5.14/2.05 YES 5.19/2.07 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.19/2.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.19/2.07 5.19/2.07 5.19/2.07 Termination w.r.t. Q of the given QTRS could be proven: 5.19/2.07 5.19/2.07 (0) QTRS 5.19/2.07 (1) QTRSRRRProof [EQUIVALENT, 321 ms] 5.19/2.07 (2) QTRS 5.19/2.07 (3) RisEmptyProof [EQUIVALENT, 0 ms] 5.19/2.07 (4) YES 5.19/2.07 5.19/2.07 5.19/2.07 ---------------------------------------- 5.19/2.07 5.19/2.07 (0) 5.19/2.07 Obligation: 5.19/2.07 Q restricted rewrite system: 5.19/2.07 The TRS R consists of the following rules: 5.19/2.07 5.19/2.07 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 5.19/2.07 a____(X, nil) -> mark(X) 5.19/2.07 a____(nil, X) -> mark(X) 5.19/2.07 a__U11(tt) -> tt 5.19/2.07 a__U21(tt, V2) -> a__U22(a__isList(V2)) 5.19/2.07 a__U22(tt) -> tt 5.19/2.07 a__U31(tt) -> tt 5.19/2.07 a__U41(tt, V2) -> a__U42(a__isNeList(V2)) 5.19/2.07 a__U42(tt) -> tt 5.19/2.07 a__U51(tt, V2) -> a__U52(a__isList(V2)) 5.19/2.07 a__U52(tt) -> tt 5.19/2.07 a__U61(tt) -> tt 5.19/2.07 a__U71(tt, P) -> a__U72(a__isPal(P)) 5.19/2.07 a__U72(tt) -> tt 5.19/2.07 a__U81(tt) -> tt 5.19/2.07 a__isList(V) -> a__U11(a__isNeList(V)) 5.19/2.07 a__isList(nil) -> tt 5.19/2.07 a__isList(__(V1, V2)) -> a__U21(a__isList(V1), V2) 5.19/2.07 a__isNeList(V) -> a__U31(a__isQid(V)) 5.19/2.07 a__isNeList(__(V1, V2)) -> a__U41(a__isList(V1), V2) 5.19/2.07 a__isNeList(__(V1, V2)) -> a__U51(a__isNeList(V1), V2) 5.19/2.07 a__isNePal(V) -> a__U61(a__isQid(V)) 5.19/2.07 a__isNePal(__(I, __(P, I))) -> a__U71(a__isQid(I), P) 5.19/2.07 a__isPal(V) -> a__U81(a__isNePal(V)) 5.19/2.07 a__isPal(nil) -> tt 5.19/2.07 a__isQid(a) -> tt 5.19/2.07 a__isQid(e) -> tt 5.19/2.07 a__isQid(i) -> tt 5.19/2.07 a__isQid(o) -> tt 5.19/2.07 a__isQid(u) -> tt 5.19/2.07 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 5.19/2.07 mark(U11(X)) -> a__U11(mark(X)) 5.19/2.07 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 5.19/2.07 mark(U22(X)) -> a__U22(mark(X)) 5.19/2.07 mark(isList(X)) -> a__isList(X) 5.19/2.07 mark(U31(X)) -> a__U31(mark(X)) 5.19/2.07 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 5.19/2.07 mark(U42(X)) -> a__U42(mark(X)) 5.19/2.07 mark(isNeList(X)) -> a__isNeList(X) 5.19/2.07 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 5.19/2.07 mark(U52(X)) -> a__U52(mark(X)) 5.19/2.07 mark(U61(X)) -> a__U61(mark(X)) 5.19/2.07 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 5.19/2.07 mark(U72(X)) -> a__U72(mark(X)) 5.19/2.07 mark(isPal(X)) -> a__isPal(X) 5.19/2.07 mark(U81(X)) -> a__U81(mark(X)) 5.19/2.07 mark(isQid(X)) -> a__isQid(X) 5.19/2.07 