6.97/2.77 YES 6.97/2.79 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 6.97/2.79 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.97/2.79 6.97/2.79 6.97/2.79 Termination w.r.t. Q of the given QTRS could be proven: 6.97/2.79 6.97/2.79 (0) QTRS 6.97/2.79 (1) QTRSRRRProof [EQUIVALENT, 515 ms] 6.97/2.79 (2) QTRS 6.97/2.79 (3) QTRSRRRProof [EQUIVALENT, 81 ms] 6.97/2.79 (4) QTRS 6.97/2.79 (5) QTRSRRRProof [EQUIVALENT, 35 ms] 6.97/2.79 (6) QTRS 6.97/2.79 (7) RisEmptyProof [EQUIVALENT, 0 ms] 6.97/2.79 (8) YES 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (0) 6.97/2.79 Obligation: 6.97/2.79 Q restricted rewrite system: 6.97/2.79 The TRS R consists of the following rules: 6.97/2.79 6.97/2.79 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 6.97/2.79 a____(X, nil) -> mark(X) 6.97/2.79 a____(nil, X) -> mark(X) 6.97/2.79 a__U11(tt, V) -> a__U12(a__isNeList(V)) 6.97/2.79 a__U12(tt) -> tt 6.97/2.79 a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) 6.97/2.79 a__U22(tt, V2) -> a__U23(a__isList(V2)) 6.97/2.79 a__U23(tt) -> tt 6.97/2.79 a__U31(tt, V) -> a__U32(a__isQid(V)) 6.97/2.79 a__U32(tt) -> tt 6.97/2.79 a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) 6.97/2.79 a__U42(tt, V2) -> a__U43(a__isNeList(V2)) 6.97/2.79 a__U43(tt) -> tt 6.97/2.79 a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) 6.97/2.79 a__U52(tt, V2) -> a__U53(a__isList(V2)) 6.97/2.79 a__U53(tt) -> tt 6.97/2.79 a__U61(tt, V) -> a__U62(a__isQid(V)) 6.97/2.79 a__U62(tt) -> tt 6.97/2.79 a__U71(tt, V) -> a__U72(a__isNePal(V)) 6.97/2.79 a__U72(tt) -> tt 6.97/2.79 a__and(tt, X) -> mark(X) 6.97/2.79 a__isList(V) -> a__U11(a__isPalListKind(V), V) 6.97/2.79 a__isList(nil) -> tt 6.97/2.79 a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 6.97/2.79 a__isNeList(V) -> a__U31(a__isPalListKind(V), V) 6.97/2.79 a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 6.97/2.79 a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 6.97/2.79 a__isNePal(V) -> a__U61(a__isPalListKind(V), V) 6.97/2.79 a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) 6.97/2.79 a__isPal(V) -> a__U71(a__isPalListKind(V), V) 6.97/2.79 a__isPal(nil) -> tt 6.97/2.79 a__isPalListKind(a) -> tt 6.97/2.79 a__isPalListKind(e) -> tt 6.97/2.79 a__isPalListKind(i) -> tt 6.97/2.79 a__isPalListKind(nil) -> tt 6.97/2.79 a__isPalListKind(o) -> tt 6.97/2.79 a__isPalListKind(u) -> tt 6.97/2.79 a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) 6.97/2.79 a__isQid(a) -> tt 6.97/2.79 a__isQid(e) -> tt 6.97/2.79 a__isQid(i) -> tt 6.97/2.79 a__isQid(o) -> tt 6.97/2.79 a__isQid(u) -> tt 6.97/2.79 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 6.97/2.79 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.97/2.79 mark(U12(X)) -> a__U12(mark(X)) 6.97/2.79 mark(isNeList(X)) -> a__isNeList(X) 6.97/2.79 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 6.97/2.79 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 6.97/2.79 mark(isList(X)) -> a__isList(X) 6.97/2.79 mark(U23(X)) -> a__U23(mark(X)) 6.97/2.79 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.97/2.79 mark(U32(X)) -> a__U32(mark(X)) 6.97/2.79 mark(isQid(X)) -> a__isQid(X) 6.97/2.79 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.97/2.79 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 6.97/2.79 mark(U43(X)) -> a__U43(mark(X)) 6.97/2.79 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 6.97/2.79 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 6.97/2.79 mark(U53(X)) -> a__U53(mark(X)) 6.97/2.79 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 6.97/2.79 mark(U62(X)) -> a__U62(mark(X)) 6.97/2.79 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 6.97/2.79 mark(U72(X)) -> a__U72(mark(X)) 6.97/2.79 mark(isNePal(X)) -> a__isNePal(X) 6.97/2.79 mark(and(X1, X2)) -> a__and(mark(X1), X2) 6.97/2.79 mark(isPalListKind(X)) -> a__isPalListKind(X) 6.97/2.79 mark(isPal(X)) -> a__isPal(X) 6.97/2.79 mark(nil) -> nil 6.97/2.79 mark(tt) -> tt 6.97/2.79 mark(a) -> a 6.97/2.79 mark(e) -> e 6.97/2.79 mark(i) -> i 6.97/2.79 mark(o) -> o 6.97/2.79 mark(u) -> u 6.97/2.79 a____(X1, X2) -> __(X1, X2) 6.97/2.79 