332.90/291.50 WORST_CASE(Omega(n^1), ?) 332.90/291.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 332.90/291.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 332.90/291.51 332.90/291.51 332.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 332.90/291.51 332.90/291.51 (0) CpxTRS 332.90/291.51 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 332.90/291.51 (2) TRS for Loop Detection 332.90/291.51 (3) DecreasingLoopProof [LOWER BOUND(ID), 150 ms] 332.90/291.51 (4) BEST 332.90/291.51 (5) proven lower bound 332.90/291.51 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 332.90/291.51 (7) BOUNDS(n^1, INF) 332.90/291.51 (8) TRS for Loop Detection 332.90/291.51 332.90/291.51 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (0) 332.90/291.51 Obligation: 332.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 332.90/291.51 332.90/291.51 332.90/291.51 The TRS R consists of the following rules: 332.90/291.51 332.90/291.51 __(__(X, Y), Z) -> __(X, __(Y, Z)) 332.90/291.51 __(X, nil) -> X 332.90/291.51 __(nil, X) -> X 332.90/291.51 U11(tt, V) -> U12(isNeList(activate(V))) 332.90/291.51 U12(tt) -> tt 332.90/291.51 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 332.90/291.51 U22(tt, V2) -> U23(isList(activate(V2))) 332.90/291.51 U23(tt) -> tt 332.90/291.51 U31(tt, V) -> U32(isQid(activate(V))) 332.90/291.51 U32(tt) -> tt 332.90/291.51 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 332.90/291.51 U42(tt, V2) -> U43(isNeList(activate(V2))) 332.90/291.51 U43(tt) -> tt 332.90/291.51 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 332.90/291.51 U52(tt, V2) -> U53(isList(activate(V2))) 332.90/291.51 U53(tt) -> tt 332.90/291.51 U61(tt, V) -> U62(isQid(activate(V))) 332.90/291.51 U62(tt) -> tt 332.90/291.51 U71(tt, V) -> U72(isNePal(activate(V))) 332.90/291.51 U72(tt) -> tt 332.90/291.51 and(tt, X) -> activate(X) 332.90/291.51 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isList(n__nil) -> tt 332.90/291.51 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNePal(n____(I, __(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P)))) 332.90/291.51 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isPal(n__nil) -> tt 332.90/291.51 isPalListKind(n__a) -> tt 332.90/291.51 isPalListKind(n__e) -> tt 332.90/291.51 isPalListKind(n__i) -> tt 332.90/291.51 isPalListKind(n__nil) -> tt 332.90/291.51 isPalListKind(n__o) -> tt 332.90/291.51 isPalListKind(n__u) -> tt 332.90/291.51 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 332.90/291.51 isQid(n__a) -> tt 332.90/291.51 isQid(n__e) -> tt 332.90/291.51 isQid(n__i) -> tt 332.90/291.51 isQid(n__o) -> tt 332.90/291.51 isQid(n__u) -> tt 332.90/291.51 nil -> n__nil 332.90/291.51 __(X1, X2) -> n____(X1, X2) 332.90/291.51 isPalListKind(X) -> n__isPalListKind(X) 332.90/291.51 and(X1, X2) -> n__and(X1, X2) 332.90/291.51 a -> n__a 332.90/291.51 e -> n__e 332.90/291.51 i -> n__i 332.90/291.51 o -> n__o 332.90/291.51 u -> n__u 332.90/291.51 activate(n__nil) -> nil 332.90/291.51 activate(n____(X1, X2)) -> __(X1, X2) 332.90/291.51 activate(n__isPalListKind(X)) -> isPalListKind(X) 332.90/291.51 activate(n__and(X1, X2)) -> and(X1, X2) 332.90/291.51 activate(n__a) -> a 332.90/291.51 activate(n__e) -> e 332.90/291.51 activate(n__i) -> i 332.90/291.51 activate(n__o) -> o 332.90/291.51 activate(n__u) -> u 332.90/291.51 activate(X) -> X 332.90/291.51 332.90/291.51 S is empty. 