff(117,plain,
% 204.24/131.13 (((~vertex(V1!10)) | (~vertex(V2!9)) | (~edge(tptp_fun_E_1(P!11, V2!9, V1!10))) | (~(V1!10 = tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))) | ![TP: $i] : ((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, TP)) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), TP))))))) | ((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))),
% 204.24/131.13 inference(tautology,[status(thm)],[])).
% 204.24/131.13 tff(118,plain,
% 204.24/131.13 ((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))))))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[117, 71])).
% 204.24/131.13 tff(119,plain,
% 204.24/131.13 ((~((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10)))))))) | (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))))))),
% 204.24/131.13 inference(tautology,[status(thm)],[])).
% 204.24/131.13 tff(120,plain,
% 204.24/131.13 ((~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))) | (~((~path(head_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, tptp_fun_TP_2(P!11, V2!9, V1!10))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), tptp_fun_TP_2(P!11, V2!9, V1!10))))))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[119, 118])).
% 204.24/131.13 tff(121,plain,
% 204.24/131.13 (~((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[120, 116])).
% 204.24/131.13 tff(122,plain,
% 204.24/131.13 (((~(V2!9 = head_of(tptp_fun_E_1(P!11, V2!9, V1!10)))) | (~(P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)))) | (P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty))),
% 204.24/131.13 inference(tautology,[status(thm)],[])).
% 204.24/131.13 tff(123,plain,
% 204.24/131.13 (P!11 = path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[122, 121])).
% 204.24/131.13 tff(124,plain,
% 204.24/131.13 (path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty) = P!11),
% 204.24/131.13 inference(symmetry,[status(thm)],[123])).
% 204.24/131.13 tff(125,plain,
% 204.24/131.13 (number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)) = number_of_in(triangles, P!11)),
% 204.24/131.13 inference(monotonicity,[status(thm)],[124])).
% 204.24/131.13 tff(126,plain,
% 204.24/131.13 (number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)) = minus(length_of(P!11), n1)),
% 204.24/131.13 inference(transitivity,[status(thm)],[125, 93, 91])).
% 204.24/131.13 tff(127,plain,
% 204.24/131.13 (less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)), number_of_in(triangles, graph)) <=> less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph))),
% 204.24/131.13 inference(monotonicity,[status(thm)],[126])).
% 204.24/131.13 tff(128,plain,
% 204.24/131.13 (less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph)) <=> less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)), number_of_in(triangles, graph))),
% 204.24/131.13 inference(symmetry,[status(thm)],[127])).
% 204.24/131.13 tff(129,plain,
% 204.24/131.13 ((~less_or_equal(minus(length_of(P!11), n1), number_of_in(triangles, graph))) <=> (~less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)), number_of_in(triangles, graph)))),
% 204.24/131.13 inference(monotonicity,[status(thm)],[128])).
% 204.24/131.13 tff(130,plain,
% 204.24/131.13 (~less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)), number_of_in(triangles, graph))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[101, 129])).
% 204.24/131.13 tff(131,plain,
% 204.24/131.13 ((~![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))) | less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)), number_of_in(triangles, graph))),
% 204.24/131.13 inference(quant_inst,[status(thm)],[])).
% 204.24/131.13 tff(132,plain,
% 204.24/131.13 (less_or_equal(number_of_in(triangles, path_cons(tptp_fun_E_1(P!11, V2!9, V1!10), empty)), number_of_in(triangles, graph))),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[131, 109])).
% 204.24/131.13 tff(133,plain,
% 204.24/131.13 ($false),
% 204.24/131.13 inference(unit_resolution,[status(thm)],[132, 130])).
% 204.24/131.13 tff(134,plain,(~(number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11))), inference(lemma,lemma(discharge,[]))).
% 204.24/131.13 tff(135,plain,
% 204.24/131.13 (^[P: $i, V1: $i, V2: $i] : refl(((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))))) <=> ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))))))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(136,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))))) <=> ![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[135])).
