TSTP Solution File: GRA002+4 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRA002+4 : TPTP v8.1.0. Bugfixed v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n022.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 20:47:26 EDT 2022
% Result : Theorem 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRA002+4 : TPTP v8.1.0. Bugfixed v3.2.0.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n022.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.19/0.34 % WCLimit : 300
% 0.19/0.34 % DateTime : Wed Aug 31 13:20:40 EDT 2022
% 0.19/0.34 % CPUTime :
% 0.19/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.19/0.34 Usage: tptp [options] [-file:]file
% 0.19/0.34 -h, -? prints this message.
% 0.19/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.19/0.34 -m, -model generate model.
% 0.19/0.34 -p, -proof generate proof.
% 0.19/0.34 -c, -core generate unsat core of named formulas.
% 0.19/0.34 -st, -statistics display statistics.
% 0.19/0.34 -t:timeout set timeout (in second).
% 0.19/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.19/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.19/0.34 -<param>:<value> configuration parameter and value.
% 0.19/0.34 -o:<output-file> file to place output in.
% 0.19/0.41 % SZS status Theorem
% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 tff(less_or_equal_type, type, (
% 0.19/0.41 less_or_equal: ( $i * $i ) > $o)).
% 0.19/0.41 tff(number_of_in_type, type, (
% 0.19/0.41 number_of_in: ( $i * $i ) > $i)).
% 0.19/0.41 tff(graph_type, type, (
% 0.19/0.41 graph: $i)).
% 0.19/0.41 tff(triangles_type, type, (
% 0.19/0.41 triangles: $i)).
% 0.19/0.41 tff(minus_type, type, (
% 0.19/0.41 minus: ( $i * $i ) > $i)).
% 0.19/0.41 tff(n1_type, type, (
% 0.19/0.41 n1: $i)).
% 0.19/0.41 tff(length_of_type, type, (
% 0.19/0.41 length_of: $i > $i)).
% 0.19/0.41 tff(tptp_fun_P_10_type, type, (
% 0.19/0.41 tptp_fun_P_10: $i)).
% 0.19/0.41 tff(sequential_pairs_type, type, (
% 0.19/0.41 sequential_pairs: $i)).
% 0.19/0.41 tff(shortest_path_type, type, (
% 0.19/0.41 shortest_path: ( $i * $i * $i ) > $o)).
% 0.19/0.41 tff(tptp_fun_V2_8_type, type, (
% 0.19/0.41 tptp_fun_V2_8: $i)).
% 0.19/0.41 tff(tptp_fun_V1_9_type, type, (
% 0.19/0.41 tptp_fun_V1_9: $i)).
% 0.19/0.41 tff(complete_type, type, (
% 0.19/0.41 complete: $o)).
% 0.19/0.41 tff(path_type, type, (
% 0.19/0.41 path: ( $i * $i * $i ) > $o)).
% 0.19/0.41 tff(tptp_fun_P_5_type, type, (
% 0.19/0.41 tptp_fun_P_5: ( $i * $i * $i ) > $i)).
% 0.19/0.41 tff(1,plain,
% 0.19/0.41 ((~![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))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(2,plain,
% 0.19/0.41 ((~(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)))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(3,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')).
% 0.19/0.41 tff(4,plain,
% 0.19/0.41 (~((~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))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.19/0.41 tff(5,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(or_elim,[status(thm)],[4])).
% 0.19/0.41 tff(6,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.19/0.41 tff(7,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.19/0.41 tff(8,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[7, 1])).
% 0.19/0.41 tff(9,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[8, 1])).
% 0.19/0.41 tff(10,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[9, 1])).
% 0.19/0.41 tff(11,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[10, 1])).
% 0.19/0.41 tff(12,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[11, 1])).
% 0.19/0.41 tff(13,plain,
% 0.19/0.41 (~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | less_or_equal(minus(length_of(P), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[12, 1])).
% 0.19/0.41 tff(14,plain,(
% 0.19/0.41 ~((~shortest_path(V1!9, V2!8, P!10)) | less_or_equal(minus(length_of(P!10), n1), number_of_in(triangles, graph)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[13])).
% 0.19/0.41 tff(15,plain,
% 0.19/0.41 (shortest_path(V1!9, V2!8, P!10)),
% 0.19/0.42 inference(or_elim,[status(thm)],[14])).