mark(isNePal(X)) -> a__isNePal(X) 5.19/2.07 mark(nil) -> nil 5.19/2.07 mark(tt) -> tt 5.19/2.07 mark(a) -> a 5.19/2.07 mark(e) -> e 5.19/2.07 mark(i) -> i 5.19/2.07 mark(o) -> o 5.19/2.07 mark(u) -> u 5.19/2.07 a____(X1, X2) -> __(X1, X2) 5.19/2.07 a__U11(X) -> U11(X) 5.19/2.07 a__U21(X1, X2) -> U21(X1, X2) 5.19/2.07 a__U22(X) -> U22(X) 5.19/2.07 a__isList(X) -> isList(X) 5.19/2.07 a__U31(X) -> U31(X) 5.19/2.07 a__U41(X1, X2) -> U41(X1, X2) 5.19/2.07 a__U42(X) -> U42(X) 5.19/2.07 a__isNeList(X) -> isNeList(X) 5.19/2.07 a__U51(X1, X2) -> U51(X1, X2) 5.19/2.07 a__U52(X) -> U52(X) 5.19/2.07 a__U61(X) -> U61(X) 5.19/2.07 a__U71(X1, X2) -> U71(X1, X2) 5.19/2.07 a__U72(X) -> U72(X) 5.19/2.07 a__isPal(X) -> isPal(X) 5.19/2.07 a__U81(X) -> U81(X) 5.19/2.07 a__isQid(X) -> isQid(X) 5.19/2.07 a__isNePal(X) -> isNePal(X) 5.19/2.07 5.19/2.07 The set Q consists of the following terms: 5.19/2.07 5.19/2.07 a__isList(x0) 5.19/2.07 a__isNeList(x0) 5.19/2.07 a__isNePal(x0) 5.19/2.07 a__isPal(x0) 5.19/2.07 mark(__(x0, x1)) 5.19/2.07 mark(U11(x0)) 5.19/2.07 mark(U21(x0, x1)) 5.19/2.07 mark(U22(x0)) 5.19/2.07 mark(isList(x0)) 5.19/2.07 mark(U31(x0)) 5.19/2.07 mark(U41(x0, x1)) 5.19/2.07 mark(U42(x0)) 5.19/2.07 mark(isNeList(x0)) 5.19/2.07 mark(U51(x0, x1)) 5.19/2.07 mark(U52(x0)) 5.19/2.07 mark(U61(x0)) 5.19/2.07 mark(U71(x0, x1)) 5.19/2.07 mark(U72(x0)) 5.19/2.07 mark(isPal(x0)) 5.19/2.07 mark(U81(x0)) 5.19/2.07 mark(isQid(x0)) 5.19/2.07 mark(isNePal(x0)) 5.19/2.07 mark(nil) 5.19/2.07 mark(tt) 5.19/2.07 mark(a) 5.19/2.07 mark(e) 5.19/2.07 mark(i) 5.19/2.07 mark(o) 5.19/2.07 mark(u) 5.19/2.07 a____(x0, x1) 5.19/2.07 a__U11(x0) 5.19/2.07 a__U21(x0, x1) 5.19/2.07 a__U22(x0) 5.19/2.07 a__U31(x0) 5.19/2.07 a__U41(x0, x1) 5.19/2.07 a__U42(x0) 5.19/2.07 a__U51(x0, x1) 5.19/2.07 a__U52(x0) 5.19/2.07 a__U61(x0) 5.19/2.07 a__U71(x0, x1) 5.19/2.07 a__U72(x0) 5.19/2.07 a__U81(x0) 5.19/2.07 a__isQid(x0) 5.19/2.07 5.19/2.07 5.19/2.07 ---------------------------------------- 5.19/2.07 5.19/2.07 (1) QTRSRRRProof (EQUIVALENT) 5.19/2.07 Used ordering: 5.19/2.07 Knuth-Bendix order [KBO] with precedence:mark_1 > tt > a__isPal_1 > a__U71_2 > a__U81_1 > a__U41_2 > U81_1 > i > a > a__isNeList_1 > o > a__U31_1 > nil > U31_1 > a__U11_1 > a__isQid_1 > a_____2 > e > U11_1 > ___2 > a__U42_1 > U42_1 > isQid_1 > U71_2 > a__U51_2 > isNeList_1 > a__U61_1 > U51_2 > a__U21_2 > a__U22_1 > a__U72_1 > U22_1 > U72_1 > isPal_1 > u > U21_2 > a__U52_1 > U52_1 > U61_1 > a__isNePal_1 > isNePal_1 > U41_2 > a__isList_1 > isList_1 5.19/2.07 5.19/2.07 and weight map: 5.19/2.07 5.19/2.07 nil=1 5.19/2.07 tt=12 5.19/2.07 a=4 5.19/2.07 e=4 5.19/2.07 i=4 5.19/2.07 