% 204.24/131.13 tff(137,plain,
% 204.24/131.13 (^[P: $i, V1: $i, V2: $i] : trans(monotonicity(rewrite((on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))) <=> (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))))), (((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) <=> ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))))))), rewrite(((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))))) <=> ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))), (((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) <=> ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(138,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) <=> ![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[137])).
% 204.24/131.13 tff(139,plain,
% 204.24/131.13 (^[P: $i, V1: $i, V2: $i] : trans(monotonicity(rewrite(((~path(V1, V2, P)) | ((~(~(on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))))) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) <=> ((~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))), ((((~path(V1, V2, P)) | ((~(~(on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))))) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> (((~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))), rewrite((((~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))), ((((~path(V1, V2, P)) | ((~(~(on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))))) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(140,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : (((~path(V1, V2, P)) | ((~(~(on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))))) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[139])).
% 204.24/131.13 tff(141,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(142,plain,
% 204.24/131.13 (^[P: $i, V1: $i, V2: $i] : trans(monotonicity(rewrite((path(V1, V2, P) & ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) <=> (path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))), (((path(V1, V2, P) & ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))), rewrite(((path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))), (((path(V1, V2, P) & ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))))),
% 204.24/131.13 inference(bind,[status(th)],[])).
% 204.24/131.13 tff(143,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((path(V1, V2, P) & ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 204.24/131.13 inference(quant_intro,[status(thm)],[142])).
% 204.24/131.13 tff(144,axiom,(![P: $i, V1: $i, V2: $i] : ((path(V1, V2, P) & ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','sequential_pairs_and_triangles')).
% 204.24/131.13 tff(145,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[144, 143])).
% 204.24/131.13 tff(146,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((~(path(V1, V2, P) & ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[145, 141])).
% 204.24/131.13 tff(147,plain,(
% 204.24/131.13 ![P: $i, V1: $i, V2: $i] : (((~path(V1, V2, P)) | ((~(~(on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))))) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3)))) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 204.24/131.13 inference(skolemize,[status(sab)],[146])).
% 204.24/131.13 tff(148,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (on_path(tptp_fun_E1_7(P), P) & on_path(tptp_fun_E2_6(P), P) & sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P)) & ![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[147, 140])).
% 204.24/131.13 tff(149,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[148, 138])).
% 204.24/131.13 tff(150,plain,
% 204.24/131.13 (![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))),
% 204.24/131.13 inference(modus_ponens,[status(thm)],[149, 136])).
% 204.24/131.13 tff(151,plain,
% 204.24/131.13 (((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | ((~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | (~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(152,plain,
% 204.24/131.13 (((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))) <=> ((~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(153,plain,
% 204.24/131.13 ((~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))) <=> (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))),
% 204.24/131.13 inference(rewrite,[status(thm)],[])).
% 204.24/131.13 tff(154,plain,
% 204.24/131.13 (((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))) <=> ((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))),
% 204.24/131.13 inference(monotonicity,[status(thm)],[153])).
% 204.24/131.13 tff(155,plain,
% 204.24/131.13 ((~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))) <=> (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))),
% 204.24/131.14 inference(monotonicity,[status(thm)],[154])).
% 204.24/131.14 tff(156,plain,
% 204.24/131.14 (((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))) <=> ((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))),
% 204.24/131.14 inference(monotonicity,[status(thm)],[155])).
% 204.24/131.14 tff(157,plain,
% 204.24/131.14 (((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))) <=> ((~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))),
% 204.24/131.14 inference(transitivity,[status(thm)],[156, 152])).
% 204.24/131.14 tff(158,plain,
% 204.24/131.14 (((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | ((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | ((~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))))),
% 204.24/131.14 inference(monotonicity,[status(thm)],[157])).
% 204.24/131.14 tff(159,plain,
% 204.24/131.14 (((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | ((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | (~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))),
% 204.24/131.14 inference(transitivity,[status(thm)],[158, 151])).