a__U11(X1, X2) -> U11(X1, X2) 6.97/2.79 a__U12(X) -> U12(X) 6.97/2.79 a__isNeList(X) -> isNeList(X) 6.97/2.79 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 6.97/2.79 a__U22(X1, X2) -> U22(X1, X2) 6.97/2.79 a__isList(X) -> isList(X) 6.97/2.79 a__U23(X) -> U23(X) 6.97/2.79 a__U31(X1, X2) -> U31(X1, X2) 6.97/2.79 a__U32(X) -> U32(X) 6.97/2.79 a__isQid(X) -> isQid(X) 6.97/2.79 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.97/2.79 a__U42(X1, X2) -> U42(X1, X2) 6.97/2.79 a__U43(X) -> U43(X) 6.97/2.79 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 6.97/2.79 a__U52(X1, X2) -> U52(X1, X2) 6.97/2.79 a__U53(X) -> U53(X) 6.97/2.79 a__U61(X1, X2) -> U61(X1, X2) 6.97/2.79 a__U62(X) -> U62(X) 6.97/2.79 a__U71(X1, X2) -> U71(X1, X2) 6.97/2.79 a__U72(X) -> U72(X) 6.97/2.79 a__isNePal(X) -> isNePal(X) 6.97/2.79 a__and(X1, X2) -> and(X1, X2) 6.97/2.79 a__isPalListKind(X) -> isPalListKind(X) 6.97/2.79 a__isPal(X) -> isPal(X) 6.97/2.79 6.97/2.79 The set Q consists of the following terms: 6.97/2.79 6.97/2.79 a__isList(x0) 6.97/2.79 a__isNeList(x0) 6.97/2.79 a__isNePal(x0) 6.97/2.79 a__isPal(x0) 6.97/2.79 mark(__(x0, x1)) 6.97/2.79 mark(U11(x0, x1)) 6.97/2.79 mark(U12(x0)) 6.97/2.79 mark(isNeList(x0)) 6.97/2.79 mark(U21(x0, x1, x2)) 6.97/2.79 mark(U22(x0, x1)) 6.97/2.79 mark(isList(x0)) 6.97/2.79 mark(U23(x0)) 6.97/2.79 mark(U31(x0, x1)) 6.97/2.79 mark(U32(x0)) 6.97/2.79 mark(isQid(x0)) 6.97/2.79 mark(U41(x0, x1, x2)) 6.97/2.79 mark(U42(x0, x1)) 6.97/2.79 mark(U43(x0)) 6.97/2.79 mark(U51(x0, x1, x2)) 6.97/2.79 mark(U52(x0, x1)) 6.97/2.79 mark(U53(x0)) 6.97/2.79 mark(U61(x0, x1)) 6.97/2.79 mark(U62(x0)) 6.97/2.79 mark(U71(x0, x1)) 6.97/2.79 mark(U72(x0)) 6.97/2.79 mark(isNePal(x0)) 6.97/2.79 mark(and(x0, x1)) 6.97/2.79 mark(isPalListKind(x0)) 6.97/2.79 mark(isPal(x0)) 6.97/2.79 mark(nil) 6.97/2.79 mark(tt) 6.97/2.79 mark(a) 6.97/2.79 mark(e) 6.97/2.79 mark(i) 6.97/2.79 mark(o) 6.97/2.79 mark(u) 6.97/2.79 a____(x0, x1) 6.97/2.79 a__U11(x0, x1) 6.97/2.79 a__U12(x0) 6.97/2.79 a__U21(x0, x1, x2) 6.97/2.79 a__U22(x0, x1) 6.97/2.79 a__U23(x0) 6.97/2.79 a__U31(x0, x1) 6.97/2.79 a__U32(x0) 6.97/2.79 a__isQid(x0) 6.97/2.79 a__U41(x0, x1, x2) 6.97/2.79 a__U42(x0, x1) 6.97/2.79 a__U43(x0) 6.97/2.79 a__U51(x0, x1, x2) 6.97/2.79 a__U52(x0, x1) 6.97/2.79 a__U53(x0) 6.97/2.79 a__U61(x0, x1) 6.97/2.79 a__U62(x0) 6.97/2.79 a__U71(x0, x1) 6.97/2.79 a__U72(x0) 6.97/2.79 a__and(x0, x1) 6.97/2.79 a__isPalListKind(x0) 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (1) QTRSRRRProof (EQUIVALENT) 6.97/2.79 Used ordering: 6.97/2.79 a____/2(YES,YES) 6.97/2.79 __/2(YES,YES) 6.97/2.79 mark/1)YES( 6.97/2.79 nil/0) 6.97/2.79 a__U11/2(YES,YES) 6.97/2.79 tt/0) 6.97/2.79 a__U12/1)YES( 6.97/2.79 a__isNeList/1(YES) 6.97/2.79 a__U21/3(YES,YES,YES) 6.97/2.79 a__U22/2(YES,YES) 6.97/2.79 a__isList/1(YES) 6.97/2.79 a__U23/1)YES( 6.97/2.79 a__U31/2(YES,YES) 6.97/2.79 a__U32/1)YES( 6.97/2.79 a__isQid/1)YES( 6.97/2.79 a__U41/3(YES,YES,YES) 6.97/2.79 a__U42/2(YES,YES) 6.97/2.79 a__U43/1)YES( 6.97/2.79 a__U51/3(YES,YES,YES) 6.97/2.79 a__U52/2(YES,YES) 6.97/2.79 a__U53/1)YES( 6.97/2.79 a__U61/2(YES,YES) 6.97/2.79 a__U62/1)YES( 6.97/2.79 a__U71/2(YES,YES) 6.97/2.79 a__U72/1)YES( 6.97/2.79 a__isNePal/1(YES) 6.97/2.79 a__and/2(YES,YES) 6.97/2.79 a__isPalListKind/1(YES) 6.97/2.79 isPalListKind/1(YES) 6.97/2.79 and/2(YES,YES) 6.97/2.79 isPal/1(YES) 6.97/2.79 a__isPal/1(YES) 6.97/2.79 a/0) 6.97/2.79 e/0) 6.97/2.79 i/0) 6.97/2.79 o/0) 6.97/2.79 u/0) 6.97/2.79 U11/2(YES,YES) 6.97/2.79 U12/1)YES( 6.97/2.79 isNeList/1(YES) 6.97/2.79 U21/3(YES,YES,YES) 6.97/2.79 U22/2(YES,YES) 6.97/2.79 isList/1(YES) 6.97/2.79 U23/1)YES( 6.97/2.79 U31/2(YES,YES) 6.97/2.79 U32/1)YES( 6.97/2.79 isQid/1)YES( 6.97/2.79 U41/3(YES,YES,YES) 6.97/2.79 U42/2(YES,YES) 6.97/2.79 U43/1)YES( 6.97/2.79 U51/3(YES,YES,YES) 6.97/2.79 U52/2(YES,YES) 6.97/2.79 U53/1)YES( 6.97/2.79 U61/2(YES,YES) 6.97/2.79 U62/1)YES( 6.97/2.79 U71/2(YES,YES) 6.97/2.79 U72/1)YES( 6.97/2.79 isNePal/1(YES) 6.97/2.79 6.97/2.79 Quasi precedence: 6.97/2.79 [a_____2, ___2] > [a__U21_3, U21_3] > [a__U22_2, U22_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 [a_____2, ___2] > [a__U21_3, U21_3] > [a__U22_2, U22_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 [a_____2, ___2] > [a__U41_3, U41_3] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 [a_____2, ___2] > [a__U41_3, U41_3] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 [a_____2, ___2] > [a__U41_3, U41_3] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 [a_____2, ___2] > [a__U41_3, U41_3] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 [a_____2, ___2] > [a__U51_3, U51_3] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 [a_____2, ___2] > [a__U51_3, U51_3] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 [a_____2, ___2] > [isPal_1, a__isPal_1] > [a__U71_2, U71_2] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] 6.97/2.79 [a_____2, ___2] > [isPal_1, a__isPal_1] > [a__U71_2, U71_2] > [a__isNePal_1, isNePal_1] > [a__and_2, and_2] 6.97/2.79 [a_____2, ___2] > [isPal_1, a__isPal_1] > [a__U71_2, U71_2] > [a__isNePal_1, isNePal_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 nil > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 nil > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 nil > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 nil > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 nil > [tt, i] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] 6.97/2.79 nil > [tt, i] > [a__isNePal_1, isNePal_1] > [a__and_2, and_2] 6.97/2.79 nil > [tt, i] > [a__isNePal_1, isNePal_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 a > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 a > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 a > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 a > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 a > [tt, i] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] 6.97/2.79 a > [tt, i] > [a__isNePal_1, isNePal_1] > [a__and_2, and_2] 6.97/2.79 a > [tt, i] > [a__isNePal_1, isNePal_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 e > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 e > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 e > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 e > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 e > [tt, i] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] 6.97/2.79 e > [tt, i] > [a__isNePal_1, isNePal_1] > [a__and_2, and_2] 6.97/2.79 e > [tt, i] > [a__isNePal_1, isNePal_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 o > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 o > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 o > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 o > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 o > [tt, i] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] 6.97/2.79 o > [tt, i] > [a__isNePal_1, isNePal_1] > [a__and_2, and_2] 6.97/2.79 o > [tt, i] > [a__isNePal_1, isNePal_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 u > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 u > [tt, i] > [a__U42_2, U42_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 u > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__U31_2, U31_2] 6.97/2.79 u > [tt, i] > [a__U52_2, U52_2] > [a__isList_1, isList_1] > [a__U11_2, U11_2] > [a__isNeList_1, isNeList_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 u > [tt, i] > [a__isNePal_1, isNePal_1] > [a__U61_2, U61_2] 6.97/2.79 u > [tt, i] > [a__isNePal_1, isNePal_1] > [a__and_2, and_2] 6.97/2.79 u > [tt, i] > [a__isNePal_1, isNePal_1] > [a__isPalListKind_1, isPalListKind_1] 6.97/2.79 6.97/2.79 6.97/2.79 Status: 6.97/2.79 a_____2: [1,2] 6.97/2.79 ___2: [1,2] 6.97/2.79 nil: multiset status 6.97/2.79 a__U11_2: multiset status 6.97/2.79 tt: multiset status 6.97/2.79 a__isNeList_1: multiset status 6.97/2.79 a__U21_3: multiset status 6.97/2.79 a__U22_2: multiset status 6.97/2.79 a__isList_1: multiset status 6.97/2.79 a__U31_2: multiset status 6.97/2.79 a__U41_3: multiset status 6.97/2.79 a__U42_2: multiset status 6.97/2.79 a__U51_3: multiset status 6.97/2.79 a__U52_2: multiset status 6.97/2.79 a__U61_2: multiset status 6.97/2.79 a__U71_2: [1,2] 6.97/2.79 a__isNePal_1: multiset status 6.97/2.79 