1133.45/293.12 WORST_CASE(Omega(n^1), ?) 1134.14/293.29 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1134.14/293.29 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1134.14/293.29 1134.14/293.29 1134.14/293.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1134.14/293.29 1134.14/293.29 (0) CpxTRS 1134.14/293.29 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1134.14/293.29 (2) TRS for Loop Detection 1134.14/293.29 (3) DecreasingLoopProof [LOWER BOUND(ID), 57 ms] 1134.14/293.29 (4) BEST 1134.14/293.29 (5) proven lower bound 1134.14/293.29 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1134.14/293.29 (7) BOUNDS(n^1, INF) 1134.14/293.29 (8) TRS for Loop Detection 1134.14/293.29 1134.14/293.29 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (0) 1134.14/293.29 Obligation: 1134.14/293.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1134.14/293.29 1134.14/293.29 1134.14/293.29 The TRS R consists of the following rules: 1134.14/293.29 1134.14/293.29 __(__(X, Y), Z) -> __(X, __(Y, Z)) 1134.14/293.29 __(X, nil) -> X 1134.14/293.29 __(nil, X) -> X 1134.14/293.29 and(tt, X) -> activate(X) 1134.14/293.29 isList(V) -> isNeList(activate(V)) 1134.14/293.29 isList(n__nil) -> tt 1134.14/293.29 isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNeList(V) -> isQid(activate(V)) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2))) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNePal(V) -> isQid(activate(V)) 1134.14/293.29 isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P))) 1134.14/293.29 isPal(V) -> isNePal(activate(V)) 1134.14/293.29 isPal(n__nil) -> tt 1134.14/293.29 isQid(n__a) -> tt 1134.14/293.29 isQid(n__e) -> tt 1134.14/293.29 isQid(n__i) -> tt 1134.14/293.29 isQid(n__o) -> tt 1134.14/293.29 isQid(n__u) -> tt 1134.14/293.29 nil -> n__nil 1134.14/293.29 __(X1, X2) -> n____(X1, X2) 1134.14/293.29 isList(X) -> n__isList(X) 1134.14/293.29 isNeList(X) -> n__isNeList(X) 1134.14/293.29 isPal(X) -> n__isPal(X) 1134.14/293.29 a -> n__a 1134.14/293.29 e -> n__e 1134.14/293.29 i -> n__i 1134.14/293.29 o -> n__o 1134.14/293.29 u -> n__u 1134.14/293.29 activate(n__nil) -> nil 1134.14/293.29 activate(n____(X1, X2)) -> __(X1, X2) 1134.14/293.29 activate(n__isList(X)) -> isList(X) 1134.14/293.29 activate(n__isNeList(X)) -> isNeList(X) 1134.14/293.29 activate(n__isPal(X)) -> isPal(X) 1134.14/293.29 activate(n__a) -> a 1134.14/293.29 activate(n__e) -> e 1134.14/293.29 activate(n__i) -> i 1134.14/293.29 activate(n__o) -> o 1134.14/293.29 activate(n__u) -> u 1134.14/293.29 activate(X) -> X 1134.14/293.29 1134.14/293.29 S is empty. 1134.14/293.29 Rewrite Strategy: FULL 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1134.14/293.29 Transformed a relative TRS into a decreasing-loop problem. 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (2) 1134.14/293.29 Obligation: 1134.14/293.29 Analyzing the following TRS for decreasing loops: 1134.14/293.29 1134.14/293.