% 204.24/131.14 tff(160,plain,
% 204.24/131.14 ((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | ((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~path(V1!10, V2!9, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))))),
% 204.24/131.14 inference(quant_inst,[status(thm)],[])).
% 204.24/131.14 tff(161,plain,
% 204.24/131.14 ((~![P: $i, V1: $i, V2: $i] : ((number_of_in(sequential_pairs, P) = number_of_in(triangles, P)) | (~path(V1, V2, P)) | (~((~on_path(tptp_fun_E1_7(P), P)) | (~on_path(tptp_fun_E2_6(P), P)) | (~sequential(tptp_fun_E1_7(P), tptp_fun_E2_6(P))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P), tptp_fun_E2_6(P), E3))))))) | (~path(V1!10, V2!9, P!11)) | (number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))),
% 204.24/131.14 inference(modus_ponens,[status(thm)],[160, 159])).
% 204.24/131.14 tff(162,plain,
% 204.24/131.14 ((number_of_in(sequential_pairs, P!11) = number_of_in(triangles, P!11)) | (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))))),
% 204.24/131.14 inference(unit_resolution,[status(thm)],[161, 150, 51])).
% 204.24/131.14 tff(163,plain,
% 204.24/131.14 (~((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))))),
% 204.24/131.14 inference(unit_resolution,[status(thm)],[162, 134])).
% 204.24/131.14 tff(164,plain,
% 204.24/131.14 (((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))) | sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))),
% 204.24/131.14 inference(tautology,[status(thm)],[])).
% 204.24/131.14 tff(165,plain,
% 204.24/131.14 (sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))),
% 204.24/131.14 inference(unit_resolution,[status(thm)],[164, 163])).
% 204.24/131.14 tff(166,plain,
% 204.24/131.14 ((sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)))),
% 204.24/131.14 inference(tautology,[status(thm)],[])).
% 204.24/131.14 tff(167,plain,
% 204.24/131.14 (sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))),
% 204.24/131.14 inference(unit_resolution,[status(thm)],[166, 165])).
% 204.24/131.14 tff(168,plain,
% 204.24/131.14 (^[P: $i, V1: $i, V2: $i] : refl(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 204.24/131.14 inference(bind,[status(th)],[])).
% 204.24/131.14 tff(169,plain,
% 204.24/131.14 (![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))))))))),
% 204.29/131.14 inference(quant_intro,[status(thm)],[168])).
% 204.29/131.14 tff(170,plain,
% 204.29/131.14 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 204.29/131.14 inference(bind,[status(th)],[])).
% 204.29/131.14 tff(171,plain,
% 204.29/131.14 (![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))))))))),
% 204.29/131.14 inference(quant_intro,[status(thm)],[170])).
% 204.29/131.14 tff(172,plain,
% 204.29/131.14 (![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))))))))),
% 204.29/131.14 inference(transitivity,[status(thm)],[171, 169])).
% 204.29/131.14 tff(173,plain,
% 204.29/131.14 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P)))))))))),
% 204.29/131.14 inference(bind,[status(th)],[])).
% 204.29/131.14 tff(174,plain,
% 204.29/131.14 (![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))))))))),
% 204.29/131.14 inference(quant_intro,[status(thm)],[173])).
% 204.29/131.14 tff(175,plain,
% 204.29/131.14 (^[P: $i, V1: $i, V2: $i] : rewrite(((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (((~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P)))))))),
% 204.29/131.14 inference(bind,[status(th)],[])).
% 204.29/131.14 tff(176,plain,
% 204.29/131.14 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (((~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))))),
% 204.29/131.14 inference(quant_intro,[status(thm)],[175])).
% 204.29/131.14 tff(177,plain,
% 204.29/131.14 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 204.29/131.14 inference(rewrite,[status(thm)],[])).