o=4 5.19/2.07 u=4 5.19/2.07 mark_1=0 5.19/2.07 a__U11_1=1 5.19/2.07 a__U22_1=1 5.19/2.07 a__isList_1=12 5.19/2.07 a__U31_1=1 5.19/2.07 a__U42_1=2 5.19/2.07 a__isNeList_1=10 5.19/2.07 a__U52_1=1 5.19/2.07 a__U61_1=1 5.19/2.07 a__U72_1=1 5.19/2.07 a__isPal_1=12 5.19/2.07 a__U81_1=1 5.19/2.07 a__isQid_1=9 5.19/2.07 a__isNePal_1=11 5.19/2.07 U11_1=1 5.19/2.07 U22_1=1 5.19/2.07 isList_1=12 5.19/2.07 U31_1=1 5.19/2.07 U42_1=2 5.19/2.07 isNeList_1=10 5.19/2.07 U52_1=1 5.19/2.07 U61_1=1 5.19/2.07 U72_1=1 5.19/2.07 isPal_1=12 5.19/2.07 U81_1=1 5.19/2.07 isQid_1=9 5.19/2.07 isNePal_1=11 5.19/2.07 ___2=3 5.19/2.07 a_____2=3 5.19/2.07 a__U21_2=2 5.19/2.07 a__U41_2=0 5.19/2.07 a__U51_2=1 5.19/2.07 a__U71_2=8 5.19/2.07 U21_2=2 5.19/2.07 U41_2=0 5.19/2.07 U51_2=1 5.19/2.07 U71_2=8 5.19/2.07 5.19/2.07 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 5.19/2.07 5.19/2.07 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 5.19/2.07 a____(X, nil) -> mark(X) 5.19/2.07 a____(nil, X) -> mark(X) 5.19/2.07 a__U11(tt) -> tt 5.19/2.07 a__U21(tt, V2) -> a__U22(a__isList(V2)) 5.19/2.07 a__U22(tt) -> tt 5.19/2.07 a__U31(tt) -> tt 5.19/2.07 a__U41(tt, V2) -> a__U42(a__isNeList(V2)) 5.19/2.07 a__U42(tt) -> tt 5.19/2.07 a__U51(tt, V2) -> a__U52(a__isList(V2)) 5.19/2.07 a__U52(tt) -> tt 5.19/2.07 a__U61(tt) -> tt 5.19/2.07 a__U71(tt, P) -> a__U72(a__isPal(P)) 5.19/2.07 a__U72(tt) -> tt 5.19/2.07 a__U81(tt) -> tt 5.19/2.07 a__isList(V) -> a__U11(a__isNeList(V)) 5.19/2.07 a__isList(nil) -> tt 5.19/2.07 a__isList(__(V1, V2)) -> a__U21(a__isList(V1), V2) 5.19/2.07 a__isNeList(V) -> a__U31(a__isQid(V)) 5.19/2.07 a__isNeList(__(V1, V2)) -> a__U41(a__isList(V1), V2) 5.19/2.07 a__isNeList(__(V1, V2)) -> a__U51(a__isNeList(V1), V2) 5.19/2.07 a__isNePal(V) -> a__U61(a__isQid(V)) 5.19/2.07 a__isNePal(__(I, __(P, I))) -> a__U71(a__isQid(I), P) 5.19/2.07 a__isPal(V) -> a__U81(a__isNePal(V)) 5.19/2.07 a__isPal(nil) -> tt 5.19/2.07 a__isQid(a) -> tt 5.19/2.07 a__isQid(e) -> tt 5.19/2.07 a__isQid(i) -> tt 5.19/2.07 a__isQid(o) -> tt 5.19/2.07 a__isQid(u) -> tt 5.19/2.07 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 5.19/2.07 mark(U11(X)) -> a__U11(mark(X)) 5.19/2.07 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 5.19/2.07 mark(U22(X)) -> a__U22(mark(X)) 5.19/2.07 mark(isList(X)) -> a__isList(X) 5.19/2.07 mark(U31(X)) -> a__U31(mark(X)) 5.19/2.07 mark(U41(X1, X2)) -> a__U41(mark(X1), X2) 5.19/2.07 mark(U42(X)) -> a__U42(mark(X)) 5.19/2.07 mark(isNeList(X)) -> a__isNeList(X) 5.19/2.07 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 5.19/2.07 mark(U52(X)) -> a__U52(mark(X)) 5.19/2.07 mark(U61(X)) -> a__U61(mark(X)) 