/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern inorder(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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, 3 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 14 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) PiDPToQDPProof [SOUND, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES ---------------------------------------- (0) Obligation: Clauses: inorder(nil, []). inorder(tree(L, V, R), I) :- ','(inorder(L, LI), ','(inorder(R, RI), append(LI, .(V, RI), I))). append([], X, X). append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs). Query: inorder(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: inorder_in_2: (b,f) append_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: inorder_in_ga(nil, []) -> inorder_out_ga(nil, []) inorder_in_ga(tree(L, V, R), I) -> U1_ga(L, V, R, I, inorder_in_ga(L, LI)) U1_ga(L, V, R, I, inorder_out_ga(L, LI)) -> U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI)) U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I)) append_in_gga([], X, X) -> append_out_gga([], X, X) append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) -> inorder_out_ga(tree(L, V, R), I) The argument filtering Pi contains the following mapping: inorder_in_ga(x1, x2) = inorder_in_ga(x1) nil = nil inorder_out_ga(x1, x2) = inorder_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: inorder_in_ga(nil, []) -> inorder_out_ga(nil, []) inorder_in_ga(tree(L, V, R), I) -> U1_ga(L, V, R, I, inorder_in_ga(L, LI)) U1_ga(L, V, R, I, inorder_out_ga(L, LI)) -> U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI)) U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I)) append_in_gga([], X, X) -> append_out_gga([], X, X) append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) -> inorder_out_ga(tree(L, V, R), I) The argument filtering Pi contains the following mapping: inorder_in_ga(x1, x2) = inorder_in_ga(x1) nil = nil inorder_out_ga(x1, x2) = inorder_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) ---------------------------------------- (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: INORDER_IN_GA(tree(L, V, R), I) -> U1_GA(L, V, R, I, inorder_in_ga(L, LI)) INORDER_IN_GA(tree(L, V, R), I) -> INORDER_IN_GA(L, LI) U1_GA(L, V, R, I, inorder_out_ga(L, LI)) -> U2_GA(L, V, R, I, LI, inorder_in_ga(R, RI)) U1_GA(L, V, R, I, inorder_out_ga(L, LI)) -> INORDER_IN_GA(R, RI) U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_GA(L, V, R, I, append_in_gga(LI, .(V, RI), I)) U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) -> APPEND_IN_GGA(LI, .(V, RI), I) APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: inorder_in_ga(nil, []) -> inorder_out_ga(nil, []) inorder_in_ga(tree(L, V, R), I) -> U1_ga(L, V, R, I, inorder_in_ga(L, LI)) U1_ga(L, V, R, I, inorder_out_ga(L, LI)) -> U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI)) U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I)) append_in_gga([], X, X) -> append_out_gga([], X, X) append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) -> inorder_out_ga(tree(L, V, R), I) The argument filtering Pi contains the following mapping: inorder_in_ga(x1, x2) = inorder_in_ga(x1) nil = nil inorder_out_ga(x1, x2) = inorder_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) INORDER_IN_GA(x1, x2) = INORDER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x3, x5) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x2, x5, x6) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: INORDER_IN_GA(tree(L, V, R), I) -> U1_GA(L, V, R, I, inorder_in_ga(L, LI)) INORDER_IN_GA(tree(L, V, R), I) -> INORDER_IN_GA(L, LI) U1_GA(L, V, R, I, inorder_out_ga(L, LI)) -> U2_GA(L, V, R, I, LI, inorder_in_ga(R, RI)) U1_GA(L, V, R, I, inorder_out_ga(L, LI)) -> INORDER_IN_GA(R, RI) U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_GA(L, V, R, I, append_in_gga(LI, .(V, RI), I)) U2_GA(L, V, R, I, LI, inorder_out_ga(R, RI)) -> APPEND_IN_GGA(LI, .