/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 185 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 71 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 29 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 19 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 3 ms] (10) QTRS (11) RisEmptyProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__U11(tt) -> tt a__U21(tt, V2) -> a__U22(a__isList(V2)) a__U22(tt) -> tt a__U31(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNeList(V2)) a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isList(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt, P) -> a__U72(a__isPal(P)) a__U72(tt) -> tt a__U81(tt) -> tt a__isList(V) -> a__U11(a__isNeList(V)) a__isList(nil) -> tt a__isList(__(V1, V2)) -> a__U21(a__isList(V1), V2) a__isNeList(V) -> a__U31(a__isQid(V)) a__isNeList(__(V1, V2)) -> a__U41(a__isList(V1), V2) a__isNeList(__(V1, V2)) -> a__U51(a__isNeList(V1), V2) a__isNePal(V) -> a__U61(a__isQid(V)) a__isNePal(__(I, __(P, I))) -> a__U71(a__isQid(I), P) a__isPal(V) -> a__U81(a__isNePal(V)) a__isPal(nil) -> tt a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(isPal(X)) -> a__isPal(X) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X) -> U11(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isList(X) -> isList(X) a__U31(X) -> U31(X) a__U41(X1, X2) -> U41(X1, X2) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) a__isNePal(X) -> isNePal(X) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U21(x_1, x_2)) = x_1 + x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 2*x_1 + x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1)) = x_1 POL(U71(x_1, x_2)) = x_1 + 2*x_2 POL(U72(x_1)) = x_1 POL(U81(x_1)) = 2*x_1 POL(__(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(a) = 0 POL(a__U11(x_1)) = x_1 POL(a__U21(x_1, x_2)) = x_1 + x_2 POL(a__U22(x_1)) = x_1 POL(a__U31(x_1)) = x_1 POL(a__U41(x_1, x_2)) = 2*x_1 + x_2 POL(a__U42(x_1)) = x_1 POL(a__U51(x_1, x_2)) = x_1 + x_2 POL(a__U52(x_1)) = x_1 POL(a__U61(x_1)) = x_1 POL(a__U71(x_1, x_2)) = x_1 + 2*x_2 POL(a__U72(x_1)) = x_1 POL(a__U81(x_1)) = 2*x_1 POL(a____(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(a__isList(x_1)) = x_1 POL(a__isNeList(x_1)) = x_1 POL(a__isNePal(x_1)) = x_1 POL(a__isPal(x_1)) = 2*x_1 POL(a__isQid(x_1)) = x_1 POL(e) = 0 POL(i) = 0 POL(isList(x_1)) = x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = x_1 POL(isPal(x_1)) = 2*x_1 POL(isQid(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(o) = 0 POL(tt) = 0 POL(u) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__isList(__(V1, V2)) -> a__U21(a__isList(V1), V2) a__isNeList(__(V1, V2)) -> a__U41(a__isList(V1), V2) a__isNeList(__(V1, V2)) -> a__U51(a__isNeList(V1), V2) a__isNePal(__(I, __(P, I))) -> a__U71(a__isQid(I), P) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__U11(tt) -> tt a__U21(tt, V2) -> a__U22(a__isList(V2)) a__U22(tt) -> tt a__U31(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNeList(V2)) a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isList(V2)) a__U52(tt) -> tt a__U61(tt) -> tt a__U71(tt, P) -> a__U72(a__isPal(P)) a__U72(tt) -> tt a__U81(tt) -> tt a__isList(V) -> a__U11(a__isNeList(V)) a__isList(nil) -> tt a__isNeList(V) -> a__U31(a__isQid(V)) a__isNePal(V) -> a__U61(a__isQid(V)) a__isPal(V) -> a__U81(a__isNePal(V)) a__isPal(nil) -> tt a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X)) -> a__U72(mark(X)) mark(isPal(X)) -> a__isPal(X) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X) -> U11(X) a__U21(X1, X2) -> U21(X1, X2) a__U22(X) -> U22(X) a__isList(X) -> isList(X) a__U31(X) -> U31(X) a__U41(X1, X2) -> U41(X1, X2) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) a__isNePal(X) -> isNePal(X) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = 