/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern preorder(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) PiDP (7) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 0 ms] (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Clauses: preorder(T, Xs) :- preorder_dl(T, -(Xs, [])). preorder_dl(nil, -(X, X)). preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))). Query: preorder(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: preorder_in_2: (b,f) preorder_dl_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X)) preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs)) U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs) The argument filtering Pi contains the following mapping: preorder_in_ga(x1, x2) = preorder_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1) nil = nil preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga tree(x1, x2, x3) = tree(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6) preorder_out_ga(x1, x2) = preorder_out_ga Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X)) preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs)) U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs) The argument filtering Pi contains the following mapping: preorder_in_ga(x1, x2) = preorder_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1) nil = nil preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga tree(x1, x2, x3) = tree(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6) preorder_out_ga(x1, x2) = preorder_out_ga ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: PREORDER_IN_GA(T, Xs) -> U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) PREORDER_IN_GA(T, Xs) -> PREORDER_DL_IN_GA(T, -(Xs, [])) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys)) U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs)) The TRS R consists of the following rules: preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X)) preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs)) U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs) The argument filtering Pi contains the following mapping: preorder_in_ga(x1, x2) = preorder_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1) nil = nil preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga tree(x1, x2, x3) = tree(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6) preorder_out_ga(x1, x2) = preorder_out_ga PREORDER_IN_GA(x1, x2) = PREORDER_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6) U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: PREORDER_IN_GA(T, Xs) -> U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) PREORDER_IN_GA(T, Xs) -> PREORDER_DL_IN_GA(T, -(Xs, [])) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys)) U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs)) The TRS R consists of the following rules: preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X)) preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs)) U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs) The argument filtering Pi contains the following mapping: preorder_in_ga(x1, x2) = preorder_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1) nil = nil preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga tree(x1, x2, x3) = tree(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6) preorder_out_ga(x1, x2) = preorder_out_ga PREORDER_IN_GA(x1, x2) = PREORDER_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6) U3_GA(x1, x2, x3, x4, x5, x6) = U3_GA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs)) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys)) The TRS R consists of the following rules: preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, []))) preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X)) preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs)) U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs) The argument filtering Pi contains the following mapping: preorder_in_ga(x1, x2) = preorder_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1) nil = nil preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga tree(x1, x2, x3) = tree(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6) preorder_out_ga(x1, x2) = preorder_out_ga PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs)) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys)) The TRS R consists of the following rules: preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X)) preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys))) U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs))) U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs)) The argument filtering Pi contains the following mapping: preorder_dl_in_ga(x1, x2) = preorder_dl_in_ga(x1) nil = nil preorder_dl_out_ga(x1, x2) = preorder_dl_out_ga tree(x1, x2, x3) = tree(x1, x2, x3) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x3, x6) U3_ga(x1, x2, x3, x4, x5, x6) = U3_ga(x6) PREORDER_DL_IN_GA(x1, x2) = PREORDER_DL_IN_GA(x1) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x3, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GA(R, preorder_dl_out_ga) -> PREORDER_DL_IN_GA(R) PREORDER_DL_IN_GA(tree(L, X, R)) -> U2_GA(R, preorder_dl_in_ga(L)) PREORDER_DL_IN_GA(tree(L, X, R)) -> PREORDER_DL_IN_GA(L) The TRS R consists of the following rules: preorder_dl_in_ga(nil) -> preorder_dl_out_ga preorder_dl_in_ga(tree(L, X, R)) -> U2_ga(R, preorder_dl_in_ga(L)) U2_ga(R, preorder_dl_out_ga) -> U3_ga(preorder_dl_in_ga(R)) U3_ga(preorder_dl_out_ga) -> preorder_dl_out_ga The set Q consists of the following terms: preorder_dl_in_ga(x0) U2_ga(x0, x1) U3_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (11) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *PREORDER_DL_IN_GA(tree(L, X, R)) -> U2_GA(R, preorder_dl_in_ga(L)) The graph contains the following edges 1 > 1 *PREORDER_DL_IN_GA(tree(L, X, R)) -> PREORDER_DL_IN_GA(L) The graph contains the following edges 1 > 1 *U2_GA(R, preorder_dl_out_ga) -> PREORDER_DL_IN_GA(R) The graph contains the following edges 1 >= 1 ---------------------------------------- (12) YES