% 204.29/131.14 tff(178,plain,
% 204.29/131.14 (^[P: $i, V1: $i, V2: $i] : trans(monotonicity(quant_intro(proof_bind(^[E1: $i, E2: $i] : trans(monotonicity(trans(monotonicity(rewrite((sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))) <=> (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))), (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) <=> ((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))), rewrite(((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) <=> (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))), (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) <=> (on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))))), ((((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)) <=> ((on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)))), rewrite(((on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)) <=> ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))), ((((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)) <=> ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))))), (![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)) <=> ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))), ((path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P))) <=> (path(V1, V2, P) => ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))))), rewrite((path(V1, V2, P) => ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))), ((path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P))) <=> ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))))),
% 204.29/131.14 inference(bind,[status(th)],[])).
% 204.29/131.14 tff(179,plain,
% 204.29/131.14 (![P: $i, V1: $i, V2: $i] : (path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 204.29/131.14 inference(quant_intro,[status(thm)],[178])).
% 204.29/131.14 tff(180,axiom,(![P: $i, V1: $i, V2: $i] : (path(V1, V2, P) => ![E1: $i, E2: $i] : (((on_path(E1, P) & on_path(E2, P)) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P)))) => precedes(E1, E2, P)))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','precedes_defn')).
% 204.29/131.14 tff(181,plain,
% 204.29/131.14 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 204.29/131.14 inference(modus_ponens,[status(thm)],[180, 179])).
% 204.29/131.14 tff(182,plain,
% 204.29/131.14 (![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : ((~(on_path(E1, P) & on_path(E2, P) & (sequential(E1, E2) | ?[E3: $i] : (sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 204.29/131.14 inference(modus_ponens,[status(thm)],[181, 177])).
% 204.29/131.14 tff(183,plain,(
% 204.29/131.14 ![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (((~on_path(E1, P)) | (~on_path(E2, P)) | ((~sequential(E1, E2)) & ![E3: $i] : (~(sequential(E1, E3) & precedes(E3, E2, P))))) | precedes(E1, E2, P)))),
% 204.29/131.14 inference(skolemize,[status(sab)],[182])).
% 204.29/131.14 tff(184,plain,
% 204.29/131.14 (![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))))))),
% 204.29/131.14 inference(modus_ponens,[status(thm)],[183, 176])).
% 204.29/131.14 tff(185,plain,
% 204.29/131.14 (![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))))))))),
% 204.29/131.14 inference(modus_ponens,[status(thm)],[184, 174])).
% 204.29/131.14 tff(186,plain,
% 204.29/131.14 (![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))))))))),
% 204.29/131.14 inference(modus_ponens,[status(thm)],[185, 172])).
% 204.29/131.14 tff(187,plain,
% 204.29/131.14 (((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) | ((~path(V1!10, V2!9, P!11)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11))))))))) <=> ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) | (~path(V1!10, V2!9, P!11)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11))))))))),
% 204.29/131.14 inference(rewrite,[status(thm)],[])).
% 204.29/131.14 tff(188,plain,
% 204.29/131.14 ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) | ((~path(V1!10, V2!9, P!11)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11))))))))),
% 204.29/131.14 inference(quant_inst,[status(thm)],[])).
% 204.29/131.14 tff(189,plain,
% 204.29/131.14 ((~![P: $i, V1: $i, V2: $i] : ((~path(V1, V2, P)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P) | (~on_path(E1, P)) | (~on_path(E2, P)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P))))))))) | (~path(V1!10, V2!9, P!11)) | ![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))),
% 204.29/131.14 inference(modus_ponens,[status(thm)],[188, 187])).
% 204.29/131.14 tff(190,plain,
% 204.29/131.14 (![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))),
% 204.29/131.14 inference(unit_resolution,[status(thm)],[189, 186, 51])).
% 204.29/131.14 tff(191,plain,
% 204.29/131.14 (((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))) | on_path(tptp_fun_E2_6(P!11), P!11)),
% 204.29/131.14 inference(tautology,[status(thm)],[])).
% 204.29/131.14 tff(192,plain,
% 204.29/131.14 (on_path(tptp_fun_E2_6(P!11), P!11)),
% 204.29/131.14 inference(unit_resolution,[status(thm)],[191, 163])).