a__and_2: multiset status 6.97/2.79 a__isPalListKind_1: multiset status 6.97/2.79 isPalListKind_1: multiset status 6.97/2.79 and_2: multiset status 6.97/2.79 isPal_1: multiset status 6.97/2.79 a__isPal_1: multiset status 6.97/2.79 a: multiset status 6.97/2.79 e: multiset status 6.97/2.79 i: multiset status 6.97/2.79 o: multiset status 6.97/2.79 u: multiset status 6.97/2.79 U11_2: multiset status 6.97/2.79 isNeList_1: multiset status 6.97/2.79 U21_3: multiset status 6.97/2.79 U22_2: multiset status 6.97/2.79 isList_1: multiset status 6.97/2.79 U31_2: multiset status 6.97/2.79 U41_3: multiset status 6.97/2.79 U42_2: multiset status 6.97/2.79 U51_3: multiset status 6.97/2.79 U52_2: multiset status 6.97/2.79 U61_2: multiset status 6.97/2.79 U71_2: [1,2] 6.97/2.79 isNePal_1: multiset status 6.97/2.79 6.97/2.79 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 6.97/2.79 6.97/2.79 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 6.97/2.79 a____(X, nil) -> mark(X) 6.97/2.79 a____(nil, X) -> mark(X) 6.97/2.79 a__U11(tt, V) -> a__U12(a__isNeList(V)) 6.97/2.79 a__U21(tt, V1, V2) -> a__U22(a__isList(V1), V2) 6.97/2.79 a__U22(tt, V2) -> a__U23(a__isList(V2)) 6.97/2.79 a__U31(tt, V) -> a__U32(a__isQid(V)) 6.97/2.79 a__U41(tt, V1, V2) -> a__U42(a__isList(V1), V2) 6.97/2.79 a__U42(tt, V2) -> a__U43(a__isNeList(V2)) 6.97/2.79 a__U51(tt, V1, V2) -> a__U52(a__isNeList(V1), V2) 6.97/2.79 a__U52(tt, V2) -> a__U53(a__isList(V2)) 6.97/2.79 a__U61(tt, V) -> a__U62(a__isQid(V)) 6.97/2.79 a__U71(tt, V) -> a__U72(a__isNePal(V)) 6.97/2.79 a__and(tt, X) -> mark(X) 6.97/2.79 a__isList(V) -> a__U11(a__isPalListKind(V), V) 6.97/2.79 a__isList(nil) -> tt 6.97/2.79 a__isList(__(V1, V2)) -> a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 6.97/2.79 a__isNeList(V) -> a__U31(a__isPalListKind(V), V) 6.97/2.79 a__isNeList(__(V1, V2)) -> a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 6.97/2.79 a__isNeList(__(V1, V2)) -> a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2) 6.97/2.79 a__isNePal(V) -> a__U61(a__isPalListKind(V), V) 6.97/2.79 a__isNePal(__(I, __(P, I))) -> a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))) 6.97/2.79 a__isPal(V) -> a__U71(a__isPalListKind(V), V) 6.97/2.79 a__isPal(nil) -> tt 6.97/2.79 a__isPalListKind(a) -> tt 6.97/2.79 a__isPalListKind(e) -> tt 6.97/2.79 a__isPalListKind(i) -> tt 6.97/2.79 a__isPalListKind(nil) -> tt 6.97/2.79 a__isPalListKind(o) -> tt 6.97/2.79 a__isPalListKind(u) -> tt 6.97/2.79 a__isPalListKind(__(V1, V2)) -> a__and(a__isPalListKind(V1), isPalListKind(V2)) 6.97/2.79 a__isQid(a) -> tt 6.97/2.79 a__isQid(e) -> tt 6.97/2.79 a__isQid(o) -> tt 6.97/2.79 a__isQid(u) -> tt 6.97/2.79 6.97/2.79 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (2) 6.97/2.79 Obligation: 6.97/2.79 Q restricted rewrite system: 6.97/2.79 The TRS R consists of the following rules: 6.97/2.79 6.97/2.79 a__U12(tt) -> tt 6.97/2.79 a__U23(tt) -> tt 6.97/2.79 a__U32(tt) -> tt 6.97/2.79 a__U43(tt) -> tt 6.97/2.79 a__U53(tt) -> tt 6.97/2.79 a__U62(tt) -> tt 6.97/2.79 a__U72(tt) -> tt 6.97/2.79 a__isQid(i) -> tt 6.97/2.79 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 6.97/2.79 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.97/2.79 mark(U12(X)) -> a__U12(mark(X)) 6.97/2.79 mark(isNeList(X)) -> a__isNeList(X) 6.97/2.79 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 6.97/2.79 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 6.97/2.79 mark(isList(X)) -> a__isList(X) 6.97/2.79 mark(U23(X)) -> a__U23(mark(X)) 6.97/2.79 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.97/2.79 mark(U32(X)) -> a__U32(mark(X)) 6.97/2.79 mark(isQid(X)) -> a__isQid(X) 6.97/2.79 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.97/2.79 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 6.97/2.79 mark(U43(X)) -> a__U43(mark(X)) 6.97/2.79 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 6.97/2.79 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 6.97/2.79 mark(U53(X)) -> a__U53(mark(X)) 6.97/2.79 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 6.97/2.79 mark(U62(X)) -> a__U62(mark(X)) 6.97/2.79 