4.98/2.00 WORST_CASE(NON_POLY, ?) 4.98/2.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.98/2.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.98/2.02 4.98/2.02 4.98/2.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.98/2.02 4.98/2.02 (0) CpxTRS 4.98/2.02 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.98/2.02 (2) TRS for Loop Detection 4.98/2.02 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.98/2.02 (4) BEST 4.98/2.02 (5) proven lower bound 4.98/2.02 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 4.98/2.02 (7) BOUNDS(n^1, INF) 4.98/2.02 (8) TRS for Loop Detection 4.98/2.02 (9) DecreasingLoopProof [FINISHED, 257 ms] 4.98/2.02 (10) BOUNDS(EXP, INF) 4.98/2.02 4.98/2.02 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (0) 4.98/2.02 Obligation: 4.98/2.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.98/2.02 4.98/2.02 4.98/2.02 The TRS R consists of the following rules: 4.98/2.02 4.98/2.02 __(__(X, Y), Z) -> __(X, __(Y, Z)) 4.98/2.02 __(X, nil) -> X 4.98/2.02 __(nil, X) -> X 4.98/2.02 U11(tt, V) -> U12(isNeList(activate(V))) 4.98/2.02 U12(tt) -> tt 4.98/2.02 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 4.98/2.02 U22(tt, V2) -> U23(isList(activate(V2))) 4.98/2.02 U23(tt) -> tt 4.98/2.02 U31(tt, V) -> U32(isQid(activate(V))) 4.98/2.02 U32(tt) -> tt 4.98/2.02 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 4.98/2.02 U42(tt, V2) -> U43(isNeList(activate(V2))) 4.98/2.02 U43(tt) -> tt 4.98/2.02 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 4.98/2.02 U52(tt, V2) -> U53(isList(activate(V2))) 4.98/2.02 U53(tt) -> tt 4.98/2.02 U61(tt, V) -> U62(isQid(activate(V))) 4.98/2.02 U62(tt) -> tt 4.98/2.02 U71(tt, V) -> U72(isNePal(activate(V))) 4.98/2.02 U72(tt) -> tt 4.98/2.02 and(tt, X) -> activate(X) 4.98/2.02 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isList(n__nil) -> tt 4.98/2.02 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNePal(n____(I, n____(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P)))) 4.98/2.02 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isPal(n__nil) -> tt 4.98/2.02 isPalListKind(n__a) -> tt 4.98/2.02 isPalListKind(n__e) -> tt 4.98/2.02 isPalListKind(n__i) -> tt 4.98/2.02 isPalListKind(n__nil) -> tt 4.98/2.02 isPalListKind(n__o) -> tt 4.98/2.02 isPalListKind(n__u) -> tt 4.98/2.02 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 4.98/2.02 isQid(n__a) -> tt 4.98/2.02 isQid(n__e) -> tt 4.98/2.02 isQid(n__i) -> tt 4.98/2.02 isQid(n__o) -> tt 4.98/2.02 isQid(n__u) -> tt 4.98/2.02 nil -> n__nil 4.98/2.02 __(X1, X2) -> n____(X1, X2) 4.98/2.02 isPalListKind(X) -> n__isPalListKind(X) 4.98/2.02 and(X1, X2) -> n__and(X1, X2) 4.98/2.02 isPal(X) -> n__isPal(X) 4.98/2.02 a -> n__a 4.98/2.02 e -> n__e 4.98/2.02 i -> n__i 4.98/2.02 o -> n__o 4.98/2.02 u -> n__u 4.98/2.02 activate(n__nil) -> nil 4.98/2.02 activate(n____(X1, X2)) -> __(activate(X1), activate(X2)) 4.98/2.02 activate(n__isPalListKind(X)) -> isPalListKind(X) 4.98/2.02 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.98/2.02 activate(n__isPal(X)) -> isPal(X) 4.98/2.02 activate(n__a) -> a 4.98/2.02 activate(n__e) -> e 4.98/2.02 activate(n__i) -> i 4.98/2.02 activate(n__o) -> o 4.98/2.02 activate(n__u) -> u 4.98/2.02 activate(X) -> X 4.98/2.02 4.98/2.02 S is empty. 4.98/2.02 Rewrite Strategy: FULL 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.98/2.02 Transformed a relative TRS into a decreasing-loop problem. 