% 0.19/0.42 tff(16,plain,
% 0.19/0.42 (^[P: $i, V1: $i, V2: $i] : refl(((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(17,plain,
% 0.19/0.42 (![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[16])).
% 0.19/0.42 tff(18,plain,
% 0.19/0.42 (![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(19,plain,
% 0.19/0.42 (($false | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(20,plain,
% 0.19/0.42 ((~$true) <=> $false),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(21,plain,
% 0.19/0.42 (complete),
% 0.19/0.42 inference(or_elim,[status(thm)],[4])).
% 0.19/0.42 tff(22,plain,
% 0.19/0.42 (complete <=> $true),
% 0.19/0.42 inference(iff_true,[status(thm)],[21])).
% 0.19/0.42 tff(23,plain,
% 0.19/0.42 ((~complete) <=> (~$true)),
% 0.19/0.42 inference(monotonicity,[status(thm)],[22])).
% 0.19/0.42 tff(24,plain,
% 0.19/0.42 ((~complete) <=> $false),
% 0.19/0.42 inference(transitivity,[status(thm)],[23, 20])).
% 0.19/0.42 tff(25,plain,
% 0.19/0.42 (((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> ($false | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[24])).
% 0.19/0.42 tff(26,plain,
% 0.19/0.42 (((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[25, 19])).
% 0.19/0.42 tff(27,plain,
% 0.19/0.42 (((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> ((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(28,plain,
% 0.19/0.42 ((complete => ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> ((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(29,plain,
% 0.19/0.42 (^[P: $i, V1: $i, V2: $i] : rewrite((shortest_path(V1, V2, P) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(30,plain,
% 0.19/0.42 (![P: $i, V1: $i, V2: $i] : (shortest_path(V1, V2, P) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))) <=> ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[29])).
% 0.19/0.42 tff(31,plain,
% 0.19/0.42 ((complete => ![P: $i, V1: $i, V2: $i] : (shortest_path(V1, V2, P) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> (complete => ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[30])).
% 0.19/0.42 tff(32,plain,
% 0.19/0.42 ((complete => ![P: $i, V1: $i, V2: $i] : (shortest_path(V1, V2, P) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) <=> ((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[31, 28])).
% 0.19/0.42 tff(33,axiom,(complete => ![P: $i, V1: $i, V2: $i] : (shortest_path(V1, V2, P) => (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','triangles_and_sequential_pairs')).
% 0.19/0.42 tff(34,plain,
% 0.19/0.42 ((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.19/0.42 tff(35,plain,
% 0.19/0.42 ((~complete) | ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.19/0.42 tff(36,plain,
% 0.19/0.42 (![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[35, 26])).
% 0.19/0.42 tff(37,plain,
% 0.19/0.42 (![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[36, 18])).
% 0.19/0.42 tff(38,plain,(
% 0.19/0.42 ![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[37])).
% 0.19/0.42 tff(39,plain,
% 0.19/0.42 (![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[38, 17])).
% 0.19/0.42 tff(40,plain,
% 0.19/0.42 (((~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) | ((~shortest_path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = number_of_in(triangles, P!10)))) <=> ((~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) | (~shortest_path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = number_of_in(triangles, P!10)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(41,plain,
% 0.19/0.42 ((~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) | ((~shortest_path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = number_of_in(triangles, P!10)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(42,plain,
% 0.19/0.42 ((~![P: $i, V1: $i, V2: $i] : ((~shortest_path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = number_of_in(triangles, P)))) | (~shortest_path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = number_of_in(triangles, P!10))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.19/0.42 tff(43,plain,
% 0.19/0.42 (number_of_in(sequential_pairs, P!10) = number_of_in(triangles, P!10)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[42, 39, 15])).
% 0.19/0.42 tff(44,plain,
% 0.19/0.42 (^[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)))))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(45,plain,
% 0.19/0.42 (![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))))))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[44])).
% 0.19/0.42 tff(46,plain,
% 0.19/0.42 (^[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))))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(47,plain,
% 0.19/0.42 (![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)))))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[46])).
% 0.19/0.42 tff(48,plain,
% 0.19/0.42 (^[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))))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(49,plain,
% 0.19/0.42 (![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)))))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[48])).