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 6.97/2.79 mark(U72(X)) -> a__U72(mark(X)) 6.97/2.79 mark(isNePal(X)) -> a__isNePal(X) 6.97/2.79 mark(and(X1, X2)) -> a__and(mark(X1), X2) 6.97/2.79 mark(isPalListKind(X)) -> a__isPalListKind(X) 6.97/2.79 mark(isPal(X)) -> a__isPal(X) 6.97/2.79 mark(nil) -> nil 6.97/2.79 mark(tt) -> tt 6.97/2.79 mark(a) -> a 6.97/2.79 mark(e) -> e 6.97/2.79 mark(i) -> i 6.97/2.79 mark(o) -> o 6.97/2.79 mark(u) -> u 6.97/2.79 a____(X1, X2) -> __(X1, X2) 6.97/2.79 a__U11(X1, X2) -> U11(X1, X2) 6.97/2.79 a__U12(X) -> U12(X) 6.97/2.79 a__isNeList(X) -> isNeList(X) 6.97/2.79 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 6.97/2.79 a__U22(X1, X2) -> U22(X1, X2) 6.97/2.79 a__isList(X) -> isList(X) 6.97/2.79 a__U23(X) -> U23(X) 6.97/2.79 a__U31(X1, X2) -> U31(X1, X2) 6.97/2.79 a__U32(X) -> U32(X) 6.97/2.79 a__isQid(X) -> isQid(X) 6.97/2.79 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.97/2.79 a__U42(X1, X2) -> U42(X1, X2) 6.97/2.79 a__U43(X) -> U43(X) 6.97/2.79 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 6.97/2.79 a__U52(X1, X2) -> U52(X1, X2) 6.97/2.79 a__U53(X) -> U53(X) 6.97/2.79 a__U61(X1, X2) -> U61(X1, X2) 6.97/2.79 a__U62(X) -> U62(X) 6.97/2.79 a__U71(X1, X2) -> U71(X1, X2) 6.97/2.79 a__U72(X) -> U72(X) 6.97/2.79 a__isNePal(X) -> isNePal(X) 6.97/2.79 a__and(X1, X2) -> and(X1, X2) 6.97/2.79 a__isPalListKind(X) -> isPalListKind(X) 6.97/2.79 a__isPal(X) -> isPal(X) 6.97/2.79 6.97/2.79 The set Q consists of the following terms: 6.97/2.79 6.97/2.79 a__isList(x0) 6.97/2.79 a__isNeList(x0) 6.97/2.79 a__isNePal(x0) 6.97/2.79 a__isPal(x0) 6.97/2.79 mark(__(x0, x1)) 6.97/2.79 mark(U11(x0, x1)) 6.97/2.79 mark(U12(x0)) 6.97/2.79 mark(isNeList(x0)) 6.97/2.79 mark(U21(x0, x1, x2)) 6.97/2.79 mark(U22(x0, x1)) 6.97/2.79 mark(isList(x0)) 6.97/2.79 mark(U23(x0)) 6.97/2.79 mark(U31(x0, x1)) 6.97/2.79 mark(U32(x0)) 6.97/2.79 mark(isQid(x0)) 6.97/2.79 mark(U41(x0, x1, x2)) 6.97/2.79 mark(U42(x0, x1)) 6.97/2.79 mark(U43(x0)) 6.97/2.79 mark(U51(x0, x1, x2)) 6.97/2.79 mark(U52(x0, x1)) 6.97/2.79 mark(U53(x0)) 6.97/2.79 mark(U61(x0, x1)) 6.97/2.79 mark(U62(x0)) 6.97/2.79 mark(U71(x0, x1)) 6.97/2.79 mark(U72(x0)) 6.97/2.79 mark(isNePal(x0)) 6.97/2.79 mark(and(x0, x1)) 6.97/2.79 mark(isPalListKind(x0)) 6.97/2.79 mark(isPal(x0)) 6.97/2.79 mark(nil) 6.97/2.79 mark(tt) 6.97/2.79 mark(a) 6.97/2.79 mark(e) 6.97/2.79 mark(i) 6.97/2.79 mark(o) 6.97/2.79 mark(u) 6.97/2.79 a____(x0, x1) 6.97/2.79 a__U11(x0, x1) 6.97/2.79 a__U12(x0) 6.97/2.79 a__U21(x0, x1, x2) 6.97/2.79 a__U22(x0, x1) 6.97/2.79 a__U23(x0) 6.97/2.79 a__U31(x0, x1) 6.97/2.79 a__U32(x0) 6.97/2.79 a__isQid(x0) 6.97/2.79 a__U41(x0, x1, x2) 6.97/2.79 a__U42(x0, x1) 6.97/2.79 a__U43(x0) 6.97/2.79 a__U51(x0, x1, x2) 6.97/2.79 a__U52(x0, x1) 6.97/2.79 a__U53(x0) 6.97/2.79 a__U61(x0, x1) 6.97/2.79 a__U62(x0) 6.97/2.79 a__U71(x0, x1) 6.97/2.79 a__U72(x0) 6.97/2.79 a__and(x0, x1) 6.97/2.79 a__isPalListKind(x0) 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (3) QTRSRRRProof (EQUIVALENT) 6.97/2.79 Used ordering: 6.97/2.79 Polynomial interpretation [POLO]: 6.97/2.79 6.97/2.79 POL(U11(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(U12(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U21(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 6.97/2.79 POL(U22(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(U23(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U31(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(U32(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U41(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(U42(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(U43(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U51(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(U52(x_1, x_2)) = 1 + x_1 + x_2 6.97/2.79 POL(U53(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U61(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(U62(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U71(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(U72(x_1)) = 1 + 2*x_1 6.97/2.79 POL(__(x_1, x_2)) = 1 + x_1 + x_2 6.97/2.79 POL(a) = 1 