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (2) 4.98/2.02 Obligation: 4.98/2.02 Analyzing the following TRS for decreasing loops: 4.98/2.02 4.98/2.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.98/2.02 4.98/2.02 4.98/2.02 The TRS R consists of the following rules: 4.98/2.02 4.98/2.02 __(__(X, Y), Z) -> __(X, __(Y, Z)) 4.98/2.02 __(X, nil) -> X 4.98/2.02 __(nil, X) -> X 4.98/2.02 U11(tt, V) -> U12(isNeList(activate(V))) 4.98/2.02 U12(tt) -> tt 4.98/2.02 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 4.98/2.02 U22(tt, V2) -> U23(isList(activate(V2))) 4.98/2.02 U23(tt) -> tt 4.98/2.02 U31(tt, V) -> U32(isQid(activate(V))) 4.98/2.02 U32(tt) -> tt 4.98/2.02 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 4.98/2.02 U42(tt, V2) -> U43(isNeList(activate(V2))) 4.98/2.02 U43(tt) -> tt 4.98/2.02 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 4.98/2.02 U52(tt, V2) -> U53(isList(activate(V2))) 4.98/2.02 U53(tt) -> tt 4.98/2.02 U61(tt, V) -> U62(isQid(activate(V))) 4.98/2.02 U62(tt) -> tt 4.98/2.02 U71(tt, V) -> U72(isNePal(activate(V))) 4.98/2.02 U72(tt) -> tt 4.98/2.02 and(tt, X) -> activate(X) 4.98/2.02 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isList(n__nil) -> tt 4.98/2.02 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNePal(n____(I, n____(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P)))) 4.98/2.02 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isPal(n__nil) -> tt 4.98/2.02 isPalListKind(n__a) -> tt 4.98/2.02 isPalListKind(n__e) -> tt 4.98/2.02 isPalListKind(n__i) -> tt 4.98/2.02 isPalListKind(n__nil) -> tt 4.98/2.02 isPalListKind(n__o) -> tt 4.98/2.02 isPalListKind(n__u) -> tt 4.98/2.02 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 4.98/2.02 isQid(n__a) -> tt 4.98/2.02 isQid(n__e) -> tt 4.98/2.02 isQid(n__i) -> tt 4.98/2.02 isQid(n__o) -> tt 4.98/2.02 isQid(n__u) -> tt 4.98/2.02 nil -> n__nil 4.98/2.02 __(X1, X2) -> n____(X1, X2) 4.98/2.02 isPalListKind(X) -> n__isPalListKind(X) 4.98/2.02 and(X1, X2) -> n__and(X1, X2) 4.98/2.02 isPal(X) -> n__isPal(X) 4.98/2.02 a -> n__a 4.98/2.02 e -> n__e 4.98/2.02 i -> n__i 4.98/2.02 o -> n__o 4.98/2.02 u -> n__u 4.98/2.02 activate(n__nil) -> nil 4.98/2.02 activate(n____(X1, X2)) -> __(activate(X1), activate(X2)) 4.98/2.02 activate(n__isPalListKind(X)) -> isPalListKind(X) 4.98/2.02 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.98/2.02 activate(n__isPal(X)) -> isPal(X) 4.98/2.02 activate(n__a) -> a 4.98/2.02 activate(n__e) -> e 4.98/2.02 activate(n__i) -> i 4.98/2.02 activate(n__o) -> o 4.98/2.02 activate(n__u) -> u 4.98/2.02 activate(X) -> X 4.98/2.02 4.98/2.02 S is empty. 4.98/2.02 Rewrite Strategy: FULL 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (3) DecreasingLoopProof (LOWER BOUND(ID)) 4.98/2.02 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.98/2.02 4.98/2.02 The rewrite sequence 4.98/2.02 4.98/2.02 activate(n____(X1, X2)) ->^+ __(activate(X1), activate(X2)) 4.98/2.02 4.98/2.02 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 4.98/2.02 4.98/2.02 The pumping substitution is [X1 / n____(X1, X2)]. 4.98/2.02 4.98/2.02 The result substitution is [ ]. 4.98/2.02 4.98/2.02 4.98/2.02 4.98/2.02 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (4) 4.98/2.02 Complex Obligation (BEST) 4.98/2.02 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (5) 4.98/2.02 Obligation: 4.98/2.02 Proved the lower bound n^1 for the following obligation: 4.98/2.02 4.98/2.