316.14/291.52 WORST_CASE(Omega(n^1), ?) 316.14/291.53 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 316.14/291.53 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 316.14/291.53 316.14/291.53 316.14/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.14/291.53 316.14/291.53 (0) CpxTRS 316.14/291.53 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 316.14/291.53 (2) TRS for Loop Detection 316.14/291.53 (3) DecreasingLoopProof [LOWER BOUND(ID), 41 ms] 316.14/291.53 (4) BEST 316.14/291.53 (5) proven lower bound 316.14/291.53 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 316.14/291.53 (7) BOUNDS(n^1, INF) 316.14/291.53 (8) TRS for Loop Detection 316.14/291.53 316.14/291.53 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (0) 316.14/291.53 Obligation: 316.14/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.14/291.53 316.14/291.53 316.14/291.53 The TRS R consists of the following rules: 316.14/291.53 316.14/291.53 zeros -> cons(0, n__zeros) 316.14/291.53 U11(tt, L) -> s(length(activate(L))) 316.14/291.53 and(tt, X) -> activate(X) 316.14/291.53 isNat(n__0) -> tt 316.14/291.53 isNat(n__length(V1)) -> isNatList(activate(V1)) 316.14/291.53 isNat(n__s(V1)) -> isNat(activate(V1)) 316.14/291.53 isNatIList(V) -> isNatList(activate(V)) 316.14/291.53 isNatIList(n__zeros) -> tt 316.14/291.53 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 316.14/291.53 isNatList(n__nil) -> tt 316.14/291.53 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 316.14/291.53 length(nil) -> 0 316.14/291.53 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 316.14/291.53 zeros -> n__zeros 316.14/291.53 0 -> n__0 316.14/291.53 length(X) -> n__length(X) 316.14/291.53 s(X) -> n__s(X) 316.14/291.53 cons(X1, X2) -> n__cons(X1, X2) 316.14/291.53 isNatIList(X) -> n__isNatIList(X) 316.14/291.53 nil -> n__nil 316.14/291.53 isNatList(X) -> n__isNatList(X) 316.14/291.53 isNat(X) -> n__isNat(X) 316.14/291.53 activate(n__zeros) -> zeros 316.14/291.53 activate(n__0) -> 0 316.14/291.53 activate(n__length(X)) -> length(X) 316.14/291.53 activate(n__s(X)) -> s(X) 316.14/291.53 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.14/291.53 activate(n__isNatIList(X)) -> isNatIList(X) 316.14/291.53 activate(n__nil) -> nil 316.14/291.53 activate(n__isNatList(X)) -> isNatList(X) 316.14/291.53 activate(n__isNat(X)) -> isNat(X) 316.14/291.53 activate(X) -> X 316.14/291.53 316.14/291.53 S is empty. 316.14/291.53 Rewrite Strategy: FULL 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 316.14/291.53 Transformed a relative TRS into a decreasing-loop problem. 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (2) 316.14/291.53 Obligation: 316.14/291.53 Analyzing the following TRS for decreasing loops: 316.14/291.53 316.14/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.14/291.53 316.14/291.53 316.14/291.53 The TRS R consists of the following rules: 316.14/291.53 316.14/291.53 zeros -> cons(0, n__zeros) 316.14/291.53 U11(tt, L) -> s(length(activate(L))) 316.14/291.53 and(tt, X) -> activate(X) 316.14/291.53 isNat(n__0) -> tt 316.14/291.53 isNat(n__length(V1)) -> isNatList(activate(V1)) 316.14/291.53 isNat(n__s(V1)) -> isNat(activate(V1)) 316.14/291.53 isNatIList(V) -> isNatList(activate(V)) 316.14/291.53 isNatIList(n__zeros) -> tt 316.14/291.53 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 316.14/291.53 isNatList(n__nil) -> tt 316.14/291.53 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 316.14/291.53 length(nil) -> 0 316.14/291.53 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 316.14/291.53 zeros -> n__zeros 316.14/291.53 0 -> n__0 316.14/291.53 length(X) -> n__length(X) 316.14/291.53 s(X) -> n__s(X) 316.14/291.53 cons(X1, X2) -> n__cons(X1, X2) 316.14/291.53 isNatIList(X) -> n__isNatIList(X) 316.14/291.53 nil -> n__nil 316.14/291.53 isNatList(X) -> n__isNatList(X) 316.14/291.53 isNat(X) -> n__isNat(X) 316.14/291.53 activate(n__zeros) -> zeros 316.14/291.53 activate(n__0) -> 0 316.14/291.53 activate(n__length(X)) -> length(X) 316.14/291.53 activate(n__s(X)) -> s(X) 316.14/291.53 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.14/291.53 activate(n__isNatIList(X)) -> isNatIList(X) 316.14/291.53 activate(n__nil) -> nil 316.14/291.53 activate(n__isNatList(X)) -> isNatList(X) 316.14/291.53 activate(n__isNat(X)) -> isNat(X) 316.14/291.53 activate(X) -> X 316.14/291.53 316.14/291.53 S is empty. 