% 204.29/131.14 tff(193,plain,
% 204.29/131.14 (((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))) | on_path(tptp_fun_E1_7(P!11), P!11)),
% 204.29/131.14 inference(tautology,[status(thm)],[])).
% 204.29/131.14 tff(194,plain,
% 204.29/131.14 (on_path(tptp_fun_E1_7(P!11), P!11)),
% 204.29/131.14 inference(unit_resolution,[status(thm)],[193, 163])).
% 204.29/131.14 tff(195,plain,
% 204.29/131.14 (((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | ((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))) <=> ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))),
% 204.29/131.14 inference(rewrite,[status(thm)],[])).
% 204.29/131.14 tff(196,plain,
% 204.29/131.14 ((precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))))) <=> ((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))),
% 204.29/131.14 inference(rewrite,[status(thm)],[])).
% 204.29/131.14 tff(197,plain,
% 204.29/131.14 ((~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))) <=> (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))))),
% 204.29/131.14 inference(rewrite,[status(thm)],[])).
% 204.29/131.14 tff(198,plain,
% 204.29/131.14 ((precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))))) <=> (precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))),
% 204.29/131.14 inference(monotonicity,[status(thm)],[197])).
% 204.29/131.14 tff(199,plain,
% 204.29/131.14 ((precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))))) <=> ((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))),
% 204.30/131.14 inference(transitivity,[status(thm)],[198, 196])).
% 204.30/131.14 tff(200,plain,
% 204.30/131.14 (((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | (precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))) <=> ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | ((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))))))),
% 204.30/131.14 inference(monotonicity,[status(thm)],[199])).
% 204.30/131.14 tff(201,plain,
% 204.30/131.14 (((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | (precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))) <=> ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))),
% 204.30/131.14 inference(transitivity,[status(thm)],[200, 195])).
% 204.30/131.14 tff(202,plain,
% 204.30/131.14 ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | (precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11)))))))),
% 204.30/131.14 inference(quant_inst,[status(thm)],[])).
% 204.30/131.14 tff(203,plain,
% 204.30/131.14 ((~![E1: $i, E2: $i] : (precedes(E1, E2, P!11) | (~on_path(E1, P!11)) | (~on_path(E2, P!11)) | (~(sequential(E1, E2) | (~![E3: $i] : ((~sequential(E1, E3)) | (~precedes(E3, E2, P!11)))))))) | (~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11) | (~(sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)) | (~![E3: $i] : ((~sequential(tptp_fun_E1_7(P!11), E3)) | (~precedes(E3, tptp_fun_E2_6(P!11), P!11))))))),
% 204.30/131.14 inference(modus_ponens,[status(thm)],[202, 201])).
% 204.30/131.14 tff(204,plain,
% 204.30/131.14 (precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)),
% 204.30/131.14 inference(unit_resolution,[status(thm)],[203, 194, 192, 190, 167])).
% 204.30/131.14 tff(205,plain,
% 204.30/131.14 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : refl((triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))) <=> (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))))),
% 204.30/131.14 inference(bind,[status(th)],[])).
% 204.30/131.14 tff(206,plain,
% 204.30/131.14 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))),
% 204.30/131.14 inference(quant_intro,[status(thm)],[205])).
% 204.30/131.14 tff(207,plain,
% 204.30/131.14 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : trans(monotonicity(trans(monotonicity(rewrite((shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2)) <=> (~((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))))), ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) <=> (~(~((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))))))), rewrite((~(~((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))), ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) <=> ((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))))), (((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | triangle(E1, E2, tptp_fun_E3_8(E2, E1))) <=> (((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))) | triangle(E1, E2, tptp_fun_E3_8(E2, E1))))), rewrite((((~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2))) | triangle(E1, E2, tptp_fun_E3_8(E2, E1))) <=> (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))), (((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | triangle(E1, E2, tptp_fun_E3_8(E2, E1))) <=> (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))))),
% 204.30/131.14 inference(bind,[status(th)],[])).