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.98/2.02 4.98/2.02 4.98/2.02 The TRS R consists of the following rules: 4.98/2.02 4.98/2.02 __(__(X, Y), Z) -> __(X, __(Y, Z)) 4.98/2.02 __(X, nil) -> X 4.98/2.02 __(nil, X) -> X 4.98/2.02 U11(tt, V) -> U12(isNeList(activate(V))) 4.98/2.02 U12(tt) -> tt 4.98/2.02 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 4.98/2.02 U22(tt, V2) -> U23(isList(activate(V2))) 4.98/2.02 U23(tt) -> tt 4.98/2.02 U31(tt, V) -> U32(isQid(activate(V))) 4.98/2.02 U32(tt) -> tt 4.98/2.02 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 4.98/2.02 U42(tt, V2) -> U43(isNeList(activate(V2))) 4.98/2.02 U43(tt) -> tt 4.98/2.02 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 4.98/2.02 U52(tt, V2) -> U53(isList(activate(V2))) 4.98/2.02 U53(tt) -> tt 4.98/2.02 U61(tt, V) -> U62(isQid(activate(V))) 4.98/2.02 U62(tt) -> tt 4.98/2.02 U71(tt, V) -> U72(isNePal(activate(V))) 4.98/2.02 U72(tt) -> tt 4.98/2.02 and(tt, X) -> activate(X) 4.98/2.02 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isList(n__nil) -> tt 4.98/2.02 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNePal(n____(I, n____(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P)))) 4.98/2.02 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isPal(n__nil) -> tt 4.98/2.02 isPalListKind(n__a) -> tt 4.98/2.02 isPalListKind(n__e) -> tt 4.98/2.02 isPalListKind(n__i) -> tt 4.98/2.02 isPalListKind(n__nil) -> tt 4.98/2.02 isPalListKind(n__o) -> tt 4.98/2.02 isPalListKind(n__u) -> tt 4.98/2.02 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 4.98/2.02 isQid(n__a) -> tt 4.98/2.02 isQid(n__e) -> tt 4.98/2.02 isQid(n__i) -> tt 4.98/2.02 isQid(n__o) -> tt 4.98/2.02 isQid(n__u) -> tt 4.98/2.02 nil -> n__nil 4.98/2.02 __(X1, X2) -> n____(X1, X2) 4.98/2.02 isPalListKind(X) -> n__isPalListKind(X) 4.98/2.02 and(X1, X2) -> n__and(X1, X2) 4.98/2.02 isPal(X) -> n__isPal(X) 4.98/2.02 a -> n__a 4.98/2.02 e -> n__e 4.98/2.02 i -> n__i 4.98/2.02 o -> n__o 4.98/2.02 u -> n__u 4.98/2.02 activate(n__nil) -> nil 4.98/2.02 activate(n____(X1, X2)) -> __(activate(X1), activate(X2)) 4.98/2.02 activate(n__isPalListKind(X)) -> isPalListKind(X) 4.98/2.02 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.98/2.02 activate(n__isPal(X)) -> isPal(X) 4.98/2.02 activate(n__a) -> a 4.98/2.02 activate(n__e) -> e 4.98/2.02 activate(n__i) -> i 4.98/2.02 activate(n__o) -> o 4.98/2.02 activate(n__u) -> u 4.98/2.02 activate(X) -> X 4.98/2.02 4.98/2.02 S is empty. 4.98/2.02 Rewrite Strategy: FULL 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (6) LowerBoundPropagationProof (FINISHED) 4.98/2.02 Propagated lower bound. 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (7) 4.98/2.02 BOUNDS(n^1, INF) 4.98/2.02 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (8) 4.98/2.02 Obligation: 4.98/2.02 Analyzing the following TRS for decreasing loops: 4.98/2.02 4.98/2.02 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.98/2.02 4.98/2.02 4.98/2.02 The TRS R consists of the following rules: 4.98/2.02 4.98/2.02 __(__(X, Y), Z) -> __(X, __(Y, Z)) 4.98/2.02 __(X, nil) -> X 4.98/2.02 __(nil, X) -> X 4.98/2.02 U11(tt, V) -> U12(isNeList(activate(V))) 4.98/2.02 U12(tt) -> tt 4.98/2.02 U21(tt, V1, V2) -> U22(isList(activate(V1)), activate(V2)) 4.98/2.02 U22(tt, V2) -> U23(isList(activate(V2))) 4.98/2.02 U23(tt) -> tt 4.98/2.02 U31(tt, V) -> U32(isQid(activate(V))) 4.98/2.02 U32(tt) -> tt 4.98/2.02 U41(tt, V1, V2) -> U42(isList(activate(V1)), activate(V2)) 