/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern ackermann(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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) PiDP (7) PiDPToQDPProof [SOUND, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Clauses: ackermann(0, N, s(N)). ackermann(s(M), 0, Val) :- ackermann(M, s(0), Val). ackermann(s(M), s(N), Val) :- ','(ackermann(s(M), N, Val1), ackermann(M, Val1, Val)). Query: ackermann(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: ackermann_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: ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) The argument filtering Pi contains the following mapping: ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 0 = 0 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3) = U1_gga(x3) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) U3_gga(x1, x2, x3, x4) = U3_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: ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) The argument filtering Pi contains the following mapping: ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 0 = 0 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3) = U1_gga(x3) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) U3_gga(x1, x2, x3, x4) = U3_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: ACKERMANN_IN_GGA(s(M), 0, Val) -> U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val)) ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val)) U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) The TRS R consists of the following rules: ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) The argument filtering Pi contains the following mapping: ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 0 = 0 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3) = U1_gga(x3) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) U3_gga(x1, x2, x3, x4) = U3_gga(x4) ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x3) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: ACKERMANN_IN_GGA(s(M), 0, Val) -> U1_GGA(M, Val, ackermann_in_gga(M, s(0), Val)) ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_GGA(M, N, Val, ackermann_in_gga(M, Val1, Val)) U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) The TRS R consists of the following rules: ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) The argument filtering Pi contains the following mapping: ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 0 = 0 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3) = U1_gga(x3) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) U3_gga(x1, x2, x3, x4) = U3_gga(x4) ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x3) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) U3_GGA(x1, x2, x3, x4) = U3_GGA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: ACKERMANN_IN_GGA(s(M), 0, Val) -> ACKERMANN_IN_GGA(M, s(0), Val) ACKERMANN_IN_GGA(s(M), s(N), Val) -> U2_GGA(M, N, Val, ackermann_in_gga(s(M), N, Val1)) U2_GGA(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> ACKERMANN_IN_GGA(M, Val1, Val) ACKERMANN_IN_GGA(s(M), s(N), Val) -> ACKERMANN_IN_GGA(s(M), N, Val1) The TRS R consists of the following rules: ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N)) ackermann_in_gga(s(M), 0, Val) -> U1_gga(M, Val, ackermann_in_gga(M, s(0), Val)) ackermann_in_gga(s(M), s(N), Val) -> U2_gga(M, N, Val, ackermann_in_gga(s(M), N, Val1)) U2_gga(M, N, Val, ackermann_out_gga(s(M), N, Val1)) -> U3_gga(M, N, Val, ackermann_in_gga(M, Val1, Val)) U3_gga(M, N, Val, ackermann_out_gga(M, Val1, Val)) -> ackermann_out_gga(s(M), s(N), Val) U1_gga(M, Val, ackermann_out_gga(M, s(0), Val)) -> ackermann_out_gga(s(M), 0, Val) The argument filtering Pi contains the following mapping: ackermann_in_gga(x1, x2, x3) = ackermann_in_gga(x1, x2) 0 = 0 ackermann_out_gga(x1, x2, x3) = ackermann_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3) = U1_gga(x3) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4) U3_gga(x1, x2, x3, x4) = U3_gga(x4) ACKERMANN_IN_GGA(x1, x2, x3) = ACKERMANN_IN_GGA(x1, x2) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0)) ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, ackermann_in_gga(s(M), N)) U2_GGA(M, ackermann_out_gga(Val1)) -> ACKERMANN_IN_GGA(M, Val1) ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N) The TRS R consists of the following rules: ackermann_in_gga(0, N) -> ackermann_out_gga(s(N)) ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0))) ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N)) U2_gga(M, ackermann_out_gga(Val1)) -> U3_gga(ackermann_in_gga(M, Val1)) U3_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) U1_gga(ackermann_out_gga(Val)) -> ackermann_out_gga(Val) The set Q consists of the following terms: ackermann_in_gga(x0, x1) U2_gga(x0, x1) U3_gga(x0) U1_gga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (9) 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: *ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N) The graph contains the following edges 1 >= 1, 2 > 2 *ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, ackermann_in_gga(s(M), N)) The graph contains the following edges 1 > 1 *U2_GGA(M, ackermann_out_gga(Val1)) -> ACKERMANN_IN_GGA(M, Val1) The graph contains the following edges 1 >= 1, 2 > 2 *ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0)) The graph contains the following edges 1 > 1 ---------------------------------------- (10) YES