316.14/291.53 Rewrite Strategy: FULL 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (3) DecreasingLoopProof (LOWER BOUND(ID)) 316.14/291.53 The following loop(s) give(s) rise to the lower bound Omega(n^1): 316.14/291.53 316.14/291.53 The rewrite sequence 316.14/291.53 316.14/291.53 activate(n__isNatList(n__cons(V11_0, V22_0))) ->^+ and(isNat(activate(V11_0)), n__isNatList(activate(V22_0))) 316.14/291.53 316.14/291.53 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 316.14/291.53 316.14/291.53 The pumping substitution is [V11_0 / n__isNatList(n__cons(V11_0, V22_0))]. 316.14/291.53 316.14/291.53 The result substitution is [ ]. 316.14/291.53 316.14/291.53 316.14/291.53 316.14/291.53 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (4) 316.14/291.53 Complex Obligation (BEST) 316.14/291.53 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (5) 316.14/291.53 Obligation: 316.14/291.53 Proved the lower bound n^1 for the following obligation: 316.14/291.53 316.14/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.14/291.53 316.14/291.53 316.14/291.53 The TRS R consists of the following rules: 316.14/291.53 316.14/291.53 zeros -> cons(0, n__zeros) 316.14/291.53 U11(tt, L) -> s(length(activate(L))) 316.14/291.53 and(tt, X) -> activate(X) 316.14/291.53 isNat(n__0) -> tt 316.14/291.53 isNat(n__length(V1)) -> isNatList(activate(V1)) 316.14/291.53 isNat(n__s(V1)) -> isNat(activate(V1)) 316.14/291.53 isNatIList(V) -> isNatList(activate(V)) 316.14/291.53 isNatIList(n__zeros) -> tt 316.14/291.53 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 316.14/291.53 isNatList(n__nil) -> tt 316.14/291.53 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 316.14/291.53 length(nil) -> 0 316.14/291.53 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 316.14/291.53 zeros -> n__zeros 316.14/291.53 0 -> n__0 316.14/291.53 length(X) -> n__length(X) 316.14/291.53 s(X) -> n__s(X) 316.14/291.53 cons(X1, X2) -> n__cons(X1, X2) 316.14/291.53 isNatIList(X) -> n__isNatIList(X) 316.14/291.53 nil -> n__nil 316.14/291.53 isNatList(X) -> n__isNatList(X) 316.14/291.53 isNat(X) -> n__isNat(X) 316.14/291.53 activate(n__zeros) -> zeros 316.14/291.53 activate(n__0) -> 0 316.14/291.53 activate(n__length(X)) -> length(X) 316.14/291.53 activate(n__s(X)) -> s(X) 316.14/291.53 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.14/291.53 activate(n__isNatIList(X)) -> isNatIList(X) 316.14/291.53 activate(n__nil) -> nil 316.14/291.53 activate(n__isNatList(X)) -> isNatList(X) 316.14/291.53 activate(n__isNat(X)) -> isNat(X) 316.14/291.53 activate(X) -> X 316.14/291.53 316.14/291.53 S is empty. 316.14/291.53 Rewrite Strategy: FULL 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (6) LowerBoundPropagationProof (FINISHED) 316.14/291.53 Propagated lower bound. 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (7) 316.14/291.53 BOUNDS(n^1, INF) 316.14/291.53 316.14/291.53 ---------------------------------------- 316.14/291.53 316.14/291.53 (8) 316.14/291.53 Obligation: 316.14/291.53 Analyzing the following TRS for decreasing loops: 316.14/291.53 316.14/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.14/291.53 316.14/291.53 316.14/291.53 The TRS R consists of the following rules: 316.14/291.53 316.14/291.53 zeros -> cons(0, n__zeros) 316.14/291.53 U11(tt, L) -> s(length(activate(L))) 316.14/291.53 and(tt, X) -> activate(X) 316.14/291.53 isNat(n__0) -> tt 316.14/291.53 isNat(n__length(V1)) -> isNatList(activate(V1)) 316.14/291.53 isNat(n__s(V1)) -> isNat(activate(V1)) 316.14/291.53 isNatIList(V) -> isNatList(activate(V)) 316.14/291.53 isNatIList(n__zeros) -> tt 316.14/291.53 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 316.14/291.53 isNatList(n__nil) -> tt 316.14/291.53 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 316.14/291.53 length(nil) -> 0 316.14/291.53 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 316.14/291.53 zeros -> n__zeros 316.14/291.53 0 -> n__0 316.14/291.53 length(X) -> n__length(X) 316.14/291.53 s(X) -> n__s(X) 316.14/291.53 cons(X1, X2) -> n__cons(X1, X2) 316.14/291.53 isNatIList(X) -> n__isNatIList(X) 316.14/291.53 nil -> n__nil 316.14/291.53 isNatList(X) -> n__isNatList(X) 316.14/291.53 isNat(X) -> n__isNat(X) 316.14/291.53 activate(n__zeros) -> zeros 316.14/291.53 activate(n__0) -> 0 316.14/291.53 activate(n__length(X)) -> length(X) 316.14/291.53 activate(n__s(X)) -> s(X) 316.14/291.53 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.14/291.53 activate(n__isNatIList(X)) -> isNatIList(X) 316.14/291.53 activate(n__nil) -> nil 316.14/291.53 activate(n__isNatList(X)) -> isNatList(X) 316.14/291.53 activate(n__isNat(X)) -> isNat(X) 316.14/291.53 activate(X) -> X 316.14/291.53 316.14/291.53 S is empty. 316.14/291.53 Rewrite Strategy: FULL 316.24/291.57 EOF