(V, RI), I) APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: inorder_in_ga(nil, []) -> inorder_out_ga(nil, []) inorder_in_ga(tree(L, V, R), I) -> U1_ga(L, V, R, I, inorder_in_ga(L, LI)) U1_ga(L, V, R, I, inorder_out_ga(L, LI)) -> U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI)) U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I)) append_in_gga([], X, X) -> append_out_gga([], X, X) append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) -> inorder_out_ga(tree(L, V, R), I) The argument filtering Pi contains the following mapping: inorder_in_ga(x1, x2) = inorder_in_ga(x1) nil = nil inorder_out_ga(x1, x2) = inorder_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) INORDER_IN_GA(x1, x2) = INORDER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x3, x5) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x2, x5, x6) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: inorder_in_ga(nil, []) -> inorder_out_ga(nil, []) inorder_in_ga(tree(L, V, R), I) -> U1_ga(L, V, R, I, inorder_in_ga(L, LI)) U1_ga(L, V, R, I, inorder_out_ga(L, LI)) -> U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI)) U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I)) append_in_gga([], X, X) -> append_out_gga([], X, X) append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) -> inorder_out_ga(tree(L, V, R), I) The argument filtering Pi contains the following mapping: inorder_in_ga(x1, x2) = inorder_in_ga(x1) nil = nil inorder_out_ga(x1, x2) = inorder_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(X, Xs), Ys) -> APPEND_IN_GGA(Xs, Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *APPEND_IN_GGA(.(X, Xs), Ys) -> APPEND_IN_GGA(Xs, Ys) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(L, V, R, I, inorder_out_ga(L, LI)) -> INORDER_IN_GA(R, RI) INORDER_IN_GA(tree(L, V, R), I) -> U1_GA(L, V, R, I, inorder_in_ga(L, LI)) INORDER_IN_GA(tree(L, V, R), I) -> INORDER_IN_GA(L, LI) The TRS R consists of the following rules: inorder_in_ga(nil, []) -> inorder_out_ga(nil, []) inorder_in_ga(tree(L, V, R), I) -> U1_ga(L, V, R, I, inorder_in_ga(L, LI)) U1_ga(L, V, R, I, inorder_out_ga(L, LI)) -> U2_ga(L, V, R, I, LI, inorder_in_ga(R, RI)) U2_ga(L, V, R, I, LI, inorder_out_ga(R, RI)) -> U3_ga(L, V, R, I, append_in_gga(LI, .(V, RI), I)) append_in_gga([], X, X) -> append_out_gga([], X, X) append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(L, V, R, I, append_out_gga(LI, .(V, RI), I)) -> inorder_out_ga(tree(L, V, R), I) The argument filtering Pi contains the following mapping: inorder_in_ga(x1, x2) = inorder_in_ga(x1) nil = nil inorder_out_ga(x1, x2) = inorder_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x2, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) INORDER_IN_GA(x1, x2) = INORDER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(V, R, inorder_out_ga(LI)) -> INORDER_IN_GA(R) INORDER_IN_GA(tree(L, V, R)) -> U1_GA(V, R, inorder_in_ga(L)) INORDER_IN_GA(tree(L, V, R)) -> INORDER_IN_GA(L) The TRS R consists of the following rules: inorder_in_ga(nil) -> inorder_out_ga([]) inorder_in_ga(tree(L, V, R)) -> U1_ga(V, R, inorder_in_ga(L)) U1_ga(V, R, inorder_out_ga(LI)) -> U2_ga(V, LI, inorder_in_ga(R)) U2_ga(V, LI, inorder_out_ga(RI)) -> U3_ga(append_in_gga(LI, .(V, RI))) append_in_gga([], X) -> append_out_gga(X) append_in_gga(.(X, Xs), Ys) -> U4_gga(X, append_in_gga(Xs, Ys)) U4_gga(X, append_out_gga(Zs)) -> append_out_gga(.(X, Zs)) U3_ga(append_out_gga(I)) -> inorder_out_ga(I) The set Q consists of the following terms: inorder_in_ga(x0) U1_ga(x0, x1, x2) U2_ga(x0, x1, x2) append_in_gga(x0, x1) U4_gga(x0, x1) U3_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (17) 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: *INORDER_IN_GA(tree(L, V, R)) -> U1_GA(V, R, inorder_in_ga(L)) The graph contains the following edges 1 > 1, 1 > 2 *INORDER_IN_GA(tree(L, V, R)) -> INORDER_IN_GA(L) The graph contains the following edges 1 > 1 *U1_GA(V, R, inorder_out_ga(LI)) -> INORDER_IN_GA(R) The graph contains the following edges 2 >= 1 ---------------------------------------- (18) YES