/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern fib(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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, 16 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: add(0, 0, 0). add(s(X), Y, s(N)) :- add(X, Y, N). add(X, s(Y), s(N)) :- add(X, Y, N). fib(0, 0). fib(s(0), s(0)). fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N))). Query: fib(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: fib_in_2: (b,f) add_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: fib_in_ga(0, 0) -> fib_out_ga(0, 0) fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0)) fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1)) U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2)) U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N)) add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0) add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N)) add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N)) U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N)) U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N)) U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N) The argument filtering Pi contains the following mapping: fib_in_ga(x1, x2) = fib_in_ga(x1) 0 = 0 fib_out_ga(x1, x2) = fib_out_ga(x2) s(x1) = s(x1) U3_ga(x1, x2, x3) = U3_ga(x1, x3) U4_ga(x1, x2, x3, x4) = U4_ga(x3, x4) U5_ga(x1, x2, x3) = U5_ga(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U1_gga(x1, x2, x3, x4) = U1_gga(x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: fib_in_ga(0, 0) -> fib_out_ga(0, 0) fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0)) fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1)) U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2)) U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N)) add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0) add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N)) add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N)) U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N)) U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N)) U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N) The argument filtering Pi contains the following mapping: fib_in_ga(x1, x2) = fib_in_ga(x1) 0 = 0 fib_out_ga(x1, x2) = fib_out_ga(x2) s(x1) = s(x1) U3_ga(x1, x2, x3) = U3_ga(x1, x3) U4_ga(x1, x2, x3, x4) = U4_ga(x3, x4) U5_ga(x1, x2, x3) = U5_ga(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U1_gga(x1, x2, x3, x4) = U1_gga(x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) ---------------------------------------- (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: FIB_IN_GA(s(s(X)), N) -> U3_GA(X, N, fib_in_ga(s(X), N1)) FIB_IN_GA(s(s(X)), N) -> FIB_IN_GA(s(X), N1) U3_GA(X, N, fib_out_ga(s(X), N1)) -> U4_GA(X, N, N1, fib_in_ga(X, N2)) U3_GA(X, N, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X, N2) U4_GA(X, N, N1, fib_out_ga(X, N2)) -> U5_GA(X, N, add_in_gga(N1, N2, N)) U4_GA(X, N, N1, fib_out_ga(X, N2)) -> ADD_IN_GGA(N1, N2, N) ADD_IN_GGA(s(X), Y, s(N)) -> U1_GGA(X, Y, N, add_in_gga(X, Y, N)) ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N) ADD_IN_GGA(X, s(Y), s(N)) -> U2_GGA(X, Y, N, add_in_gga(X, Y, N)) ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N) The TRS R consists of the following rules: fib_in_ga(0, 0) -> fib_out_ga(0, 0) fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0)) fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1)) U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2)) U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N)) add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0) add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N)) add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N)) U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N)) U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N)) U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N) The argument filtering Pi contains the following mapping: fib_in_ga(x1, x2) = fib_in_ga(x1) 0 = 0 fib_out_ga(x1, x2) = fib_out_ga(x2) s(x1) = s(x1) U3_ga(x1, x2, x3) = U3_ga(x1, x3) U4_ga(x1, x2, x3, x4) = U4_ga(x3, x4) U5_ga(x1, x2, x3) = U5_ga(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U1_gga(x1, x2, x3, x4) = U1_gga(x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) FIB_IN_GA(x1, x2) = FIB_IN_GA(x1) U3_GA(x1, x2, x3) = U3_GA(x1, x3) U4_GA(x1, x2, x3, x4) = U4_GA(x3, x4) U5_GA(x1, x2, x3) = U5_GA(x3) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) U2_GGA(x1, x2, x3, x4) = U2_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: FIB_IN_GA(s(s(X)), N) -> U3_GA(X, N, fib_in_ga(s(X), N1)) FIB_IN_GA(s(s(X)), N) -> FIB_IN_GA(s(X), N1) U3_GA(X, N, fib_out_ga(s(X), N1)) -> U4_GA(X, N, N1, fib_in_ga(X, N2)) U3_GA(X, N, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X, N2) U4_GA(X, N, N1, fib_out_ga(X, N2)) -> U5_GA(X, N, add_in_gga(N1, N2, N)) U4_GA(X, N, N1, fib_out_ga(X, N2)) -> ADD_IN_GGA(N1, N2, N) ADD_IN_GGA(s(X), Y, s(N)) -> U1_GGA(X, Y, N, add_in_gga(X, Y, N)) ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N) ADD_IN_GGA(X, s(Y), s(N)) -> U2_GGA(X, Y, N, add_in_gga(X, Y, N)) ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N) The TRS R consists of the following rules: fib_in_ga(0, 0) -> fib_out_ga(0, 0) fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0)) fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1)) U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2)) U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N)) add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0) add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N)) add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N)) U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N)) U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N)) U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N) The argument filtering Pi contains the following mapping: fib_in_ga(x1, x2) = fib_in_ga(x1) 0 = 0 fib_out_ga(x1, x2) = fib_out_ga(x2) s(x1) = s(x1) U3_ga(x1, x2, x3) = U3_ga(x1, x3) U4_ga(x1, x2, x3, x4) = U4_ga(x3, x4) U5_ga(x1, x2, x3) = U5_ga(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U1_gga(x1, x2, x3, x4) = U1_gga(x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) FIB_IN_GA(x1, x2) = FIB_IN_GA(x1) U3_GA(x1, x2, x3) = U3_GA(x1, x3) U4_GA(x1, x2, x3, x4) = U4_GA(x3, x4) U5_GA(x1, x2, x3) = U5_GA(x3) ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) U2_GGA(x1, x2, x3, x4) = U2_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N) ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N) The TRS R consists of the following rules: fib_in_ga(0, 0) -> fib_out_ga(0, 0) fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0)) fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1)) U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2)) U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N)) add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0) add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N)) add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N)) U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N)) U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N)) U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N) The argument filtering Pi contains the following mapping: fib_in_ga(x1, x2) = fib_in_ga(x1) 0 = 0 fib_out_ga(x1, x2) = fib_out_ga(x2) s(x1) = s(x1) U3_ga(x1, x2, x3) = U3_ga(x1, x3) U4_ga(x1, x2, x3, x4) = U4_ga(x3, x4) U5_ga(x1, x2, x3) = U5_ga(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U1_gga(x1, x2, x3, x4) = U1_gga(x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) ADD_IN_GGA(x1, x2, x3) = ADD_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: ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N) ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) ADD_IN_GGA(x1, x2, x3) = ADD_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: ADD_IN_GGA(X, s(Y)) -> ADD_IN_GGA(X, Y) ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y) 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: *ADD_IN_GGA(X, s(Y)) -> ADD_IN_GGA(X, Y) The graph contains the following edges 1 >= 1, 2 > 2 *ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y) 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: U3_GA(X, N, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X, N2) FIB_IN_GA(s(s(X)), N) -> U3_GA(X, N, fib_in_ga(s(X), N1)) FIB_IN_GA(s(s(X)), N) -> FIB_IN_GA(s(X), N1) The TRS R consists of the following rules: fib_in_ga(0, 0) -> fib_out_ga(0, 0) fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0)) fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1)) U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2)) U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N)) add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0) add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N)) add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N)) U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N)) U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N)) U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N) The argument filtering Pi contains the following mapping: fib_in_ga(x1, x2) = fib_in_ga(x1) 0 = 0 fib_out_ga(x1, x2) = fib_out_ga(x2) s(x1) = s(x1) U3_ga(x1, x2, x3) = U3_ga(x1, x3) U4_ga(x1, x2, x3, x4) = U4_ga(x3, x4) U5_ga(x1, x2, x3) = U5_ga(x3) add_in_gga(x1, x2, x3) = add_in_gga(x1, x2) add_out_gga(x1, x2, x3) = add_out_gga(x3) U1_gga(x1, x2, x3, x4) = U1_gga(x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) FIB_IN_GA(x1, x2) = FIB_IN_GA(x1) U3_GA(x1, x2, x3) = U3_GA(x1, x3) 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: U3_GA(X, fib_out_ga(N1)) -> FIB_IN_GA(X) FIB_IN_GA(s(s(X))) -> U3_GA(X, fib_in_ga(s(X))) FIB_IN_GA(s(s(X))) -> FIB_IN_GA(s(X)) The TRS R consists of the following rules: fib_in_ga(0) -> fib_out_ga(0) fib_in_ga(s(0)) -> fib_out_ga(s(0)) fib_in_ga(s(s(X))) -> U3_ga(X, fib_in_ga(s(X))) U3_ga(X, fib_out_ga(N1)) -> U4_ga(N1, fib_in_ga(X)) U4_ga(N1, fib_out_ga(N2)) -> U5_ga(add_in_gga(N1, N2)) add_in_gga(0, 0) -> add_out_gga(0) add_in_gga(s(X), Y) -> U1_gga(add_in_gga(X, Y)) add_in_gga(X, s(Y)) -> U2_gga(add_in_gga(X, Y)) U2_gga(add_out_gga(N)) -> add_out_gga(s(N)) U1_gga(add_out_gga(N)) -> add_out_gga(s(N)) U5_ga(add_out_gga(N)) -> fib_out_ga(N) The set Q consists of the following terms: fib_in_ga(x0) U3_ga(x0, x1) U4_ga(x0, x1) add_in_gga(x0, x1) U2_gga(x0) U1_gga(x0) U5_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: *FIB_IN_GA(s(s(X))) -> U3_GA(X, fib_in_ga(s(X))) The graph contains the following edges 1 > 1 *FIB_IN_GA(s(s(X))) -> FIB_IN_GA(s(X)) The graph contains the following edges 1 > 1 *U3_GA(X, fib_out_ga(N1)) -> FIB_IN_GA(X) The graph contains the following edges 1 >= 1 ---------------------------------------- (18) YES