% 0.19/0.42 tff(50,plain,
% 0.19/0.42 (![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)))))))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[49, 47])).
% 0.19/0.42 tff(51,plain,
% 0.19/0.42 (^[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)))))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(52,plain,
% 0.19/0.42 (![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)))))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[51])).
% 0.19/0.42 tff(53,plain,
% 0.19/0.42 (^[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))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(54,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[53])).
% 0.19/0.42 tff(55,plain,
% 0.19/0.42 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))) <=> ![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(56,plain,
% 0.19/0.42 (^[V1: $i, V2: $i, SP: $i] : rewrite((shortest_path(V1, V2, SP) <=> ((path(V1, V2, SP) & (~(V1 = V2))) & ![P: $i] : (path(V1, V2, P) => less_or_equal(length_of(SP), length_of(P))))) <=> (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(57,plain,
% 0.19/0.42 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> ((path(V1, V2, SP) & (~(V1 = V2))) & ![P: $i] : (path(V1, V2, P) => less_or_equal(length_of(SP), length_of(P))))) <=> ![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[56])).
% 0.19/0.42 tff(58,axiom,(![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> ((path(V1, V2, SP) & (~(V1 = V2))) & ![P: $i] : (path(V1, V2, P) => less_or_equal(length_of(SP), length_of(P)))))), file('/export/starexec/sandbox2/benchmark/Axioms/GRA001+0.ax','shortest_path_defn')).
% 0.19/0.42 tff(59,plain,
% 0.19/0.42 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.42 tff(60,plain,
% 0.19/0.42 (![V1: $i, V2: $i, SP: $i] : (shortest_path(V1, V2, SP) <=> (path(V1, V2, SP) & (~(V1 = V2)) & ![P: $i] : ((~path(V1, V2, P)) | less_or_equal(length_of(SP), length_of(P)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.19/0.42 tff(61,plain,(
% 0.19/0.42 ![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))))))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[60])).
% 0.19/0.42 tff(62,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[61, 54])).
% 0.19/0.43 tff(63,plain,
% 0.19/0.43 (![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)))))))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[62, 52])).
% 0.19/0.43 tff(64,plain,
% 0.19/0.43 (![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)))))))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[63, 50])).
% 0.19/0.43 tff(65,plain,
% 0.19/0.43 (![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))))))))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[64, 45])).
% 0.19/0.43 tff(66,plain,
% 0.19/0.43 (((~![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!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), 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!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(67,plain,
% 0.19/0.43 ((~((~((V1!9 = V2!8) | shortest_path(V1!9, V2!8, P!10) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))))) <=> (~((~(shortest_path(V1!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(68,plain,
% 0.19/0.43 (((~![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!9 = V2!8) | shortest_path(V1!9, V2!8, P!10) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), 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!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[67])).
% 0.19/0.43 tff(69,plain,
% 0.19/0.43 (((~![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!9 = V2!8) | shortest_path(V1!9, V2!8, P!10) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), 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!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[68, 66])).
% 0.19/0.43 tff(70,plain,
% 0.19/0.43 ((~![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!9 = V2!8) | shortest_path(V1!9, V2!8, P!10) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))))))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(71,plain,
% 0.19/0.44 ((~![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!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))))))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.19/0.44 tff(72,plain,
% 0.19/0.44 (~((~(shortest_path(V1!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[71, 65])).
% 0.19/0.44 tff(73,plain,
% 0.19/0.44 (((~(shortest_path(V1!9, V2!8, P!10) | (V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~((~path(V1!9, V2!8, tptp_fun_P_5(P!10, V2!8, V1!9))) | less_or_equal(length_of(P!10), length_of(tptp_fun_P_5(P!10, V2!8, V1!9))))))) | (~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))))) | ((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(74,plain,
% 0.19/0.44 ((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.19/0.44 tff(75,plain,
% 0.19/0.44 ((~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))) | (~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(76,plain,
% 0.19/0.44 ((~((~shortest_path(V1!9, V2!8, P!10)) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))))) | (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[75, 15])).
% 0.19/0.44 tff(77,plain,
% 0.19/0.44 (~((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P)))))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[76, 74])).
