320.51/291.55 WORST_CASE(Omega(n^1), ?) 320.51/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 320.51/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 320.51/291.56 320.51/291.56 320.51/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 320.51/291.56 320.51/291.56 (0) CpxTRS 320.51/291.56 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 320.51/291.56 (2) TRS for Loop Detection 320.51/291.56 (3) DecreasingLoopProof [LOWER BOUND(ID), 36 ms] 320.51/291.56 (4) BEST 320.51/291.56 (5) proven lower bound 320.51/291.56 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 320.51/291.56 (7) BOUNDS(n^1, INF) 320.51/291.56 (8) TRS for Loop Detection 320.51/291.56 320.51/291.56 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (0) 320.51/291.56 Obligation: 320.51/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 320.51/291.56 320.51/291.56 320.51/291.56 The TRS R consists of the following rules: 320.51/291.56 320.51/291.56 zeros -> cons(0, n__zeros) 320.51/291.56 U11(tt, L) -> s(length(activate(L))) 320.51/291.56 U21(tt) -> nil 320.51/291.56 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 320.51/291.56 and(tt, X) -> activate(X) 320.51/291.56 isNat(n__0) -> tt 320.51/291.56 isNat(n__length(V1)) -> isNatList(activate(V1)) 320.51/291.56 isNat(n__s(V1)) -> isNat(activate(V1)) 320.51/291.56 isNatIList(V) -> isNatList(activate(V)) 320.51/291.56 isNatIList(n__zeros) -> tt 320.51/291.56 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 isNatList(n__nil) -> tt 320.51/291.56 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 320.51/291.56 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 length(nil) -> 0 320.51/291.56 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 320.51/291.56 take(0, IL) -> U21(isNatIList(IL)) 320.51/291.56 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) 320.51/291.56 zeros -> n__zeros 320.51/291.56 take(X1, X2) -> n__take(X1, X2) 320.51/291.56 0 -> n__0 320.51/291.56 length(X) -> n__length(X) 320.51/291.56 s(X) -> n__s(X) 320.51/291.56 cons(X1, X2) -> n__cons(X1, X2) 320.51/291.56 isNatIList(X) -> n__isNatIList(X) 320.51/291.56 nil -> n__nil 320.51/291.56 isNatList(X) -> n__isNatList(X) 320.51/291.56 isNat(X) -> n__isNat(X) 320.51/291.56 and(X1, X2) -> n__and(X1, X2) 320.51/291.56 activate(n__zeros) -> zeros 320.51/291.56 activate(n__take(X1, X2)) -> take(X1, X2) 320.51/291.56 activate(n__0) -> 0 320.51/291.56 activate(n__length(X)) -> length(X) 320.51/291.56 activate(n__s(X)) -> s(X) 320.51/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 320.51/291.56 activate(n__isNatIList(X)) -> isNatIList(X) 320.51/291.56 activate(n__nil) -> nil 320.51/291.56 activate(n__isNatList(X)) -> isNatList(X) 320.51/291.56 activate(n__isNat(X)) -> isNat(X) 320.51/291.56 activate(n__and(X1, X2)) -> and(X1, X2) 320.51/291.56 activate(X) -> X 320.51/291.56 320.51/291.56 S is empty. 320.51/291.56 Rewrite Strategy: FULL 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 320.51/291.56 Transformed a relative TRS into a decreasing-loop problem. 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (2) 320.51/291.56 Obligation: 320.51/291.56 Analyzing the following TRS for decreasing loops: 320.51/291.56 320.51/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 320.51/291.56 320.51/291.56 320.51/291.56 The TRS R consists of the following rules: 320.51/291.56 320.51/291.56 zeros -> cons(0, n__zeros) 320.51/291.56 U11(tt, L) -> s(length(activate(L))) 320.51/291.56 U21(tt) -> nil 320.51/291.56 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 320.51/291.56 and(tt, X) -> activate(X) 320.51/291.56 isNat(n__0) -> tt 320.51/291.56 isNat(n__length(V1)) -> isNatList(activate(V1)) 