5.19/2.07 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 5.19/2.07 mark(U72(X)) -> a__U72(mark(X)) 5.19/2.07 mark(isPal(X)) -> a__isPal(X) 5.19/2.07 mark(U81(X)) -> a__U81(mark(X)) 5.19/2.07 mark(isQid(X)) -> a__isQid(X) 5.19/2.07 mark(isNePal(X)) -> a__isNePal(X) 5.19/2.07 mark(nil) -> nil 5.19/2.07 mark(tt) -> tt 5.19/2.07 mark(a) -> a 5.19/2.07 mark(e) -> e 5.19/2.07 mark(i) -> i 5.19/2.07 mark(o) -> o 5.19/2.07 mark(u) -> u 5.19/2.07 a____(X1, X2) -> __(X1, X2) 5.19/2.07 a__U11(X) -> U11(X) 5.19/2.07 a__U21(X1, X2) -> U21(X1, X2) 5.19/2.07 a__U22(X) -> U22(X) 5.19/2.07 a__isList(X) -> isList(X) 5.19/2.07 a__U31(X) -> U31(X) 5.19/2.07 a__U41(X1, X2) -> U41(X1, X2) 5.19/2.07 a__U42(X) -> U42(X) 5.19/2.07 a__isNeList(X) -> isNeList(X) 5.19/2.07 a__U51(X1, X2) -> U51(X1, X2) 5.19/2.07 a__U52(X) -> U52(X) 5.19/2.07 a__U61(X) -> U61(X) 5.19/2.07 a__U71(X1, X2) -> U71(X1, X2) 5.19/2.07 a__U72(X) -> U72(X) 5.19/2.07 a__isPal(X) -> isPal(X) 5.19/2.07 a__U81(X) -> U81(X) 5.19/2.07 a__isQid(X) -> isQid(X) 5.19/2.07 a__isNePal(X) -> isNePal(X) 5.19/2.07 5.19/2.07 5.19/2.07 5.19/2.07 5.19/2.07 ---------------------------------------- 5.19/2.07 5.19/2.07 (2) 5.19/2.07 Obligation: 5.19/2.07 Q restricted rewrite system: 5.19/2.07 R is empty. 5.19/2.07 The set Q consists of the following terms: 5.19/2.07 5.19/2.07 a__isList(x0) 5.19/2.07 a__isNeList(x0) 5.19/2.07 a__isNePal(x0) 5.19/2.07 a__isPal(x0) 5.19/2.07 mark(__(x0, x1)) 5.19/2.07 mark(U11(x0)) 5.19/2.07 mark(U21(x0, x1)) 5.19/2.07 mark(U22(x0)) 5.19/2.07 mark(isList(x0)) 5.19/2.07 mark(U31(x0)) 5.19/2.07 mark(U41(x0, x1)) 5.19/2.07 mark(U42(x0)) 5.19/2.07 mark(isNeList(x0)) 5.19/2.07 mark(U51(x0, x1)) 5.19/2.07 mark(U52(x0)) 5.19/2.07 mark(U61(x0)) 5.19/2.07 mark(U71(x0, x1)) 5.19/2.07 mark(U72(x0)) 5.19/2.07 mark(isPal(x0)) 5.19/2.07 mark(U81(x0)) 5.19/2.07 mark(isQid(x0)) 5.19/2.07 mark(isNePal(x0)) 5.19/2.07 mark(nil) 5.19/2.07 mark(tt) 5.19/2.07 mark(a) 5.19/2.07 mark(e) 5.19/2.07 mark(i) 5.19/2.07 mark(o) 5.19/2.07 mark(u) 5.19/2.07 a____(x0, x1) 5.19/2.07 a__U11(x0) 5.19/2.07 a__U21(x0, x1) 5.19/2.07 a__U22(x0) 5.19/2.07 a__U31(x0) 5.19/2.07 a__U41(x0, x1) 5.19/2.07 a__U42(x0) 5.19/2.07 a__U51(x0, x1) 5.19/2.07 a__U52(x0) 5.19/2.07 a__U61(x0) 5.19/2.07 a__U71(x0, x1) 5.19/2.07 a__U72(x0) 5.19/2.07 a__U81(x0) 5.19/2.07 a__isQid(x0) 5.19/2.07 5.19/2.07 5.19/2.07 ---------------------------------------- 5.19/2.07 5.19/2.07 (3) RisEmptyProof (EQUIVALENT) 5.19/2.07 The TRS R is empty. Hence, termination is trivially proven. 5.19/2.07 ---------------------------------------- 5.19/2.07 5.19/2.07 (4) 5.19/2.07 YES 5.19/2.10 EOF