332.90/291.51 Rewrite Strategy: FULL 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 332.90/291.51 Transformed a relative TRS into a decreasing-loop problem. 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (2) 332.90/291.51 Obligation: 332.90/291.51 Analyzing the following TRS for decreasing loops: 332.90/291.51 332.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 332.90/291.51 332.90/291.51 332.90/291.51 The TRS R consists of the following rules: 332.90/291.51 332.90/291.51 __(__(X, Y), Z) -> __(X, __(Y, Z)) 332.90/291.51 __(X, nil) -> X 332.90/291.51 __(nil, X) -> X 332.90/291.51 U11(tt, V) -> U12(isNeList(activate(V))) 332.90/291.51 U12(tt) -> tt 332.90/291.51 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 332.90/291.51 U22(tt, V2) -> U23(isList(activate(V2))) 332.90/291.51 U23(tt) -> tt 332.90/291.51 U31(tt, V) -> U32(isQid(activate(V))) 332.90/291.51 U32(tt) -> tt 332.90/291.51 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 332.90/291.51 U42(tt, V2) -> U43(isNeList(activate(V2))) 332.90/291.51 U43(tt) -> tt 332.90/291.51 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 332.90/291.51 U52(tt, V2) -> U53(isList(activate(V2))) 332.90/291.51 U53(tt) -> tt 332.90/291.51 U61(tt, V) -> U62(isQid(activate(V))) 332.90/291.51 U62(tt) -> tt 332.90/291.51 U71(tt, V) -> U72(isNePal(activate(V))) 332.90/291.51 U72(tt) -> tt 332.90/291.51 and(tt, X) -> activate(X) 332.90/291.51 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isList(n__nil) -> tt 332.90/291.51 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNePal(n____(I, __(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P)))) 332.90/291.51 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isPal(n__nil) -> tt 332.90/291.51 isPalListKind(n__a) -> tt 332.90/291.51 isPalListKind(n__e) -> tt 332.90/291.51 isPalListKind(n__i) -> tt 332.90/291.51 isPalListKind(n__nil) -> tt 332.90/291.51 isPalListKind(n__o) -> tt 332.90/291.51 isPalListKind(n__u) -> tt 332.90/291.51 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 332.90/291.51 isQid(n__a) -> tt 332.90/291.51 isQid(n__e) -> tt 332.90/291.51 isQid(n__i) -> tt 332.90/291.51 isQid(n__o) -> tt 332.90/291.51 isQid(n__u) -> tt 332.90/291.51 nil -> n__nil 332.90/291.51 __(X1, X2) -> n____(X1, X2) 332.90/291.51 isPalListKind(X) -> n__isPalListKind(X) 332.90/291.51 and(X1, X2) -> n__and(X1, X2) 332.90/291.51 a -> n__a 332.90/291.51 e -> n__e 332.90/291.51 i -> n__i 332.90/291.51 o -> n__o 332.90/291.51 u -> n__u 332.90/291.51 activate(n__nil) -> nil 332.90/291.51 activate(n____(X1, X2)) -> __(X1, X2) 332.90/291.51 activate(n__isPalListKind(X)) -> isPalListKind(X) 332.90/291.51 activate(n__and(X1, X2)) -> and(X1, X2) 332.90/291.51 activate(n__a) -> a 332.90/291.51 activate(n__e) -> e 332.90/291.51 activate(n__i) -> i 332.90/291.51 activate(n__o) -> o 332.90/291.51 activate(n__u) -> u 332.90/291.51 activate(X) -> X 332.90/291.51 332.90/291.51 S is empty. 