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1134.14/293.29 1134.14/293.29 1134.14/293.29 The TRS R consists of the following rules: 1134.14/293.29 1134.14/293.29 __(__(X, Y), Z) -> __(X, __(Y, Z)) 1134.14/293.29 __(X, nil) -> X 1134.14/293.29 __(nil, X) -> X 1134.14/293.29 and(tt, X) -> activate(X) 1134.14/293.29 isList(V) -> isNeList(activate(V)) 1134.14/293.29 isList(n__nil) -> tt 1134.14/293.29 isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNeList(V) -> isQid(activate(V)) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2))) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNePal(V) -> isQid(activate(V)) 1134.14/293.29 isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P))) 1134.14/293.29 isPal(V) -> isNePal(activate(V)) 1134.14/293.29 isPal(n__nil) -> tt 1134.14/293.29 isQid(n__a) -> tt 1134.14/293.29 isQid(n__e) -> tt 1134.14/293.29 isQid(n__i) -> tt 1134.14/293.29 isQid(n__o) -> tt 1134.14/293.29 isQid(n__u) -> tt 1134.14/293.29 nil -> n__nil 1134.14/293.29 __(X1, X2) -> n____(X1, X2) 1134.14/293.29 isList(X) -> n__isList(X) 1134.14/293.29 isNeList(X) -> n__isNeList(X) 1134.14/293.29 isPal(X) -> n__isPal(X) 1134.14/293.29 a -> n__a 1134.14/293.29 e -> n__e 1134.14/293.29 i -> n__i 1134.14/293.29 o -> n__o 1134.14/293.29 u -> n__u 1134.14/293.29 activate(n__nil) -> nil 1134.14/293.29 activate(n____(X1, X2)) -> __(X1, X2) 1134.14/293.29 activate(n__isList(X)) -> isList(X) 1134.14/293.29 activate(n__isNeList(X)) -> isNeList(X) 1134.14/293.29 activate(n__isPal(X)) -> isPal(X) 1134.14/293.29 activate(n__a) -> a 1134.14/293.29 activate(n__e) -> e 1134.14/293.29 activate(n__i) -> i 1134.14/293.29 activate(n__o) -> o 1134.14/293.29 activate(n__u) -> u 1134.14/293.29 activate(X) -> X 1134.14/293.29 1134.14/293.29 S is empty. 1134.14/293.29 Rewrite Strategy: FULL 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1134.14/293.29 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1134.14/293.29 1134.14/293.29 The rewrite sequence 1134.14/293.29 1134.14/293.29 isList(n____(n__isList(X1_0), V2)) ->^+ and(isList(isList(X1_0)), n__isList(activate(V2))) 1134.14/293.29 1134.14/293.29 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 1134.14/293.29 1134.14/293.29 The pumping substitution is [X1_0 / n____(n__isList(X1_0), V2)]. 1134.14/293.29 1134.14/293.29 The result substitution is [ ]. 1134.14/293.29 1134.14/293.29 1134.14/293.29 1134.14/293.29 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (4) 1134.14/293.29 Complex Obligation (BEST) 1134.14/293.29 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (5) 1134.14/293.29 Obligation: 1134.14/293.29 Proved the lower bound n^1 for the following obligation: 1134.14/293.29 1134.14/293.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1134.14/293.29 1134.14/293.29 1134.14/293.29 The TRS R consists of the following rules: 1134.14/293.29 1134.14/293.29 __(__(X, Y), Z) -> __(X, __(Y, Z)) 1134.14/293.29 __(X, nil) -> X 1134.14/293.29 __(nil, X) -> X 1134.14/293.29 and(tt, X) -> activate(X) 1134.14/293.29 isList(V) -> isNeList(activate(V)) 1134.14/293.29 isList(n__nil) -> tt 1134.14/293.29 isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNeList(V) -> isQid(activate(V)) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2))) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNePal(V) -> isQid(activate(V)) 1134.14/293.29 isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P))) 1134.14/293.29 isPal(V) -> isNePal(activate(V)) 1134.14/293.29 isPal(n__nil) -> tt 1134.14/293.29 isQid(n__a) -> tt 1134.14/293.29 isQid(n__e) -> tt 1134.14/293.29 isQid(n__i) -> tt 1134.14/293.29 isQid(n__o) -> tt 1134.14/293.29 isQid(n__u) -> tt 