/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern samefringe(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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, 0 ms] (11) QDP (12) MRRProof [EQUIVALENT, 3 ms] (13) QDP (14) PisEmptyProof [EQUIVALENT, 0 ms] (15) YES (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDPToQDPProof [SOUND, 3 ms] (20) QDP (21) MRRProof [EQUIVALENT, 22 ms] (22) QDP (23) PisEmptyProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Clauses: gopher(nil, nil). gopher(cons(nil, Y), cons(nil, Y)). gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X). samefringe(nil, nil). samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1))). Query: samefringe(g,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: samefringe_in_2: (b,b) gopher_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: samefringe_in_gg(nil, nil) -> samefringe_out_gg(nil, nil) samefringe_in_gg(cons(U, V), cons(X, Y)) -> U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil) gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1)) U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) -> samefringe_out_gg(cons(U, V), cons(X, Y)) The argument filtering Pi contains the following mapping: samefringe_in_gg(x1, x2) = samefringe_in_gg(x1, x2) nil = nil samefringe_out_gg(x1, x2) = samefringe_out_gg cons(x1, x2) = cons(x1, x2) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) U3_gg(x1, x2, x3, x4, x5, x6, x7) = U3_gg(x6, x7) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: samefringe_in_gg(nil, nil) -> samefringe_out_gg(nil, nil) samefringe_in_gg(cons(U, V), cons(X, Y)) -> U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil) gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1)) U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) -> samefringe_out_gg(cons(U, V), cons(X, Y)) The argument filtering Pi contains the following mapping: samefringe_in_gg(x1, x2) = samefringe_in_gg(x1, x2) nil = nil samefringe_out_gg(x1, x2) = samefringe_out_gg cons(x1, x2) = cons(x1, x2) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) U3_gg(x1, x2, x3, x4, x5, x6, x7) = U3_gg(x6, x7) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) ---------------------------------------- (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: SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> GOPHER_IN_GA(cons(U, V), cons(U1, V1)) GOPHER_IN_GA(cons(cons(U, V), W), X) -> U1_GA(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X) U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> GOPHER_IN_GA(cons(X, Y), cons(X1, Y1)) U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_GG(U, V, X, Y, samefringe_in_gg(V1, Y1)) U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> SAMEFRINGE_IN_GG(V1, Y1) The TRS R consists of the following rules: samefringe_in_gg(nil, nil) -> samefringe_out_gg(nil, nil) samefringe_in_gg(cons(U, V), cons(X, Y)) -> U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil) gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1)) U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) -> samefringe_out_gg(cons(U, V), cons(X, Y)) The argument filtering Pi contains the following mapping: samefringe_in_gg(x1, x2) = samefringe_in_gg(x1, x2) nil = nil samefringe_out_gg(x1, x2) = samefringe_out_gg cons(x1, x2) = cons(x1, x2) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) U3_gg(x1, x2, x3, x4, x5, x6, x7) = U3_gg(x6, x7) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) SAMEFRINGE_IN_GG(x1, x2) = SAMEFRINGE_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) GOPHER_IN_GA(x1, x2) = GOPHER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) U3_GG(x1, x2, x3, x4, x5, x6, x7) = U3_GG(x6, x7) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> GOPHER_IN_GA(cons(U, V), cons(U1, V1)) GOPHER_IN_GA(cons(cons(U, V), W), X) -> U1_GA(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X) U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> GOPHER_IN_GA(cons(X, Y), cons(X1, Y1)) U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_GG(U, V, X, Y, samefringe_in_gg(V1, Y1)) U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> SAMEFRINGE_IN_GG(V1, Y1) The TRS R consists of the following rules: samefringe_in_gg(nil, nil) -> samefringe_out_gg(nil, nil) samefringe_in_gg(cons(U, V), cons(X, Y)) -> U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil) gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1)) U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) -> samefringe_out_gg(cons(U, V), cons(X, Y)) The argument filtering Pi contains the following mapping: samefringe_in_gg(x1, x2) = samefringe_in_gg(x1, x2) nil = nil samefringe_out_gg(x1, x2) = samefringe_out_gg cons(x1, x2) = cons(x1, x2) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) U3_gg(x1, x2, x3, x4, x5, x6, x7) = U3_gg(x6, x7) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) SAMEFRINGE_IN_GG(x1, x2) = SAMEFRINGE_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) GOPHER_IN_GA(x1, x2) = GOPHER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x5) U3_GG(x1, x2, x3, x4, x5, x6, x7) = U3_GG(x6, x7) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) 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: GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X) The