4.98/2.02 U42(tt, V2) -> U43(isNeList(activate(V2))) 4.98/2.02 U43(tt) -> tt 4.98/2.02 U51(tt, V1, V2) -> U52(isNeList(activate(V1)), activate(V2)) 4.98/2.02 U52(tt, V2) -> U53(isList(activate(V2))) 4.98/2.02 U53(tt) -> tt 4.98/2.02 U61(tt, V) -> U62(isQid(activate(V))) 4.98/2.02 U62(tt) -> tt 4.98/2.02 U71(tt, V) -> U72(isNePal(activate(V))) 4.98/2.02 U72(tt) -> tt 4.98/2.02 and(tt, X) -> activate(X) 4.98/2.02 isList(V) -> U11(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isList(n__nil) -> tt 4.98/2.02 isList(n____(V1, V2)) -> U21(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(V) -> U31(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNeList(n____(V1, V2)) -> U41(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNeList(n____(V1, V2)) -> U51(and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))), activate(V1), activate(V2)) 4.98/2.02 isNePal(V) -> U61(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isNePal(n____(I, n____(P, I))) -> and(and(isQid(activate(I)), n__isPalListKind(activate(I))), n__and(n__isPal(activate(P)), n__isPalListKind(activate(P)))) 4.98/2.02 isPal(V) -> U71(isPalListKind(activate(V)), activate(V)) 4.98/2.02 isPal(n__nil) -> tt 4.98/2.02 isPalListKind(n__a) -> tt 4.98/2.02 isPalListKind(n__e) -> tt 4.98/2.02 isPalListKind(n__i) -> tt 4.98/2.02 isPalListKind(n__nil) -> tt 4.98/2.02 isPalListKind(n__o) -> tt 4.98/2.02 isPalListKind(n__u) -> tt 4.98/2.02 isPalListKind(n____(V1, V2)) -> and(isPalListKind(activate(V1)), n__isPalListKind(activate(V2))) 4.98/2.02 isQid(n__a) -> tt 4.98/2.02 isQid(n__e) -> tt 4.98/2.02 isQid(n__i) -> tt 4.98/2.02 isQid(n__o) -> tt 4.98/2.02 isQid(n__u) -> tt 4.98/2.02 nil -> n__nil 4.98/2.02 __(X1, X2) -> n____(X1, X2) 4.98/2.02 isPalListKind(X) -> n__isPalListKind(X) 4.98/2.02 and(X1, X2) -> n__and(X1, X2) 4.98/2.02 isPal(X) -> n__isPal(X) 4.98/2.02 a -> n__a 4.98/2.02 e -> n__e 4.98/2.02 i -> n__i 4.98/2.02 o -> n__o 4.98/2.02 u -> n__u 4.98/2.02 activate(n__nil) -> nil 4.98/2.02 activate(n____(X1, X2)) -> __(activate(X1), activate(X2)) 4.98/2.02 activate(n__isPalListKind(X)) -> isPalListKind(X) 4.98/2.02 activate(n__and(X1, X2)) -> and(activate(X1), X2) 4.98/2.02 activate(n__isPal(X)) -> isPal(X) 4.98/2.02 activate(n__a) -> a 4.98/2.02 activate(n__e) -> e 4.98/2.02 activate(n__i) -> i 4.98/2.02 activate(n__o) -> o 4.98/2.02 activate(n__u) -> u 4.98/2.02 activate(X) -> X 4.98/2.02 4.98/2.02 S is empty. 4.98/2.02 Rewrite Strategy: FULL 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (9) DecreasingLoopProof (FINISHED) 4.98/2.02 The following loop(s) give(s) rise to the lower bound EXP: 4.98/2.02 4.98/2.02 The rewrite sequence 4.98/2.02 4.98/2.02 activate(n__isPal(X)) ->^+ U71(isPalListKind(activate(X)), activate(X)) 4.98/2.02 4.98/2.02 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 4.98/2.02 4.98/2.02 The pumping substitution is [X / n__isPal(X)]. 4.98/2.02 4.98/2.02 The result substitution is [ ]. 4.98/2.02 4.98/2.02 4.98/2.02 4.98/2.02 The rewrite sequence 4.98/2.02 4.98/2.02 activate(n__isPal(X)) ->^+ U71(isPalListKind(activate(X)), activate(X)) 4.98/2.02 4.98/2.02 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 4.98/2.02 4.98/2.02 The pumping substitution is [X / n__isPal(X)]. 4.98/2.02 4.98/2.02 The result substitution is [ ]. 4.98/2.02 4.98/2.02 4.98/2.02 4.98/2.02 4.98/2.02 ---------------------------------------- 4.98/2.02 4.98/2.02 (10) 4.98/2.02 BOUNDS(EXP, INF) 5.21/2.10 EOF