/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern f(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 21 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: f(A, [], RES) :- g(A, [], RES). f(.(A, As), .(B, Bs), RES) :- f(.(B, .(A, As)), Bs, RES). g([], RES, RES). g(.(C, Cs), D, RES) :- g(Cs, .(C, D), RES). Query: f(g,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: f_in_3: (b,b,f) g_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: f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES)) g_in_gga([], RES, RES) -> g_out_gga([], RES, RES) g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES) U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES) f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES) The argument filtering Pi contains the following mapping: f_in_gga(x1, x2, x3) = f_in_gga(x1, x2) [] = [] U1_gga(x1, x2, x3) = U1_gga(x1, x3) g_in_gga(x1, x2, x3) = g_in_gga(x1, x2) g_out_gga(x1, x2, x3) = g_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) f_out_gga(x1, x2, x3) = f_out_gga(x1, x2, x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES)) g_in_gga([], RES, RES) -> g_out_gga([], RES, RES) g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES) U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES) f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES) The argument filtering Pi contains the following mapping: f_in_gga(x1, x2, x3) = f_in_gga(x1, x2) [] = [] U1_gga(x1, x2, x3) = U1_gga(x1, x3) g_in_gga(x1, x2, x3) = g_in_gga(x1, x2) g_out_gga(x1, x2, x3) = g_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) f_out_gga(x1, x2, x3) = f_out_gga(x1, x2, x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) ---------------------------------------- (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: F_IN_GGA(A, [], RES) -> U1_GGA(A, RES, g_in_gga(A, [], RES)) F_IN_GGA(A, [], RES) -> G_IN_GGA(A, [], RES) G_IN_GGA(.(C, Cs), D, RES) -> U3_GGA(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES) F_IN_GGA(.(A, As), .(B, Bs), RES) -> U2_GGA(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES) The TRS R consists of the following rules: f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES)) g_in_gga([], RES, RES) -> g_out_gga([], RES, RES) g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES) U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES) f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES) The argument filtering Pi contains the following mapping: f_in_gga(x1, x2, x3) = f_in_gga(x1, x2) [] = [] U1_gga(x1, x2, x3) = U1_gga(x1, x3) g_in_gga(x1, x2, x3) = g_in_gga(x1, x2) g_out_gga(x1, x2, x3) = g_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) f_out_gga(x1, x2, x3) = f_out_gga(x1, x2, x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) F_IN_GGA(x1, x2, x3) = F_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) G_IN_GGA(x1, x2, x3) = G_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: F_IN_GGA(A, [], RES) -> U1_GGA(A, RES, g_in_gga(A, [], RES)) F_IN_GGA(A, [], RES) -> G_IN_GGA(A, [], RES) G_IN_GGA(.(C, Cs), D, RES) -> U3_GGA(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES) F_IN_GGA(.(A, As), .(B, Bs), RES) -> U2_GGA(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES) The TRS R consists of the following rules: f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES)) g_in_gga([], RES, RES) -> g_out_gga([], RES, RES) g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES) U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES) f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES) The argument filtering Pi contains the following mapping: f_in_gga(x1, x2, x3) = f_in_gga(x1, x2) [] = [] U1_gga(x1, x2, x3) = U1_gga(x1, x3) g_in_gga(x1, x2, x3) = g_in_gga(x1, x2) g_out_gga(x1, x2, x3) = g_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) f_out_gga(x1, x2, x3) = f_out_gga(x1, x2, x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) F_IN_GGA(x1, x2, x3) = F_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) G_IN_GGA(x1, x2, x3) = G_IN_GGA(x1, x2) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) 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: G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES) The TRS R consists of the following rules: f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES)) g_in_gga([], RES, RES) -> g_out_gga([], RES, RES) g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES) U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES) f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES) The argument filtering Pi contains the following mapping: f_in_gga(x1, x2, x3) = f_in_gga(x1, x2) [] = [] U1_gga(x1, x2, x3) = U1_gga(x1, x3) g_in_gga(x1, x2, x3) = g_in_gga(x1, x2) g_out_gga(x1, x2, x3) = g_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) f_out_gga(x1, x2, x3) = f_out_gga(x1, x2, x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) G_IN_GGA(x1, x2, x3) = G_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: G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) G_IN_GGA(x1, x2, x3) = G_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: G_IN_GGA(.(C, Cs), D) -> G_IN_GGA(Cs, .(C, D)) 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: *G_IN_GGA(.(C, Cs), D) -> G_IN_GGA(Cs, .(C, D)) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES) The TRS R consists of the following rules: f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES)) g_in_gga([], RES, RES) -> g_out_gga([], RES, RES) g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES)) U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES) U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES) f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES)) U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES) The argument filtering Pi contains the following mapping: f_in_gga(x1, x2, x3) = f_in_gga(x1, x2) [] = [] U1_gga(x1, x2, x3) = U1_gga(x1, x3) g_in_gga(x1, x2, x3) = g_in_gga(x1, x2) g_out_gga(x1, x2, x3) = g_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) f_out_gga(x1, x2, x3) = f_out_gga(x1, x2, x3) U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) F_IN_GGA(x1, x2, x3) = F_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) F_IN_GGA(x1, x2, x3) = F_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: F_IN_GGA(.(A, As), .(B, Bs)) -> F_IN_GGA(.(B, .(A, As)), Bs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *F_IN_GGA(.(A, As), .(B, Bs)) -> F_IN_GGA(.(B, .(A, As)), Bs) The graph contains the following edges 2 > 2 ---------------------------------------- (20) YES