/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern times(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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) UsableRulesProof [EQUIVALENT, 0 ms] (8) PiDP (9) PiDPToQDPProof [SOUND, 0 ms] (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Clauses: times(X, Y, Z) :- mult(X, Y, 0, Z). mult(0, Y, 0, 0). mult(s(U), Y, 0, Z) :- mult(U, Y, Y, Z). mult(X, Y, s(W), s(Z)) :- mult(X, Y, W, Z). Query: times(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: times_in_3: (b,b,f) mult_in_4: (b,b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0) mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z)) U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U2_ggga(x1, x2, x3, x4) = U2_ggga(x4) U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0) mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z)) U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U2_ggga(x1, x2, x3, x4) = U2_ggga(x4) U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) ---------------------------------------- (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: TIMES_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z) MULT_IN_GGGA(s(U), Y, 0, Z) -> U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z) MULT_IN_GGGA(X, Y, s(W), s(Z)) -> U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0) mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z)) U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U2_ggga(x1, x2, x3, x4) = U2_ggga(x4) U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3, x4) = U2_GGGA(x4) U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z) MULT_IN_GGGA(s(U), Y, 0, Z) -> U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z) MULT_IN_GGGA(X, Y, s(W), s(Z)) -> U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0) mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z)) U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U2_ggga(x1, x2, x3, x4) = U2_ggga(x4) U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x4) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) U2_GGGA(x1, x2, x3, x4) = U2_GGGA(x4) U3_GGGA(x1, x2, x3, x4, x5) = U3_GGGA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes. ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z) MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z) The TRS R consists of the following rules: times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z)) mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0) mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z)) mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z)) U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z)) U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z) U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z) The argument filtering Pi contains the following mapping: times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) U1_gga(x1, x2, x3, x4) = U1_gga(x4) mult_in_ggga(x1, x2, x3, x4) = mult_in_ggga(x1, x2, x3) 0 = 0 mult_out_ggga(x1, x2, x3, x4) = mult_out_ggga(x4) s(x1) = s(x1) U2_ggga(x1, x2, x3, x4) = U2_ggga(x4) U3_ggga(x1, x2, x3, x4, x5) = U3_ggga(x5) times_out_gga(x1, x2, x3) = times_out_gga(x3) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) 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: MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z) MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z) R is empty. The argument filtering Pi contains the following mapping: 0 = 0 s(x1) = s(x1) MULT_IN_GGGA(x1, x2, x3, x4) = MULT_IN_GGGA(x1, x2, x3) 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: MULT_IN_GGGA(X, Y, s(W)) -> MULT_IN_GGGA(X, Y, W) MULT_IN_GGGA(s(U), Y, 0) -> MULT_IN_GGGA(U, Y, Y) R is empty. Q is empty. 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: *MULT_IN_GGGA(X, Y, s(W)) -> MULT_IN_GGGA(X, Y, W) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *MULT_IN_GGGA(s(U), Y, 0) -> MULT_IN_GGGA(U, Y, Y) The graph contains the following edges 1 > 1, 2 >= 2, 2 >= 3 ---------------------------------------- (12) YES