1 + 2*x_1 POL(U21(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U42(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1)) = 1 + x_1 POL(U71(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = x_1 + 2*x_2 POL(a) = 2 POL(a__U11(x_1)) = 1 + 2*x_1 POL(a__U21(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__U22(x_1)) = x_1 POL(a__U31(x_1)) = x_1 POL(a__U41(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__U42(x_1)) = 2*x_1 POL(a__U51(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__U52(x_1)) = x_1 POL(a__U61(x_1)) = 1 + x_1 POL(a__U71(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(a__U72(x_1)) = x_1 POL(a__U81(x_1)) = x_1 POL(a____(x_1, x_2)) = x_1 + 2*x_2 POL(a__isList(x_1)) = 2 + 2*x_1 POL(a__isNeList(x_1)) = x_1 POL(a__isNePal(x_1)) = 2 + x_1 POL(a__isPal(x_1)) = 2 + x_1 POL(a__isQid(x_1)) = x_1 POL(e) = 1 POL(i) = 2 POL(isList(x_1)) = 1 + 2*x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = 1 + x_1 POL(isPal(x_1)) = 2 + x_1 POL(isQid(x_1)) = x_1 POL(mark(x_1)) = 2*x_1 POL(nil) = 1 POL(o) = 1 POL(tt) = 0 POL(u) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__U11(tt) -> tt a__U41(tt, V2) -> a__U42(a__isNeList(V2)) a__U61(tt) -> tt a__isList(V) -> a__U11(a__isNeList(V)) a__isList(nil) -> tt a__isNePal(V) -> a__U61(a__isQid(V)) a__isPal(nil) -> tt a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt mark(U11(X)) -> a__U11(mark(X)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o a__U21(X1, X2) -> U21(X1, X2) a__isList(X) -> isList(X) a__U41(X1, X2) -> U41(X1, X2) a__U51(X1, X2) -> U51(X1, X2) a__isNePal(X) -> isNePal(X) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__U21(tt, V2) -> a__U22(a__isList(V2)) a__U22(tt) -> tt a__U31(tt) -> tt a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isList(V2)) a__U52(tt) -> tt a__U71(tt, P) -> a__U72(a__isPal(P)) a__U72(tt) -> tt a__U81(tt) -> tt a__isNeList(V) -> a__U31(a__isQid(V)) a__isPal(V) -> a__U81(a__isNePal(V)) a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(tt) -> tt mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X) -> U11(X) a__U22(X) -> U22(X) a__U31(X) -> U31(X) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U52(X) -> U52(X) a__U61(X) -> U61(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = 2 + x_1 POL(U21(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U22(x_1)) = 2*x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1)) = x_1 POL(U71(x_1, x_2)) = x_1 + 2*x_2 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = x_1 + x_2 POL(a__U11(x_1)) = 2 + x_1 POL(a__U21(x_1, x_2)) = x_1 + 2*x_2 POL(a__U22(x_1)) = 2*x_1 POL(a__U31(x_1)) = x_1 POL(a__U41(x_1, x_2)) = x_1 + 2*x_2 POL(a__U42(x_1)) = x_1 POL(a__U51(x_1, x_2)) = x_1 + 2*x_2 POL(a__U52(x_1)) = x_1 POL(a__U61(x_1)) = 1 + x_1 POL(a__U71(x_1, x_2)) = x_1 + 2*x_2 POL(a__U72(x_1)) = x_1 POL(a__U81(x_1)) = x_1 POL(a____(x_1, x_2)) = x_1 + x_2 POL(a__isList(x_1)) = x_1 POL(a__isNeList(x_1)) = x_1 POL(a__isNePal(x_1)) = x_1 POL(a__isPal(x_1)) = 2*x_1 POL(a__isQid(x_1)) = x_1 POL(isList(x_1)) = 2*x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = 2*x_1 POL(isPal(x_1)) = 2*x_1 POL(isQid(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(tt) = 0 POL(u) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__U61(X) -> U61(X) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__U21(tt, V2) -> a__U22(a__isList(V2)) a__U22(tt) -> tt a__U31(tt) -> tt a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isList(V2)) a__U52(tt) -> tt a__U71(tt, P) -> a__U72(a__isPal(P)) a__U72(tt) -> tt a__U81(tt) -> tt a__isNeList(V) -> a__U31(a__isQid(V)) a__isPal(V) -> a__U81(a__isNePal(V)) a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(tt) -> tt mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U11(X) -> U11(X) a__U22(X) -> U22(X) a__U31(X) -> U31(X) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U52(X) -> U52(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U21(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U22(x_1)) = 2*x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U52(x_1)) = x_1 POL(U71(x_1, x_2)) = x_1 + 2*x_2 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = x_1 + x_2 POL(a__U11(x_1)) = 1 + x_1 POL(a__U21(x_1, x_2)) = x_1 + 2*x_2 POL(a__U22(x_1)) = 2*x_1 POL(a__U31(x_1)) = x_1 POL(a__U41(x_1, x_2)) = x_1 + 2*x_2 POL(a__U42(x_1)) = x_1 POL(a__U51(x_1, x_2)) = x_1 + 2*x_2 POL(a__U52(x_1)) = x_1 POL(a__U71(x_1, x_2)) = x_1 + 2*x_2 POL(a__U72(x_1)) = x_1 POL(a__U81(x_1)) = x_1 POL(a____(x_1, x_2)) = x_1 + x_2 POL(a__isList(x_1)) = x_1 POL(a__isNeList(x_1)) = x_1 POL(a__isNePal(x_1)) = x_1 POL(a__isPal(x_1)) = 2*x_1 POL(a__isQid(x_1)) = x_1 POL(isList(x_1)) = 2*x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = 2*x_1 POL(isPal(x_1)) = x_1 POL(isQid(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(tt) = 0 POL(u) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__U11(X) -> U11(X) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__U21(tt, V2) -> a__U22(a__isList(V2)) a__U22(tt) -> tt a__U31(tt) -> tt a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isList(V2)) a__U52(tt) -> tt a__U71(tt, P) -> a__U72(a__isPal(P)) a__U72(tt) -> tt a__U81(tt) -> tt a__isNeList(V) -> a__U31(a__isQid(V)) a__isPal(V) -> a__U81(a__isNePal(V)) a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(tt) -> tt mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U22(X) -> U22(X) a__U31(X) -> U31(X) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U52(X) -> U52(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) Q is empty. ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:mark_1 > a__U51_2 > a__U52_1 > isPal_1 > U71_2 > isNePal_1 > u > a__U21_2 > a__U22_1 > U52_1 > U22_1 > a__U71_2 > a__U72_1 > a_____2 > ___2 > a__isNeList_1 > isNeList_1 > a__U42_1 > a__U31_1 > a__isPal_1 > U42_1 > U21_2 > U31_1 > a__isQid_1 > tt > isQid_1 > a__isList_1 > a__U81_1 > U72_1 > a__U41_2 > U81_1 > U51_2 > isList_1 > a__isNePal_1 > U41_2 and weight map: tt=2 u=1 a__U22_1=1 a__isList_1=1 a__U31_1=1 a__U42_1=1 a__U52_1=1 a__U72_1=1 a__isPal_1=2 a__U81_1=1 a__isNeList_1=2 a__isQid_1=1 a__isNePal_1=1 mark_1=0 U22_1=1 isList_1=1 U31_1=1 U42_1=1 isNeList_1=2 U52_1=1 U72_1=1 U81_1=1 isQid_1=1 isNePal_1=1 isPal_1=1 a__U21_2=0 a__U51_2=0 a__U71_2=1 ___2=0 a_____2=0 U21_2=1 U41_2=1 a__U41_2=0 U51_2=1 U71_2=0 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__U21(tt, V2) -> a__U22(a__isList(V2)) a__U22(tt) -> tt a__U31(tt) -> tt a__U42(tt) -> tt a__U51(tt, V2) -> a__U52(a__isList(V2)) a__U52(tt) -> tt a__U71(tt, P) -> a__U72(a__isPal(P)) a__U72(tt) -> tt a__U81(tt) -> tt a__isNeList(V) -> a__U31(a__isQid(V)) a__isPal(V) -> a__U81(a__isNePal(V)) a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X)) -> a__U22(mark(X)) mark(isList(X)) -> a__isList(X) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2)) -> a__U41(mark(X1), X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNeList(X)) -> a__isNeList(X) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U72(X)) -> a__U72(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(tt) -> tt mark(u) -> u a____(X1, X2) -> __(X1, X2) a__U22(X) -> U22(X) a__U31(X) -> U31(X) a__U42(X) -> U42(X) a__isNeList(X) -> isNeList(X) a__U52(X) -> U52(X) a__U71(X1, X2) -> U71(X1, X2) a__U72(X) -> U72(X) a__isPal(X) -> isPal(X) a__U81(X) -> U81(X) a__isQid(X) -> isQid(X) ---------------------------------------- (10) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (11) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (12) YES