1134.14/293.29 nil -> n__nil 1134.14/293.29 __(X1, X2) -> n____(X1, X2) 1134.14/293.29 isList(X) -> n__isList(X) 1134.14/293.29 isNeList(X) -> n__isNeList(X) 1134.14/293.29 isPal(X) -> n__isPal(X) 1134.14/293.29 a -> n__a 1134.14/293.29 e -> n__e 1134.14/293.29 i -> n__i 1134.14/293.29 o -> n__o 1134.14/293.29 u -> n__u 1134.14/293.29 activate(n__nil) -> nil 1134.14/293.29 activate(n____(X1, X2)) -> __(X1, X2) 1134.14/293.29 activate(n__isList(X)) -> isList(X) 1134.14/293.29 activate(n__isNeList(X)) -> isNeList(X) 1134.14/293.29 activate(n__isPal(X)) -> isPal(X) 1134.14/293.29 activate(n__a) -> a 1134.14/293.29 activate(n__e) -> e 1134.14/293.29 activate(n__i) -> i 1134.14/293.29 activate(n__o) -> o 1134.14/293.29 activate(n__u) -> u 1134.14/293.29 activate(X) -> X 1134.14/293.29 1134.14/293.29 S is empty. 1134.14/293.29 Rewrite Strategy: FULL 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (6) LowerBoundPropagationProof (FINISHED) 1134.14/293.29 Propagated lower bound. 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (7) 1134.14/293.29 BOUNDS(n^1, INF) 1134.14/293.29 1134.14/293.29 ---------------------------------------- 1134.14/293.29 1134.14/293.29 (8) 1134.14/293.29 Obligation: 1134.14/293.29 Analyzing the following TRS for decreasing loops: 1134.14/293.29 1134.14/293.29 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1134.14/293.29 1134.14/293.29 1134.14/293.29 The TRS R consists of the following rules: 1134.14/293.29 1134.14/293.29 __(__(X, Y), Z) -> __(X, __(Y, Z)) 1134.14/293.29 __(X, nil) -> X 1134.14/293.29 __(nil, X) -> X 1134.14/293.29 and(tt, X) -> activate(X) 1134.14/293.29 isList(V) -> isNeList(activate(V)) 1134.14/293.29 isList(n__nil) -> tt 1134.14/293.29 isList(n____(V1, V2)) -> and(isList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNeList(V) -> isQid(activate(V)) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isList(activate(V1)), n__isNeList(activate(V2))) 1134.14/293.29 isNeList(n____(V1, V2)) -> and(isNeList(activate(V1)), n__isList(activate(V2))) 1134.14/293.29 isNePal(V) -> isQid(activate(V)) 1134.14/293.29 isNePal(n____(I, __(P, I))) -> and(isQid(activate(I)), n__isPal(activate(P))) 1134.14/293.29 isPal(V) -> isNePal(activate(V)) 1134.14/293.29 isPal(n__nil) -> tt 1134.14/293.29 isQid(n__a) -> tt 1134.14/293.29 isQid(n__e) -> tt 1134.14/293.29 isQid(n__i) -> tt 1134.14/293.29 isQid(n__o) -> tt 1134.14/293.29 isQid(n__u) -> tt 1134.14/293.29 nil -> n__nil 1134.14/293.29 __(X1, X2) -> n____(X1, X2) 1134.14/293.29 isList(X) -> n__isList(X) 1134.14/293.29 isNeList(X) -> n__isNeList(X) 1134.14/293.29 isPal(X) -> n__isPal(X) 1134.14/293.29 a -> n__a 1134.14/293.29 e -> n__e 1134.14/293.29 i -> n__i 1134.14/293.29 o -> n__o 1134.14/293.29 u -> n__u 1134.14/293.29 activate(n__nil) -> nil 1134.14/293.29 activate(n____(X1, X2)) -> __(X1, X2) 1134.14/293.29 activate(n__isList(X)) -> isList(X) 1134.14/293.29 activate(n__isNeList(X)) -> isNeList(X) 1134.14/293.29 activate(n__isPal(X)) -> isPal(X) 1134.14/293.29 activate(n__a) -> a 1134.14/293.29 activate(n__e) -> e 1134.14/293.29 activate(n__i) -> i 1134.14/293.29 activate(n__o) -> o 1134.14/293.29 activate(n__u) -> u 1134.14/293.29 activate(X) -> X 1134.14/293.29 1134.14/293.29 S is empty. 1134.14/293.29 Rewrite Strategy: FULL 1134.14/293.36 EOF