6.97/2.79 POL(a__U11(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U12(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__U21(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(a__U22(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U23(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__U31(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U32(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(a__U42(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U43(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__U51(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(a__U52(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U53(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__U61(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U62(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__U71(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__U72(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a____(x_1, x_2)) = 2 + x_1 + x_2 6.97/2.79 POL(a__and(x_1, x_2)) = 2 + x_1 + 2*x_2 6.97/2.79 POL(a__isList(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__isNeList(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__isNePal(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__isPal(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__isPalListKind(x_1)) = 2 + 2*x_1 6.97/2.79 POL(a__isQid(x_1)) = 2 + 2*x_1 6.97/2.79 POL(and(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(e) = 1 6.97/2.79 POL(i) = 1 6.97/2.79 POL(isList(x_1)) = 1 + x_1 6.97/2.79 POL(isNeList(x_1)) = 1 + x_1 6.97/2.79 POL(isNePal(x_1)) = 1 + x_1 6.97/2.79 POL(isPal(x_1)) = 1 + 2*x_1 6.97/2.79 POL(isPalListKind(x_1)) = 1 + x_1 6.97/2.79 POL(isQid(x_1)) = 1 + 2*x_1 6.97/2.79 POL(mark(x_1)) = 2*x_1 6.97/2.79 POL(nil) = 1 6.97/2.79 POL(o) = 1 6.97/2.79 POL(tt) = 1 6.97/2.79 POL(u) = 1 6.97/2.79 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 6.97/2.79 6.97/2.79 a__U12(tt) -> tt 6.97/2.79 a__U23(tt) -> tt 6.97/2.79 a__U32(tt) -> tt 6.97/2.79 a__U43(tt) -> tt 6.97/2.79 a__U53(tt) -> tt 6.97/2.79 a__U62(tt) -> tt 6.97/2.79 a__U72(tt) -> tt 6.97/2.79 a__isQid(i) -> tt 6.97/2.79 mark(nil) -> nil 6.97/2.79 mark(tt) -> tt 6.97/2.79 mark(a) -> a 6.97/2.79 mark(e) -> e 6.97/2.79 mark(i) -> i 6.97/2.79 mark(o) -> o 6.97/2.79 mark(u) -> u 6.97/2.79 a____(X1, X2) -> __(X1, X2) 6.97/2.79 a__U11(X1, X2) -> U11(X1, X2) 6.97/2.79 a__U12(X) -> U12(X) 6.97/2.79 a__isNeList(X) -> isNeList(X) 6.97/2.79 a__U21(X1, X2, X3) -> U21(X1, X2, X3) 6.97/2.79 a__U22(X1, X2) -> U22(X1, X2) 6.97/2.79 a__isList(X) -> isList(X) 6.97/2.79 a__U23(X) -> U23(X) 6.97/2.79 a__U31(X1, X2) -> U31(X1, X2) 6.97/2.79 a__U32(X) -> U32(X) 6.97/2.79 a__isQid(X) -> isQid(X) 6.97/2.79 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.97/2.79 a__U42(X1, X2) -> U42(X1, X2) 6.97/2.79 a__U43(X) -> U43(X) 6.97/2.79 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 6.97/2.79 a__U52(X1, X2) -> U52(X1, X2) 6.97/2.79 a__U53(X) -> U53(X) 6.97/2.79 a__U61(X1, X2) -> U61(X1, X2) 6.97/2.79 a__U62(X) -> U62(X) 6.97/2.79 a__U71(X1, X2) -> U71(X1, X2) 6.97/2.79 a__U72(X) -> U72(X) 6.97/2.79 a__isNePal(X) -> isNePal(X) 6.97/2.79 a__and(X1, X2) -> and(X1, X2) 6.97/2.79 a__isPalListKind(X) -> isPalListKind(X) 6.97/2.79 a__isPal(X) -> isPal(X) 6.97/2.79 6.97/2.79 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (4) 6.97/2.79 Obligation: 6.97/2.79 Q restricted rewrite system: 6.97/2.79 The TRS R consists of the following rules: 6.97/2.79 6.97/2.79 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 6.97/2.79 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.97/2.79 mark(U12(X)) -> a__U12(mark(X)) 6.97/2.79 mark(isNeList(X)) -> a__isNeList(X) 6.97/2.79 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 6.97/2.79 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 6.97/2.79 mark(isList(X)) -> a__isList(X) 6.97/2.79 mark(U23(X)) -> a__U23(mark(X)) 6.97/2.79 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.97/2.79 mark(U32(X)) -> a__U32(mark(X)) 6.97/2.79 mark(isQid(X)) -> a__isQid(X) 6.97/2.79 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.97/2.79 