/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 1140 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 236 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 16 ms] (6) QTRS (7) RisEmptyProof [EQUIVALENT, 0 ms] (8) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U12(tt)) -> mark(tt) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U23(tt)) -> mark(tt) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U32(tt)) -> mark(tt) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U43(tt)) -> mark(tt) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U53(tt)) -> mark(tt) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U62(tt)) -> mark(tt) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(U72(tt)) -> mark(tt) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(o)) -> mark(tt) active(isQid(u)) -> mark(tt) mark(__(X1, X2)) -> active(__(mark(X1), mark(X2))) mark(nil) -> active(nil) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNeList(X)) -> active(isNeList(X)) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(isList(X)) -> active(isList(X)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U32(X)) -> active(U32(mark(X))) mark(isQid(X)) -> active(isQid(X)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2)) -> active(U42(mark(X1), X2)) mark(U43(X)) -> active(U43(mark(X))) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2)) -> active(U52(mark(X1), X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X)) -> active(U72(mark(X))) mark(isNePal(X)) -> active(isNePal(X)) mark(and(X1, X2)) -> active(and(mark(X1), X2)) mark(isPalListKind(X)) -> active(isPalListKind(X)) mark(isPal(X)) -> active(isPal(X)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(o) -> active(o) mark(u) -> active(u) __(mark(X1), X2) -> __(X1, X2) __(X1, mark(X2)) -> __(X1, X2) __(active(X1), X2) -> __(X1, X2) __(X1, active(X2)) -> __(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNeList(mark(X)) -> isNeList(X) isNeList(active(X)) -> isNeList(X) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) isList(mark(X)) -> isList(X) isList(active(X)) -> isList(X) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) isQid(mark(X)) -> isQid(X) isQid(active(X)) -> isQid(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2) -> U42(X1, X2) U42(X1, mark(X2)) -> U42(X1, X2) U42(active(X1), X2) -> U42(X1, X2) U42(X1, active(X2)) -> U42(X1, X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2) -> U52(X1, X2) U52(X1, mark(X2)) -> U52(X1, X2) U52(active(X1), X2) -> U52(X1, X2) U52(X1, active(X2)) -> U52(X1, X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X)) -> U72(X) U72(active(X)) -> U72(X) isNePal(mark(X)) -> isNePal(X) isNePal(active(X)) -> isNePal(X) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isPalListKind(mark(X)) -> isPalListKind(X) isPalListKind(active(X)) -> isPalListKind(X) isPal(mark(X)) -> isPal(X) isPal(active(X)) -> isPal(X) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: active/1)YES( __/2(YES,YES) mark/1)YES( nil/0) U11/2(YES,YES) tt/0) U12/1)YES( isNeList/1(YES) U21/3(YES,YES,YES) U22/2(YES,YES) isList/1(YES) U23/1)YES( U31/2(YES,YES) U32/1)YES( isQid/1)YES( U41/3(YES,YES,YES) U42/2(YES,YES) U43/1)YES( U51/3(YES,YES,YES) U52/2(YES,YES) U53/1)YES( U61/2(YES,YES) U62/1)YES( U71/2(YES,YES) U72/1)YES( isNePal/1(YES) and/2(YES,YES) isPalListKind/1)YES( isPal/1(YES) a/0) e/0) i/0) o/0) u/0) Quasi precedence: ___2 > U21_3 > U22_2 > [isList_1, U52_2] > [U11_2, isNeList_1, U42_2] > U31_2 ___2 > U41_3 > [isList_1, U52_2] > [U11_2, isNeList_1, U42_2] > U31_2 ___2 > U51_3 > [isList_1, U52_2] > [U11_2, isNeList_1, U42_2] > U31_2 ___2 > and_2 ___2 > isPal_1 > U71_2 > isNePal_1 > U61_2 nil > [tt, e, i, u] > U22_2 > [isList_1, U52_2] > [U11_2, isNeList_1, U42_2] > U31_2 nil > [tt, e, i, u] > isNePal_1 > U61_2 a > [tt, e, i, u] > U22_2 > [isList_1, U52_2] > [U11_2, isNeList_1, U42_2] > U31_2 a > [tt, e, i, u] > isNePal_1 > U61_2 o > [tt, e, i, u] > U22_2 > [isList_1, U52_2] > [U11_2, isNeList_1, U42_2] > U31_2 o > [tt, e, i, u] > isNePal_1 > U61_2 Status: ___2: [1,2] nil: multiset status U11_2: multiset status tt: multiset status isNeList_1: multiset status U21_3: [1,2,3] U22_2: multiset status isList_1: multiset status U31_2: multiset status U41_3: [1,3,2] U42_2: multiset status U51_3: [1,2,3] U52_2: multiset status U61_2: multiset status U71_2: multiset status isNePal_1: multiset status and_2: [2,1] isPal_1: multiset status a: multiset status e: multiset status i: multiset status o: multiset status u: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) active(__(X, nil)) -> mark(X) active(__(nil, X)) -> mark(X) active(U11(tt, V)) -> mark(U12(isNeList(V))) active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2)) active(U22(tt, V2)) -> mark(U23(isList(V2))) active(U31(tt, V)) -> mark(U32(isQid(V))) active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2)) active(U42(tt, V2)) -> mark(U43(isNeList(V2))) active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2)) active(U52(tt, V2)) -> mark(U53(isList(V2))) active(U61(tt, V)) -> mark(U62(isQid(V))) active(U71(tt, V)) -> mark(U72(isNePal(V))) active(and(tt, X)) -> mark(X) active(isList(V)) -> mark(U11(isPalListKind(V), V)) active(isList(nil)) -> mark(tt) active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(V)) -> mark(U31(isPalListKind(V), V)) active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2)) active(isNePal(V)) -> mark(U61(isPalListKind(V), V)) active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))) active(isPal(V)) -> mark(U71(isPalListKind(V), V)) active(isPal(nil)) -> mark(tt) active(isPalListKind(a)) -> mark(tt) active(isPalListKind(nil)) -> mark(tt) active(isPalListKind(o)) -> mark(tt) active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2))) active(isQid(a)) -> mark(tt) active(isQid(o)) -> mark(tt) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U12(tt)) -> mark(tt) active(U23(tt)) -> mark(tt) active(U32(tt)) -> mark(tt) active(U43(tt)) -> mark(tt) active(U53(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U72(tt)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(u)) -> mark(tt) mark(__(X1, X2)) -> active(__(mark(X1), mark(X2))) mark(nil) -> active(nil) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNeList(X)) -> active(isNeList(X)) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(isList(X)) -> active(isList(X)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U32(X)) -> active(U32(mark(X))) mark(isQid(X)) -> active(isQid(X)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2)) -> active(U42(mark(X1), X2)) mark(U43(X)) -> active(U43(mark(X))) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2)) -> active(U52(mark(X1), X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X)) -> active(U72(mark(X))) mark(isNePal(X)) -> active(isNePal(X)) mark(and(X1, X2)) -> active(and(mark(X1), X2)) mark(isPalListKind(X)) -> active(isPalListKind(X)) mark(isPal(X)) -> active(isPal(X)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(o) -> active(o) mark(u) -> active(u) __(mark(X1), X2) -> __(X1, X2) __(X1, mark(X2)) -> __(X1, X2) __(active(X1), X2) -> __(X1, X2) __(X1, active(X2)) -> __(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNeList(mark(X)) -> isNeList(X) isNeList(active(X)) -> isNeList(X) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) isList(mark(X)) -> isList(X) isList(active(X)) -> isList(X) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) isQid(mark(X)) -> isQid(X) isQid(active(X)) -> isQid(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2) -> U42(X1, X2) U42(X1, mark(X2)) -> U42(X1, X2) U42(active(X1), X2) -> U42(X1, X2) U42(X1, active(X2)) -> U42(X1, X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2) -> U52(X1, X2) U52(X1, mark(X2)) -> U52(X1, X2) U52(active(X1), X2) -> U52(X1, X2) U52(X1, active(X2)) -> U52(X1, X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X)) -> U72(X) U72(active(X)) -> U72(X) isNePal(mark(X)) -> isNePal(X) isNePal(active(X)) -> isNePal(X) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isPalListKind(mark(X)) -> isPalListKind(X) isPalListKind(active(X)) -> isPalListKind(X) isPal(mark(X)) -> isPal(X) isPal(active(X)) -> isPal(X) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1, x_2)) = 