% 204.30/131.14 tff(208,plain,
% 204.30/131.14 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | triangle(E1, E2, tptp_fun_E3_8(E2, E1))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))),
% 204.30/131.14 inference(quant_intro,[status(thm)],[207])).
% 204.30/131.14 tff(209,plain,
% 204.30/131.14 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.14 inference(rewrite,[status(thm)],[])).
% 204.30/131.14 tff(210,plain,
% 204.30/131.14 (($false | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.14 inference(rewrite,[status(thm)],[])).
% 204.30/131.14 tff(211,plain,
% 204.30/131.14 ((~$true) <=> $false),
% 204.30/131.14 inference(rewrite,[status(thm)],[])).
% 204.30/131.14 tff(212,plain,
% 204.30/131.14 (complete),
% 204.30/131.14 inference(or_elim,[status(thm)],[35])).
% 204.30/131.14 tff(213,plain,
% 204.30/131.14 (complete <=> $true),
% 204.30/131.14 inference(iff_true,[status(thm)],[212])).
% 204.30/131.14 tff(214,plain,
% 204.30/131.14 ((~complete) <=> (~$true)),
% 204.30/131.14 inference(monotonicity,[status(thm)],[213])).
% 204.30/131.14 tff(215,plain,
% 204.30/131.14 ((~complete) <=> $false),
% 204.30/131.14 inference(transitivity,[status(thm)],[214, 211])).
% 204.30/131.14 tff(216,plain,
% 204.30/131.14 (((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) <=> ($false | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))),
% 204.30/131.14 inference(monotonicity,[status(thm)],[215])).
% 204.30/131.14 tff(217,plain,
% 204.30/131.14 (((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.15 inference(transitivity,[status(thm)],[216, 210])).
% 204.30/131.15 tff(218,plain,
% 204.30/131.15 (((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) <=> ((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))),
% 204.30/131.15 inference(rewrite,[status(thm)],[])).
% 204.30/131.15 tff(219,plain,
% 204.30/131.15 ((complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))) <=> ((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))),
% 204.30/131.15 inference(rewrite,[status(thm)],[])).
% 204.30/131.15 tff(220,plain,
% 204.30/131.15 (^[V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : trans(monotonicity(rewrite(((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) <=> (shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))), ((((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3)) <=> ((shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3)))), rewrite(((shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3)) <=> ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))), ((((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3)) <=> ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))))),
% 204.30/131.15 inference(bind,[status(th)],[])).
% 204.30/131.15 tff(221,plain,
% 204.30/131.15 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3)) <=> ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.15 inference(quant_intro,[status(thm)],[220])).
% 204.30/131.15 tff(222,plain,
% 204.30/131.15 ((complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) <=> (complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))),
% 204.30/131.15 inference(monotonicity,[status(thm)],[221])).
% 204.30/131.15 tff(223,plain,
% 204.30/131.15 ((complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))) <=> ((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3)))),
% 204.30/131.15 inference(transitivity,[status(thm)],[222, 219])).
% 204.30/131.15 tff(224,axiom,(complete => ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (((shortest_path(V1, V2, P) & precedes(E1, E2, P)) & sequential(E1, E2)) => ?[E3: $i] : triangle(E1, E2, E3))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','sequential_is_triangle')).
% 204.30/131.15 tff(225,plain,
% 204.30/131.15 ((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.15 inference(modus_ponens,[status(thm)],[224, 223])).
% 204.30/131.15 tff(226,plain,
% 204.30/131.15 ((~complete) | ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.15 inference(modus_ponens,[status(thm)],[225, 218])).
% 204.30/131.15 tff(227,plain,
% 204.30/131.15 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.15 inference(modus_ponens,[status(thm)],[226, 217])).
% 204.30/131.15 tff(228,plain,
% 204.30/131.15 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | ?[E3: $i] : triangle(E1, E2, E3))),
% 204.30/131.15 inference(modus_ponens,[status(thm)],[227, 209])).