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 6.97/2.79 mark(U43(X)) -> a__U43(mark(X)) 6.97/2.79 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 6.97/2.79 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 6.97/2.79 mark(U53(X)) -> a__U53(mark(X)) 6.97/2.79 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 6.97/2.79 mark(U62(X)) -> a__U62(mark(X)) 6.97/2.79 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 6.97/2.79 mark(U72(X)) -> a__U72(mark(X)) 6.97/2.79 mark(isNePal(X)) -> a__isNePal(X) 6.97/2.79 mark(and(X1, X2)) -> a__and(mark(X1), X2) 6.97/2.79 mark(isPalListKind(X)) -> a__isPalListKind(X) 6.97/2.79 mark(isPal(X)) -> a__isPal(X) 6.97/2.79 6.97/2.79 The set Q consists of the following terms: 6.97/2.79 6.97/2.79 a__isList(x0) 6.97/2.79 a__isNeList(x0) 6.97/2.79 a__isNePal(x0) 6.97/2.79 a__isPal(x0) 6.97/2.79 mark(__(x0, x1)) 6.97/2.79 mark(U11(x0, x1)) 6.97/2.79 mark(U12(x0)) 6.97/2.79 mark(isNeList(x0)) 6.97/2.79 mark(U21(x0, x1, x2)) 6.97/2.79 mark(U22(x0, x1)) 6.97/2.79 mark(isList(x0)) 6.97/2.79 mark(U23(x0)) 6.97/2.79 mark(U31(x0, x1)) 6.97/2.79 mark(U32(x0)) 6.97/2.79 mark(isQid(x0)) 6.97/2.79 mark(U41(x0, x1, x2)) 6.97/2.79 mark(U42(x0, x1)) 6.97/2.79 mark(U43(x0)) 6.97/2.79 mark(U51(x0, x1, x2)) 6.97/2.79 mark(U52(x0, x1)) 6.97/2.79 mark(U53(x0)) 6.97/2.79 mark(U61(x0, x1)) 6.97/2.79 mark(U62(x0)) 6.97/2.79 mark(U71(x0, x1)) 6.97/2.79 mark(U72(x0)) 6.97/2.79 mark(isNePal(x0)) 6.97/2.79 mark(and(x0, x1)) 6.97/2.79 mark(isPalListKind(x0)) 6.97/2.79 mark(isPal(x0)) 6.97/2.79 mark(nil) 6.97/2.79 mark(tt) 6.97/2.79 mark(a) 6.97/2.79 mark(e) 6.97/2.79 mark(i) 6.97/2.79 mark(o) 6.97/2.79 mark(u) 6.97/2.79 a____(x0, x1) 6.97/2.79 a__U11(x0, x1) 6.97/2.79 a__U12(x0) 6.97/2.79 a__U21(x0, x1, x2) 6.97/2.79 a__U22(x0, x1) 6.97/2.79 a__U23(x0) 6.97/2.79 a__U31(x0, x1) 6.97/2.79 a__U32(x0) 6.97/2.79 a__isQid(x0) 6.97/2.79 a__U41(x0, x1, x2) 6.97/2.79 a__U42(x0, x1) 6.97/2.79 a__U43(x0) 6.97/2.79 a__U51(x0, x1, x2) 6.97/2.79 a__U52(x0, x1) 6.97/2.79 a__U53(x0) 6.97/2.79 a__U61(x0, x1) 6.97/2.79 a__U62(x0) 6.97/2.79 a__U71(x0, x1) 6.97/2.79 a__U72(x0) 6.97/2.79 a__and(x0, x1) 6.97/2.79 a__isPalListKind(x0) 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (5) QTRSRRRProof (EQUIVALENT) 6.97/2.79 Used ordering: 6.97/2.79 Polynomial interpretation [POLO]: 6.97/2.79 6.97/2.79 POL(U11(x_1, x_2)) = 2 + 2*x_1 + x_2 6.97/2.79 POL(U12(x_1)) = 2 + 2*x_1 6.97/2.79 POL(U21(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(U22(x_1, x_2)) = 1 + 2*x_1 + x_2 6.97/2.79 POL(U23(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U31(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 6.97/2.79 POL(U32(x_1)) = 2 + 2*x_1 6.97/2.79 POL(U41(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(U42(x_1, x_2)) = 2 + 2*x_1 + x_2 6.97/2.79 POL(U43(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U51(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(U52(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 6.97/2.79 POL(U53(x_1)) = 2 + 2*x_1 6.97/2.79 POL(U61(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 6.97/2.79 POL(U62(x_1)) = 1 + 2*x_1 6.97/2.79 POL(U71(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(U72(x_1)) = 2 + 2*x_1 6.97/2.79 POL(__(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__U11(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__U12(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a__U21(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(a__U22(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(a__U23(x_1)) = 1 + x_1 6.97/2.79 POL(a__U31(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__U32(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a__U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(a__U42(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__U43(x_1)) = 1 + x_1 6.97/2.79 POL(a__U51(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 6.97/2.79 POL(a__U52(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__U53(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a__U61(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__U62(x_1)) = 1 + x_1 6.97/2.79 POL(a__U71(x_1, x_2)) = 