332.90/291.51 Rewrite Strategy: FULL 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (3) DecreasingLoopProof (LOWER BOUND(ID)) 332.90/291.51 The following loop(s) give(s) rise to the lower bound Omega(n^1): 332.90/291.51 332.90/291.51 The rewrite sequence 332.90/291.51 332.90/291.51 isPalListKind(n____(V1, V2)) ->^+ and(isPalListKind(V1), n__isPalListKind(activate(V2))) 332.90/291.51 332.90/291.51 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 332.90/291.51 332.90/291.51 The pumping substitution is [V1 / n____(V1, V2)]. 332.90/291.51 332.90/291.51 The result substitution is [ ]. 332.90/291.51 332.90/291.51 332.90/291.51 332.90/291.51 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (4) 332.90/291.51 Complex Obligation (BEST) 332.90/291.51 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (5) 332.90/291.51 Obligation: 332.90/291.51 Proved the lower bound n^1 for the following obligation: 332.90/291.51 332.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 332.90/291.51 332.90/291.51 332.90/291.51 The TRS R consists of the following rules: 332.90/291.51 332.90/291.51 __(__(X, Y), Z) -> __(X, __(Y, Z)) 332.90/291.51 __(X, nil) -> X 332.90/291.51 __(nil, X) -> X 332.90/291.51 U11(tt, V) -> U12(isNeList(activate(V))) 332.90/291.51 U12(tt) -> tt 332.90/291.51 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 332.90/291.51 U22(tt, V2) -> U23(isList(activate(V2))) 332.90/291.51 U23(tt) -> tt 332.90/291.51 U31(tt, V) -> U32(isQid(activate(V))) 332.90/291.51 U32(tt) -> tt 332.90/291.51 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 332.90/291.51 U42(tt, V2) -> U43(isNeList(activate(V2))) 332.90/291.51 U43(tt) -> tt 332.90/291.51 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 332.90/291.51 U52(tt, V2) -> U53(isList(activate(V2))) 332.90/291.51 U53(tt) -> tt 332.90/291.51 U61(tt, V) -> U62(isQid(activate(V))) 332.90/291.51 U62(tt) -> tt 332.90/291.51 U71(tt, V) -> U72(isNePal(activate(V))) 332.90/291.51 U72(tt) -> tt 332.90/291.51 and(tt, X) -> activate(X) 332.90/291.51 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isList(n__nil) -> tt 332.90/291.51 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNePal(n____(I, __(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P)))) 332.90/291.51 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isPal(n__nil) -> tt 332.90/291.51 isPalListKind(n__a) -> tt 332.90/291.51 isPalListKind(n__e) -> tt 332.90/291.51 isPalListKind(n__i) -> tt 332.90/291.51 isPalListKind(n__nil) -> tt 332.90/291.51 isPalListKind(n__o) -> tt 332.90/291.51 isPalListKind(n__u) -> tt 332.90/291.51 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 332.90/291.51 isQid(n__a) -> tt 332.90/291.51 isQid(n__e) -> tt 332.90/291.51 isQid(n__i) -> tt 332.90/291.51 isQid(n__o) -> tt 332.90/291.51 isQid(n__u) -> tt 332.90/291.51 nil -> n__nil 332.90/291.51 __(X1, X2) -> n____(X1, X2) 332.90/291.51 isPalListKind(X) -> n__isPalListKind(X) 332.90/291.51 and(X1, X2) -> n__and(X1, X2) 332.90/291.51 a -> n__a 332.90/291.51 e -> n__e 332.90/291.51 i -> n__i 332.90/291.51 o -> n__o 332.90/291.51 u -> n__u 332.90/291.51 activate(n__nil) -> nil 332.90/291.51 activate(n____(X1, X2)) -> __(X1, X2) 332.90/291.51 activate(n__isPalListKind(X)) -> isPalListKind(X) 332.90/291.51 activate(n__and(X1, X2)) -> and(X1, X2) 332.90/291.51 activate(n__a) -> a 332.90/291.51 activate(n__e) -> e 332.90/291.51 activate(n__i) -> i 332.90/291.51 activate(n__o) -> o 332.90/291.51 activate(n__u) -> u 332.90/291.51 activate(X) -> X 332.90/291.51 332.90/291.51 S is empty. 