320.51/291.56 isNat(n__s(V1)) -> isNat(activate(V1)) 320.51/291.56 isNatIList(V) -> isNatList(activate(V)) 320.51/291.56 isNatIList(n__zeros) -> tt 320.51/291.56 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 isNatList(n__nil) -> tt 320.51/291.56 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 320.51/291.56 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 length(nil) -> 0 320.51/291.56 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 320.51/291.56 take(0, IL) -> U21(isNatIList(IL)) 320.51/291.56 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) 320.51/291.56 zeros -> n__zeros 320.51/291.56 take(X1, X2) -> n__take(X1, X2) 320.51/291.56 0 -> n__0 320.51/291.56 length(X) -> n__length(X) 320.51/291.56 s(X) -> n__s(X) 320.51/291.56 cons(X1, X2) -> n__cons(X1, X2) 320.51/291.56 isNatIList(X) -> n__isNatIList(X) 320.51/291.56 nil -> n__nil 320.51/291.56 isNatList(X) -> n__isNatList(X) 320.51/291.56 isNat(X) -> n__isNat(X) 320.51/291.56 and(X1, X2) -> n__and(X1, X2) 320.51/291.56 activate(n__zeros) -> zeros 320.51/291.56 activate(n__take(X1, X2)) -> take(X1, X2) 320.51/291.56 activate(n__0) -> 0 320.51/291.56 activate(n__length(X)) -> length(X) 320.51/291.56 activate(n__s(X)) -> s(X) 320.51/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 320.51/291.56 activate(n__isNatIList(X)) -> isNatIList(X) 320.51/291.56 activate(n__nil) -> nil 320.51/291.56 activate(n__isNatList(X)) -> isNatList(X) 320.51/291.56 activate(n__isNat(X)) -> isNat(X) 320.51/291.56 activate(n__and(X1, X2)) -> and(X1, X2) 320.51/291.56 activate(X) -> X 320.51/291.56 320.51/291.56 S is empty. 320.51/291.56 Rewrite Strategy: FULL 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (3) DecreasingLoopProof (LOWER BOUND(ID)) 320.51/291.56 The following loop(s) give(s) rise to the lower bound Omega(n^1): 320.51/291.56 320.51/291.56 The rewrite sequence 320.51/291.56 320.51/291.56 isNatList(n__take(n__isNatList(X1_0), V2)) ->^+ and(isNat(isNatList(X1_0)), n__isNatIList(activate(V2))) 320.51/291.56 320.51/291.56 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 320.51/291.56 320.51/291.56 The pumping substitution is [X1_0 / n__take(n__isNatList(X1_0), V2)]. 320.51/291.56 320.51/291.56 The result substitution is [ ]. 320.51/291.56 320.51/291.56 320.51/291.56 320.51/291.56 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (4) 320.51/291.56 Complex Obligation (BEST) 320.51/291.56 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (5) 320.51/291.56 Obligation: 320.51/291.56 Proved the lower bound n^1 for the following obligation: 320.51/291.56 320.51/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 320.51/291.56 320.51/291.56 320.51/291.56 The TRS R consists of the following rules: 320.51/291.56 320.51/291.56 zeros -> cons(0, n__zeros) 320.51/291.56 U11(tt, L) -> s(length(activate(L))) 320.51/291.56 U21(tt) -> nil 320.51/291.56 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 320.51/291.56 and(tt, X) -> activate(X) 320.51/291.56 isNat(n__0) -> tt 320.51/291.56 isNat(n__length(V1)) -> isNatList(activate(V1)) 320.51/291.56 isNat(n__s(V1)) -> isNat(activate(V1)) 320.51/291.56 isNatIList(V) -> isNatList(activate(V)) 320.51/291.56 isNatIList(n__zeros) -> tt 320.51/291.56 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 isNatList(n__nil) -> tt 320.51/291.56 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 320.51/291.56 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 length(nil) -> 0 320.51/291.56 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 320.51/291.56 take(0, IL) -> U21(isNatIList(IL)) 320.51/291.56 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) 