TRS R consists of the following rules: samefringe_in_gg(nil, nil) -> samefringe_out_gg(nil, nil) samefringe_in_gg(cons(U, V), cons(X, Y)) -> U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil) gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1)) U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) -> samefringe_out_gg(cons(U, V), cons(X, Y)) The argument filtering Pi contains the following mapping: samefringe_in_gg(x1, x2) = samefringe_in_gg(x1, x2) nil = nil samefringe_out_gg(x1, x2) = samefringe_out_gg cons(x1, x2) = cons(x1, x2) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) U3_gg(x1, x2, x3, x4, x5, x6, x7) = U3_gg(x6, x7) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) GOPHER_IN_GA(x1, x2) = GOPHER_IN_GA(x1) 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: GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X) R is empty. The argument filtering Pi contains the following mapping: cons(x1, x2) = cons(x1, x2) GOPHER_IN_GA(x1, x2) = GOPHER_IN_GA(x1) 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: GOPHER_IN_GA(cons(cons(U, V), W)) -> GOPHER_IN_GA(cons(U, cons(V, W))) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: GOPHER_IN_GA(cons(cons(U, V), W)) -> GOPHER_IN_GA(cons(U, cons(V, W))) Used ordering: Knuth-Bendix order [KBO] with precedence:cons_2 > GOPHER_IN_GA_1 and weight map: GOPHER_IN_GA_1=1 cons_2=0 The variable weight is 1 ---------------------------------------- (13) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> SAMEFRINGE_IN_GG(V1, Y1) SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) The TRS R consists of the following rules: samefringe_in_gg(nil, nil) -> samefringe_out_gg(nil, nil) samefringe_in_gg(cons(U, V), cons(X, Y)) -> U2_gg(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil) gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) U2_gg(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_gg(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_gg(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> U4_gg(U, V, X, Y, samefringe_in_gg(V1, Y1)) U4_gg(U, V, X, Y, samefringe_out_gg(V1, Y1)) -> samefringe_out_gg(cons(U, V), cons(X, Y)) The argument filtering Pi contains the following mapping: samefringe_in_gg(x1, x2) = samefringe_in_gg(x1, x2) nil = nil samefringe_out_gg(x1, x2) = samefringe_out_gg cons(x1, x2) = cons(x1, x2) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x3, x4, x5) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) U3_gg(x1, x2, x3, x4, x5, x6, x7) = U3_gg(x6, x7) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) SAMEFRINGE_IN_GG(x1, x2) = SAMEFRINGE_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) U3_GG(x1, x2, x3, x4, x5, x6, x7) = U3_GG(x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_GG(U, V, X, Y, gopher_out_ga(cons(U, V), cons(U1, V1))) -> U3_GG(U, V, X, Y, U1, V1, gopher_in_ga(cons(X, Y), cons(X1, Y1))) U3_GG(U, V, X, Y, U1, V1, gopher_out_ga(cons(X, Y), cons(X1, Y1))) -> SAMEFRINGE_IN_GG(V1, Y1) SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> U2_GG(U, V, X, Y, gopher_in_ga(cons(U, V), cons(U1, V1))) The TRS R consists of the following rules: gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X)) U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X) The argument filtering Pi contains the following mapping: nil = nil cons(x1, x2) = cons(x1, x2) gopher_in_ga(x1, x2) = gopher_in_ga(x1) gopher_out_ga(x1, x2) = gopher_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x5) SAMEFRINGE_IN_GG(x1, x2) = SAMEFRINGE_IN_GG(x1, x2) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x3, x4, x5) U3_GG(x1, x2, x3, x4, x5, x6, x7) = U3_GG(x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U2_GG(X, Y, gopher_out_ga(cons(U1, V1))) -> U3_GG(V1, gopher_in_ga(cons(X, Y))) U3_GG(V1, gopher_out_ga(cons(X1, Y1))) -> SAMEFRINGE_IN_GG(V1, Y1) SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> U2_GG(X, Y, gopher_in_ga(cons(U, V))) The TRS R consists of the following rules: gopher_in_ga(cons(nil, Y)) -> gopher_out_ga(cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W)) -> U1_ga(gopher_in_ga(cons(U, cons(V, W)))) U1_ga(gopher_out_ga(X)) -> gopher_out_ga(X) The set Q consists of the following terms: gopher_in_ga(x0) U1_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: U2_GG(X, Y, gopher_out_ga(cons(U1, V1))) -> U3_GG(V1, gopher_in_ga(cons(X, Y))) U3_GG(V1, gopher_out_ga(cons(X1, Y1))) -> SAMEFRINGE_IN_GG(V1, Y1) SAMEFRINGE_IN_GG(cons(U, V), cons(X, Y)) -> U2_GG(X, Y, gopher_in_ga(cons(U, V))) Used ordering: Polynomial interpretation [POLO]: POL(SAMEFRINGE_IN_GG(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U1_ga(x_1)) = x_1 POL(U2_GG(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U3_GG(x_1, x_2)) = 2*x_1 + x_2 POL(cons(x_1, x_2)) = 2 + x_1 + x_2 POL(gopher_in_ga(x_1)) = 2*x_1 POL(gopher_out_ga(x_1)) = 2*x_1 POL(nil) = 0 ---------------------------------------- (22) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: gopher_in_ga(cons(nil, Y)) -> gopher_out_ga(cons(nil, Y)) gopher_in_ga(cons(cons(U, V), W)) -> U1_ga(gopher_in_ga(cons(U, cons(V, W)))) U1_ga(gopher_out_ga(X)) -> gopher_out_ga(X) The set Q consists of the following terms: gopher_in_ga(x0) U1_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (24) YES