/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern mult(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND,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, 0 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: mult(X1, 0, 0). mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)). sum(X, 0, X). sum(X, s(Y), s(Z)) :- sum(X, Y, Z). Query: mult(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: mult_in_3: (b,b,f) sum_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: mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0) mult_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, W)) U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_gga(X, Y, Z, sum_in_gga(W, X, Z)) sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X) sum_in_gga(X, s(Y), s(Z)) -> U3_gga(X, Y, Z, sum_in_gga(X, Y, Z)) U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z)) U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 0 = 0 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2) sum_out_gga(x1, x2, x3) = sum_out_gga(x3) 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: mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0) mult_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, W)) U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_gga(X, Y, Z, sum_in_gga(W, X, Z)) sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X) sum_in_gga(X, s(Y), s(Z)) -> U3_gga(X, Y, Z, sum_in_gga(X, Y, Z)) U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z)) U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 0 = 0 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2) sum_out_gga(x1, x2, x3) = sum_out_gga(x3) 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: MULT_IN_GGA(X, s(Y), Z) -> U1_GGA(X, Y, Z, mult_in_gga(X, Y, W)) MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W) U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_GGA(X, Y, Z, sum_in_gga(W, X, Z)) U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> SUM_IN_GGA(W, X, Z) SUM_IN_GGA(X, s(Y), s(Z)) -> U3_GGA(X, Y, Z, sum_in_gga(X, Y, Z)) SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z) The TRS R consists of the following rules: mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0) mult_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, W)) U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_gga(X, Y, Z, sum_in_gga(W, X, Z)) sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X) sum_in_gga(X, s(Y), s(Z)) -> U3_gga(X, Y, Z, sum_in_gga(X, Y, Z)) U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z)) U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 0 = 0 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2) sum_out_gga(x1, x2, x3) = sum_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x4) U2_GGA(x1, x2, x3, x4) = U2_GGA(x4) SUM_IN_GGA(x1, x2, x3) = SUM_IN_GGA(x1, x2) 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: MULT_IN_GGA(X, s(Y), Z) -> U1_GGA(X, Y, Z, mult_in_gga(X, Y, W)) MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W) U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_GGA(X, Y, Z, sum_in_gga(W, X, Z)) U1_GGA(X, Y, Z, mult_out_gga(X, Y, W)) -> SUM_IN_GGA(W, X, Z) SUM_IN_GGA(X, s(Y), s(Z)) -> U3_GGA(X, Y, Z, sum_in_gga(X, Y, Z)) SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z) The TRS R consists of the following rules: mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0) mult_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, W)) U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_gga(X, Y, Z, sum_in_gga(W, X, Z)) sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X) sum_in_gga(X, s(Y), s(Z)) -> U3_gga(X, Y, Z, sum_in_gga(X, Y, Z)) U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z)) U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 0 = 0 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2) sum_out_gga(x1, x2, x3) = sum_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x4) U2_GGA(x1, x2, x3, x4) = U2_GGA(x4) SUM_IN_GGA(x1, x2, x3) = SUM_IN_GGA(x1, x2) 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 2 SCCs with 4 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z) The TRS R consists of the following rules: mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0) mult_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, W)) U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_gga(X, Y, Z, sum_in_gga(W, X, Z)) sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X) sum_in_gga(X, s(Y), s(Z)) -> U3_gga(X, Y, Z, sum_in_gga(X, Y, Z)) U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z)) U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 0 = 0 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2) sum_out_gga(x1, x2, x3) = sum_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) SUM_IN_GGA(x1, x2, x3) = SUM_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: SUM_IN_GGA(X, s(Y), s(Z)) -> SUM_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) SUM_IN_GGA(x1, x2, x3) = SUM_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: SUM_IN_GGA(X, s(Y)) -> SUM_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: *SUM_IN_GGA(X, s(Y)) -> SUM_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: MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W) The TRS R consists of the following rules: mult_in_gga(X1, 0, 0) -> mult_out_gga(X1, 0, 0) mult_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, W)) U1_gga(X, Y, Z, mult_out_gga(X, Y, W)) -> U2_gga(X, Y, Z, sum_in_gga(W, X, Z)) sum_in_gga(X, 0, X) -> sum_out_gga(X, 0, X) sum_in_gga(X, s(Y), s(Z)) -> U3_gga(X, Y, Z, sum_in_gga(X, Y, Z)) U3_gga(X, Y, Z, sum_out_gga(X, Y, Z)) -> sum_out_gga(X, s(Y), s(Z)) U2_gga(X, Y, Z, sum_out_gga(W, X, Z)) -> mult_out_gga(X, s(Y), Z) The argument filtering Pi contains the following mapping: mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2) 0 = 0 mult_out_gga(x1, x2, x3) = mult_out_gga(x3) s(x1) = s(x1) U1_gga(x1, x2, x3, x4) = U1_gga(x1, x4) U2_gga(x1, x2, x3, x4) = U2_gga(x4) sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2) sum_out_gga(x1, x2, x3) = sum_out_gga(x3) U3_gga(x1, x2, x3, x4) = U3_gga(x4) MULT_IN_GGA(x1, x2, x3) = MULT_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: MULT_IN_GGA(X, s(Y), Z) -> MULT_IN_GGA(X, Y, W) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) MULT_IN_GGA(x1, x2, x3) = MULT_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: MULT_IN_GGA(X, s(Y)) -> MULT_IN_GGA(X, Y) 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: *MULT_IN_GGA(X, s(Y)) -> MULT_IN_GGA(X, Y) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (20) YES