% 0.19/0.44 tff(78,plain,
% 0.19/0.44 (((V1!9 = V2!8) | (~path(V1!9, V2!8, P!10)) | (~![P: $i] : ((~path(V1!9, V2!8, P)) | less_or_equal(length_of(P!10), length_of(P))))) | path(V1!9, V2!8, P!10)),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(79,plain,
% 0.19/0.44 (path(V1!9, V2!8, P!10)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[78, 77])).
% 0.19/0.44 tff(80,plain,
% 0.19/0.44 (^[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))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(81,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[80])).
% 0.19/0.44 tff(82,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(83,plain,
% 0.19/0.44 (^[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))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(84,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[83])).
% 0.19/0.44 tff(85,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')).
% 0.19/0.44 tff(86,plain,
% 0.19/0.44 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.44 tff(87,plain,
% 0.19/0.44 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.19/0.44 tff(88,plain,(
% 0.19/0.44 ![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 0.19/0.44 inference(skolemize,[status(sab)],[87])).
% 0.19/0.44 tff(89,plain,
% 0.19/0.44 (![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.19/0.44 tff(90,plain,
% 0.19/0.44 (((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | ((~path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = minus(length_of(P!10), n1)))) <=> ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | (~path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = minus(length_of(P!10), n1)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(91,plain,
% 0.19/0.44 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | ((~path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = minus(length_of(P!10), n1)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(92,plain,
% 0.19/0.44 ((~![V1: $i, V2: $i, P: $i] : ((~path(V1, V2, P)) | (number_of_in(sequential_pairs, P) = minus(length_of(P), n1)))) | (~path(V1!9, V2!8, P!10)) | (number_of_in(sequential_pairs, P!10) = minus(length_of(P!10), n1))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.44 tff(93,plain,
% 0.19/0.44 (number_of_in(sequential_pairs, P!10) = minus(length_of(P!10), n1)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[92, 89, 79])).
% 0.19/0.44 tff(94,plain,
% 0.19/0.44 (minus(length_of(P!10), n1) = number_of_in(sequential_pairs, P!10)),
% 0.19/0.44 inference(symmetry,[status(thm)],[93])).
% 0.19/0.44 tff(95,plain,
% 0.19/0.44 (minus(length_of(P!10), n1) = number_of_in(triangles, P!10)),
% 0.19/0.44 inference(transitivity,[status(thm)],[94, 43])).
% 0.19/0.44 tff(96,plain,
% 0.19/0.44 (less_or_equal(minus(length_of(P!10), n1), number_of_in(triangles, graph)) <=> less_or_equal(number_of_in(triangles, P!10), number_of_in(triangles, graph))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[95])).
% 0.19/0.44 tff(97,plain,
% 0.19/0.44 (less_or_equal(number_of_in(triangles, P!10), number_of_in(triangles, graph)) <=> less_or_equal(minus(length_of(P!10), n1), number_of_in(triangles, graph))),
% 0.19/0.44 inference(symmetry,[status(thm)],[96])).
% 0.19/0.44 tff(98,plain,
% 0.19/0.44 (^[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)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(99,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[98])).
% 0.19/0.44 tff(100,plain,
% 0.19/0.44 (![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))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(101,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')).
% 0.19/0.44 tff(102,plain,
% 0.19/0.44 (![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.19/0.44 tff(103,plain,(
% 0.19/0.44 ![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 0.19/0.44 inference(skolemize,[status(sab)],[102])).
% 0.19/0.44 tff(104,plain,
% 0.19/0.44 (![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.19/0.44 tff(105,plain,
% 0.19/0.44 ((~![Things: $i, InThese: $i] : less_or_equal(number_of_in(Things, InThese), number_of_in(Things, graph))) | less_or_equal(number_of_in(triangles, P!10), number_of_in(triangles, graph))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(106,plain,
% 0.19/0.44 (less_or_equal(number_of_in(triangles, P!10), number_of_in(triangles, graph))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[105, 104])).
% 0.19/0.44 tff(107,plain,
% 0.19/0.44 (less_or_equal(minus(length_of(P!10), n1), number_of_in(triangles, graph))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[106, 97])).
% 0.19/0.44 tff(108,plain,
% 0.19/0.44 (~less_or_equal(minus(length_of(P!10), n1), number_of_in(triangles, graph))),
% 0.19/0.44 inference(or_elim,[status(thm)],[14])).
% 0.19/0.44 tff(109,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.19/0.44 % SZS output end Proof
%------------------------------------------------------------------------------