% 204.30/131.15 tff(229,plain,(
% 204.30/131.15 ![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : ((~(shortest_path(V1, V2, P) & precedes(E1, E2, P) & sequential(E1, E2))) | triangle(E1, E2, tptp_fun_E3_8(E2, E1)))),
% 204.30/131.15 inference(skolemize,[status(sab)],[228])).
% 204.30/131.15 tff(230,plain,
% 204.30/131.15 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))),
% 204.30/131.15 inference(modus_ponens,[status(thm)],[229, 208])).
% 204.30/131.15 tff(231,plain,
% 204.30/131.15 (![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))),
% 204.30/131.15 inference(modus_ponens,[status(thm)],[230, 206])).
% 204.30/131.15 tff(232,plain,
% 204.30/131.15 (((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | ((~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)))) <=> ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)))),
% 204.30/131.15 inference(rewrite,[status(thm)],[])).
% 204.30/131.15 tff(233,plain,
% 204.30/131.15 ((triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11)))) <=> ((~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)))),
% 204.30/131.15 inference(rewrite,[status(thm)],[])).
% 204.30/131.15 tff(234,plain,
% 204.30/131.15 (((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | (triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))))) <=> ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | ((~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11))))),
% 204.30/131.15 inference(monotonicity,[status(thm)],[233])).
% 204.30/131.15 tff(235,plain,
% 204.30/131.15 (((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | (triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))))) <=> ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)))),
% 204.30/131.16 inference(transitivity,[status(thm)],[234, 232])).
% 204.30/131.16 tff(236,plain,
% 204.30/131.16 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | (triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11)) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))))),
% 204.30/131.16 inference(quant_inst,[status(thm)],[])).
% 204.30/131.16 tff(237,plain,
% 204.30/131.16 ((~![V1: $i, V2: $i, E1: $i, E2: $i, P: $i] : (triangle(E1, E2, tptp_fun_E3_8(E2, E1)) | (~shortest_path(V1, V2, P)) | (~precedes(E1, E2, P)) | (~sequential(E1, E2)))) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~precedes(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), P!11)) | triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11))),
% 204.30/131.16 inference(modus_ponens,[status(thm)],[236, 235])).
% 204.30/131.16 tff(238,plain,
% 204.30/131.16 (triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))) | (~shortest_path(tail_of(tptp_fun_E_1(P!11, V2!9, V1!10)), V2!9, P!11))),
% 204.30/131.16 inference(unit_resolution,[status(thm)],[237, 231, 165, 204])).
% 204.30/131.16 tff(239,plain,
% 204.30/131.16 (triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11)))),
% 204.30/131.16 inference(unit_resolution,[status(thm)],[238, 77])).
% 204.30/131.16 tff(240,plain,
% 204.30/131.16 (((~on_path(tptp_fun_E1_7(P!11), P!11)) | (~on_path(tptp_fun_E2_6(P!11), P!11)) | (~sequential(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11))) | (~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3)))) | ![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))),
% 204.30/131.16 inference(tautology,[status(thm)],[])).
% 204.30/131.16 tff(241,plain,
% 204.30/131.16 (![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))),
% 204.30/131.16 inference(unit_resolution,[status(thm)],[240, 163])).
% 204.30/131.16 tff(242,plain,
% 204.30/131.16 ((~![E3: $i] : (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), E3))) | (~triangle(tptp_fun_E1_7(P!11), tptp_fun_E2_6(P!11), tptp_fun_E3_8(tptp_fun_E2_6(P!11), tptp_fun_E1_7(P!11))))),
% 204.30/131.16 inference(quant_inst,[status(thm)],[])).
% 204.30/131.16 tff(243,plain,
% 204.30/131.16 ($false),
% 204.30/131.16 inference(unit_resolution,[status(thm)],[242, 241, 239])).
% 204.30/131.16 % SZS output end Proof
%------------------------------------------------------------------------------