1 + x_1 + x_2 POL(U12(x_1)) = 1 + x_1 POL(U21(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U22(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U23(x_1)) = 1 + 2*x_1 POL(U31(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U32(x_1)) = 1 + 2*x_1 POL(U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U42(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U43(x_1)) = 1 + x_1 POL(U51(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U52(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U53(x_1)) = 1 + 2*x_1 POL(U61(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U62(x_1)) = 1 + 2*x_1 POL(U71(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U72(x_1)) = 1 + 2*x_1 POL(__(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(a) = 1 POL(active(x_1)) = x_1 POL(and(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(e) = 1 POL(i) = 1 POL(isList(x_1)) = 1 + 2*x_1 POL(isNeList(x_1)) = 1 + 2*x_1 POL(isNePal(x_1)) = 1 + 2*x_1 POL(isPal(x_1)) = 1 + 2*x_1 POL(isPalListKind(x_1)) = 1 + 2*x_1 POL(isQid(x_1)) = 1 + 2*x_1 POL(mark(x_1)) = 2*x_1 POL(nil) = 2 POL(o) = 1 POL(tt) = 1 POL(u) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: active(U23(tt)) -> mark(tt) active(U32(tt)) -> mark(tt) active(U53(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U72(tt)) -> mark(tt) active(isPalListKind(e)) -> mark(tt) active(isPalListKind(i)) -> mark(tt) active(isPalListKind(u)) -> mark(tt) active(isQid(e)) -> mark(tt) active(isQid(i)) -> mark(tt) active(isQid(u)) -> mark(tt) mark(__(X1, X2)) -> active(__(mark(X1), mark(X2))) mark(nil) -> active(nil) mark(U11(X1, X2)) -> active(U11(mark(X1), X2)) mark(tt) -> active(tt) mark(U12(X)) -> active(U12(mark(X))) mark(isNeList(X)) -> active(isNeList(X)) mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3)) mark(U22(X1, X2)) -> active(U22(mark(X1), X2)) mark(isList(X)) -> active(isList(X)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1, X2)) -> active(U31(mark(X1), X2)) mark(U32(X)) -> active(U32(mark(X))) mark(isQid(X)) -> active(isQid(X)) mark(U41(X1, X2, X3)) -> active(U41(mark(X1), X2, X3)) mark(U42(X1, X2)) -> active(U42(mark(X1), X2)) mark(U43(X)) -> active(U43(mark(X))) mark(U51(X1, X2, X3)) -> active(U51(mark(X1), X2, X3)) mark(U52(X1, X2)) -> active(U52(mark(X1), X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2)) -> active(U71(mark(X1), X2)) mark(U72(X)) -> active(U72(mark(X))) mark(isNePal(X)) -> active(isNePal(X)) mark(and(X1, X2)) -> active(and(mark(X1), X2)) mark(isPalListKind(X)) -> active(isPalListKind(X)) mark(isPal(X)) -> active(isPal(X)) mark(a) -> active(a) mark(e) -> active(e) mark(i) -> active(i) mark(o) -> active(o) mark(u) -> active(u) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(U12(tt)) -> mark(tt) active(U43(tt)) -> mark(tt) __(mark(X1), X2) -> __(X1, X2) __(X1, mark(X2)) -> __(X1, X2) __(active(X1), X2) -> __(X1, X2) __(X1, active(X2)) -> __(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNeList(mark(X)) -> isNeList(X) isNeList(active(X)) -> isNeList(X) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) isList(mark(X)) -> isList(X) isList(active(X)) -> isList(X) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) isQid(mark(X)) -> isQid(X) isQid(active(X)) -> isQid(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2) -> U42(X1, X2) U42(X1, mark(X2)) -> U42(X1, X2) U42(active(X1), X2) -> U42(X1, X2) U42(X1, active(X2)) -> U42(X1, X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2) -> U52(X1, X2) U52(X1, mark(X2)) -> U52(X1, X2) U52(active(X1), X2) -> U52(X1, X2) U52(X1, active(X2)) -> U52(X1, X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X)) -> U72(X) U72(active(X)) -> U72(X) isNePal(mark(X)) -> isNePal(X) isNePal(active(X)) -> isNePal(X) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isPalListKind(mark(X)) -> isPalListKind(X) isPalListKind(active(X)) -> isPalListKind(X) isPal(mark(X)) -> isPal(X) isPal(active(X)) -> isPal(X) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:active_1 > isPal_1 > isPalListKind_1 > and_2 > isNePal_1 > U72_1 > U71_2 > U62_1 > U61_2 > U53_1 > U52_2 > U51_3 > U42_2 > U41_3 > isQid_1 > U32_1 > U31_2 > U23_1 > isList_1 > U22_2 > U21_3 > isNeList_1 > U11_2 > ___2 > U43_1 > mark_1 > tt > U12_1 and weight map: tt=1 active_1=0 U12_1=1 mark_1=1 U43_1=1 isNeList_1=1 isList_1=1 U23_1=1 U32_1=1 isQid_1=1 U53_1=1 U62_1=1 U72_1=1 isNePal_1=1 isPalListKind_1=1 isPal_1=1 ___2=0 U11_2=0 U21_3=0 U22_2=0 U31_2=0 U41_3=0 U42_2=0 U51_3=0 U52_2=0 U61_2=0 U71_2=0 and_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: active(U12(tt)) -> mark(tt) active(U43(tt)) -> mark(tt) __(mark(X1), X2) -> __(X1, X2) __(X1, mark(X2)) -> __(X1, X2) __(active(X1), X2) -> __(X1, X2) __(X1, active(X2)) -> __(X1, X2) U11(mark(X1), X2) -> U11(X1, X2) U11(X1, mark(X2)) -> U11(X1, X2) U11(active(X1), X2) -> U11(X1, X2) U11(X1, active(X2)) -> U11(X1, X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNeList(mark(X)) -> isNeList(X) isNeList(active(X)) -> isNeList(X) U21(mark(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, mark(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, mark(X3)) -> U21(X1, X2, X3) U21(active(X1), X2, X3) -> U21(X1, X2, X3) U21(X1, active(X2), X3) -> U21(X1, X2, X3) U21(X1, X2, active(X3)) -> U21(X1, X2, X3) U22(mark(X1), X2) -> U22(X1, X2) U22(X1, mark(X2)) -> U22(X1, X2) U22(active(X1), X2) -> U22(X1, X2) U22(X1, active(X2)) -> U22(X1, X2) isList(mark(X)) -> isList(X) isList(active(X)) -> isList(X) U23(mark(X)) -> U23(X) U23(active(X)) -> U23(X) U31(mark(X1), X2) -> U31(X1, X2) U31(X1, mark(X2)) -> U31(X1, X2) U31(active(X1), X2) -> U31(X1, X2) U31(X1, active(X2)) -> U31(X1, X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) isQid(mark(X)) -> isQid(X) isQid(active(X)) -> isQid(X) U41(mark(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, mark(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, mark(X3)) -> U41(X1, X2, X3) U41(active(X1), X2, X3) -> U41(X1, X2, X3) U41(X1, active(X2), X3) -> U41(X1, X2, X3) U41(X1, X2, active(X3)) -> U41(X1, X2, X3) U42(mark(X1), X2) -> U42(X1, X2) U42(X1, mark(X2)) -> U42(X1, X2) U42(active(X1), X2) -> U42(X1, X2) U42(X1, active(X2)) -> U42(X1, X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) U51(mark(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, mark(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, mark(X3)) -> U51(X1, X2, X3) U51(active(X1), X2, X3) -> U51(X1, X2, X3) U51(X1, active(X2), X3) -> U51(X1, X2, X3) U51(X1, X2, active(X3)) -> U51(X1, X2, X3) U52(mark(X1), X2) -> U52(X1, X2) U52(X1, mark(X2)) -> U52(X1, X2) U52(active(X1), X2) -> U52(X1, X2) U52(X1, active(X2)) -> U52(X1, X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2) -> U71(X1, X2) U71(X1, mark(X2)) -> U71(X1, X2) U71(active(X1), X2) -> U71(X1, X2) U71(X1, active(X2)) -> U71(X1, X2) U72(mark(X)) -> U72(X) U72(active(X)) -> U72(X) isNePal(mark(X)) -> isNePal(X) isNePal(active(X)) -> isNePal(X) and(mark(X1), X2) -> and(X1, X2) and(X1, mark(X2)) -> and(X1, X2) and(active(X1), X2) -> and(X1, X2) and(X1, active(X2)) -> and(X1, X2) isPalListKind(mark(X)) -> isPalListKind(X) isPalListKind(active(X)) -> isPalListKind(X) isPal(mark(X)) -> isPal(X) isPal(active(X)) -> isPal(X) ---------------------------------------- (6) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (7) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (8) YES