320.51/291.56 zeros -> n__zeros 320.51/291.56 take(X1, X2) -> n__take(X1, X2) 320.51/291.56 0 -> n__0 320.51/291.56 length(X) -> n__length(X) 320.51/291.56 s(X) -> n__s(X) 320.51/291.56 cons(X1, X2) -> n__cons(X1, X2) 320.51/291.56 isNatIList(X) -> n__isNatIList(X) 320.51/291.56 nil -> n__nil 320.51/291.56 isNatList(X) -> n__isNatList(X) 320.51/291.56 isNat(X) -> n__isNat(X) 320.51/291.56 and(X1, X2) -> n__and(X1, X2) 320.51/291.56 activate(n__zeros) -> zeros 320.51/291.56 activate(n__take(X1, X2)) -> take(X1, X2) 320.51/291.56 activate(n__0) -> 0 320.51/291.56 activate(n__length(X)) -> length(X) 320.51/291.56 activate(n__s(X)) -> s(X) 320.51/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 320.51/291.56 activate(n__isNatIList(X)) -> isNatIList(X) 320.51/291.56 activate(n__nil) -> nil 320.51/291.56 activate(n__isNatList(X)) -> isNatList(X) 320.51/291.56 activate(n__isNat(X)) -> isNat(X) 320.51/291.56 activate(n__and(X1, X2)) -> and(X1, X2) 320.51/291.56 activate(X) -> X 320.51/291.56 320.51/291.56 S is empty. 320.51/291.56 Rewrite Strategy: FULL 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (6) LowerBoundPropagationProof (FINISHED) 320.51/291.56 Propagated lower bound. 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (7) 320.51/291.56 BOUNDS(n^1, INF) 320.51/291.56 320.51/291.56 ---------------------------------------- 320.51/291.56 320.51/291.56 (8) 320.51/291.56 Obligation: 320.51/291.56 Analyzing the following TRS for decreasing loops: 320.51/291.56 320.51/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 320.51/291.56 320.51/291.56 320.51/291.56 The TRS R consists of the following rules: 320.51/291.56 320.51/291.56 zeros -> cons(0, n__zeros) 320.51/291.56 U11(tt, L) -> s(length(activate(L))) 320.51/291.56 U21(tt) -> nil 320.51/291.56 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 320.51/291.56 and(tt, X) -> activate(X) 320.51/291.56 isNat(n__0) -> tt 320.51/291.56 isNat(n__length(V1)) -> isNatList(activate(V1)) 320.51/291.56 isNat(n__s(V1)) -> isNat(activate(V1)) 320.51/291.56 isNatIList(V) -> isNatList(activate(V)) 320.51/291.56 isNatIList(n__zeros) -> tt 320.51/291.56 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 isNatList(n__nil) -> tt 320.51/291.56 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 320.51/291.56 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 320.51/291.56 length(nil) -> 0 320.51/291.56 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 320.51/291.56 take(0, IL) -> U21(isNatIList(IL)) 320.51/291.56 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) 320.51/291.56 zeros -> n__zeros 320.51/291.56 take(X1, X2) -> n__take(X1, X2) 320.51/291.56 0 -> n__0 320.51/291.56 length(X) -> n__length(X) 320.51/291.56 s(X) -> n__s(X) 320.51/291.56 cons(X1, X2) -> n__cons(X1, X2) 320.51/291.56 isNatIList(X) -> n__isNatIList(X) 320.51/291.56 nil -> n__nil 320.51/291.56 isNatList(X) -> n__isNatList(X) 320.51/291.56 isNat(X) -> n__isNat(X) 320.51/291.56 and(X1, X2) -> n__and(X1, X2) 320.51/291.56 activate(n__zeros) -> zeros 320.51/291.56 activate(n__take(X1, X2)) -> take(X1, X2) 320.51/291.56 activate(n__0) -> 0 320.51/291.56 activate(n__length(X)) -> length(X) 320.51/291.56 activate(n__s(X)) -> s(X) 320.51/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 320.51/291.56 activate(n__isNatIList(X)) -> isNatIList(X) 320.51/291.56 activate(n__nil) -> nil 320.51/291.56 activate(n__isNatList(X)) -> isNatList(X) 320.51/291.56 activate(n__isNat(X)) -> isNat(X) 320.51/291.56 activate(n__and(X1, X2)) -> and(X1, X2) 320.51/291.56 activate(X) -> X 320.51/291.56 320.51/291.56 S is empty. 320.51/291.56 Rewrite Strategy: FULL 320.57/291.60 EOF