332.90/291.51 Rewrite Strategy: FULL 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (6) LowerBoundPropagationProof (FINISHED) 332.90/291.51 Propagated lower bound. 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (7) 332.90/291.51 BOUNDS(n^1, INF) 332.90/291.51 332.90/291.51 ---------------------------------------- 332.90/291.51 332.90/291.51 (8) 332.90/291.51 Obligation: 332.90/291.51 Analyzing the following TRS for decreasing loops: 332.90/291.51 332.90/291.51 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 332.90/291.51 332.90/291.51 332.90/291.51 The TRS R consists of the following rules: 332.90/291.51 332.90/291.51 __(__(X, Y), Z) -> __(X, __(Y, Z)) 332.90/291.51 __(X, nil) -> X 332.90/291.51 __(nil, X) -> X 332.90/291.51 U11(tt, V) -> U12(isNeList(activate(V))) 332.90/291.51 U12(tt) -> tt 332.90/291.51 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 332.90/291.51 U22(tt, V2) -> U23(isList(activate(V2))) 332.90/291.51 U23(tt) -> tt 332.90/291.51 U31(tt, V) -> U32(isQid(activate(V))) 332.90/291.51 U32(tt) -> tt 332.90/291.51 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 332.90/291.51 U42(tt, V2) -> U43(isNeList(activate(V2))) 332.90/291.51 U43(tt) -> tt 332.90/291.51 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 332.90/291.51 U52(tt, V2) -> U53(isList(activate(V2))) 332.90/291.51 U53(tt) -> tt 332.90/291.51 U61(tt, V) -> U62(isQid(activate(V))) 332.90/291.51 U62(tt) -> tt 332.90/291.51 U71(tt, V) -> U72(isNePal(activate(V))) 332.90/291.51 U72(tt) -> tt 332.90/291.51 and(tt, X) -> activate(X) 332.90/291.51 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isList(n__nil) -> tt 332.90/291.51 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 332.90/291.51 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isNePal(n____(I, __(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(isPal(activate(P)), n__isPalListKind(activate(P)))) 332.90/291.51 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 332.90/291.51 isPal(n__nil) -> tt 332.90/291.51 isPalListKind(n__a) -> tt 332.90/291.51 isPalListKind(n__e) -> tt 332.90/291.51 isPalListKind(n__i) -> tt 332.90/291.51 isPalListKind(n__nil) -> tt 332.90/291.51 isPalListKind(n__o) -> tt 332.90/291.51 isPalListKind(n__u) -> tt 332.90/291.51 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 332.90/291.51 isQid(n__a) -> tt 332.90/291.51 isQid(n__e) -> tt 332.90/291.51 isQid(n__i) -> tt 332.90/291.51 isQid(n__o) -> tt 332.90/291.51 isQid(n__u) -> tt 332.90/291.51 nil -> n__nil 332.90/291.51 __(X1, X2) -> n____(X1, X2) 332.90/291.51 isPalListKind(X) -> n__isPalListKind(X) 332.90/291.51 and(X1, X2) -> n__and(X1, X2) 332.90/291.51 a -> n__a 332.90/291.51 e -> n__e 332.90/291.51 i -> n__i 332.90/291.51 o -> n__o 332.90/291.51 u -> n__u 332.90/291.51 activate(n__nil) -> nil 332.90/291.51 activate(n____(X1, X2)) -> __(X1, X2) 332.90/291.51 activate(n__isPalListKind(X)) -> isPalListKind(X) 332.90/291.51 activate(n__and(X1, X2)) -> and(X1, X2) 332.90/291.51 activate(n__a) -> a 332.90/291.51 activate(n__e) -> e 332.90/291.51 activate(n__i) -> i 332.90/291.51 activate(n__o) -> o 332.90/291.51 activate(n__u) -> u 332.90/291.51 activate(X) -> X 332.90/291.51 332.90/291.51 S is empty. 332.90/291.51 Rewrite Strategy: FULL 333.00/291.54 EOF