1 + x_1 + 2*x_2 6.97/2.79 POL(a__U72(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a____(x_1, x_2)) = 1 + x_1 + x_2 6.97/2.79 POL(a__and(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 6.97/2.79 POL(a__isList(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a__isNeList(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a__isNePal(x_1)) = 2*x_1 6.97/2.79 POL(a__isPal(x_1)) = 1 + 2*x_1 6.97/2.79 POL(a__isPalListKind(x_1)) = 2*x_1 6.97/2.79 POL(a__isQid(x_1)) = 1 + 2*x_1 6.97/2.79 POL(and(x_1, x_2)) = 2 + 2*x_1 + x_2 6.97/2.79 POL(isList(x_1)) = 2*x_1 6.97/2.79 POL(isNeList(x_1)) = 2*x_1 6.97/2.79 POL(isNePal(x_1)) = 1 + 2*x_1 6.97/2.79 POL(isPal(x_1)) = x_1 6.97/2.79 POL(isPalListKind(x_1)) = 1 + 2*x_1 6.97/2.79 POL(isQid(x_1)) = x_1 6.97/2.79 POL(mark(x_1)) = 2 + 2*x_1 6.97/2.79 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 6.97/2.79 6.97/2.79 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 6.97/2.79 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.97/2.79 mark(U12(X)) -> a__U12(mark(X)) 6.97/2.79 mark(isNeList(X)) -> a__isNeList(X) 6.97/2.79 mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) 6.97/2.79 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 6.97/2.79 mark(isList(X)) -> a__isList(X) 6.97/2.79 mark(U23(X)) -> a__U23(mark(X)) 6.97/2.79 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.97/2.79 mark(U32(X)) -> a__U32(mark(X)) 6.97/2.79 mark(isQid(X)) -> a__isQid(X) 6.97/2.79 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.97/2.79 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 6.97/2.79 mark(U43(X)) -> a__U43(mark(X)) 6.97/2.79 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 6.97/2.79 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 6.97/2.79 mark(U53(X)) -> a__U53(mark(X)) 6.97/2.79 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 6.97/2.79 mark(U62(X)) -> a__U62(mark(X)) 6.97/2.79 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 6.97/2.79 mark(U72(X)) -> a__U72(mark(X)) 6.97/2.79 mark(isNePal(X)) -> a__isNePal(X) 6.97/2.79 mark(and(X1, X2)) -> a__and(mark(X1), X2) 6.97/2.79 mark(isPalListKind(X)) -> a__isPalListKind(X) 6.97/2.79 mark(isPal(X)) -> a__isPal(X) 6.97/2.79 6.97/2.79 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (6) 6.97/2.79 Obligation: 6.97/2.79 Q restricted rewrite system: 6.97/2.79 R is empty. 6.97/2.79 The set Q consists of the following terms: 6.97/2.79 6.97/2.79 a__isList(x0) 6.97/2.79 a__isNeList(x0) 6.97/2.79 a__isNePal(x0) 6.97/2.79 a__isPal(x0) 6.97/2.79 mark(__(x0, x1)) 6.97/2.79 mark(U11(x0, x1)) 6.97/2.79 mark(U12(x0)) 6.97/2.79 mark(isNeList(x0)) 6.97/2.79 mark(U21(x0, x1, x2)) 6.97/2.79 mark(U22(x0, x1)) 6.97/2.79 mark(isList(x0)) 6.97/2.79 mark(U23(x0)) 6.97/2.79 mark(U31(x0, x1)) 6.97/2.79 mark(U32(x0)) 6.97/2.79 mark(isQid(x0)) 6.97/2.79 mark(U41(x0, x1, x2)) 6.97/2.79 mark(U42(x0, x1)) 6.97/2.79 mark(U43(x0)) 6.97/2.79 mark(U51(x0, x1, x2)) 6.97/2.79 mark(U52(x0, x1)) 6.97/2.79 mark(U53(x0)) 6.97/2.79 mark(U61(x0, x1)) 6.97/2.79 mark(U62(x0)) 6.97/2.79 mark(U71(x0, x1)) 6.97/2.79 mark(U72(x0)) 6.97/2.79 mark(isNePal(x0)) 6.97/2.79 mark(and(x0, x1)) 6.97/2.79 mark(isPalListKind(x0)) 6.97/2.79 mark(isPal(x0)) 6.97/2.79 mark(nil) 6.97/2.79 mark(tt) 6.97/2.79 mark(a) 6.97/2.79 mark(e) 6.97/2.79 mark(i) 6.97/2.79 mark(o) 6.97/2.79 mark(u) 6.97/2.79 a____(x0, x1) 6.97/2.79 a__U11(x0, x1) 6.97/2.79 a__U12(x0) 6.97/2.79 a__U21(x0, x1, x2) 6.97/2.79 a__U22(x0, x1) 6.97/2.79 a__U23(x0) 6.97/2.79 a__U31(x0, x1) 6.97/2.79 a__U32(x0) 6.97/2.79 a__isQid(x0) 6.97/2.79 a__U41(x0, x1, x2) 6.97/2.79 a__U42(x0, x1) 6.97/2.79 a__U43(x0) 6.97/2.79 a__U51(x0, x1, x2) 6.97/2.79 a__U52(x0, x1) 6.97/2.79 a__U53(x0) 6.97/2.79 a__U61(x0, x1) 6.97/2.79 a__U62(x0) 6.97/2.79 a__U71(x0, x1) 6.97/2.79 a__U72(x0) 6.97/2.79 a__and(x0, x1) 6.97/2.79 a__isPalListKind(x0) 6.97/2.79 6.97/2.79 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (7) RisEmptyProof (EQUIVALENT) 6.97/2.79 The TRS R is empty. Hence, termination is trivially proven. 6.97/2.79 ---------------------------------------- 6.97/2.79 6.97/2.79 (8) 6.97/2.79 YES 7.22/2.84 EOF