6.80/2.57 WORST_CASE(NON_POLY, ?) 7.07/2.59 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 7.07/2.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.07/2.59 7.07/2.59 7.07/2.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.07/2.59 7.07/2.59 (0) CpxTRS 7.07/2.59 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 7.07/2.59 (2) TRS for Loop Detection 7.07/2.59 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 7.07/2.59 (4) BEST 7.07/2.59 (5) proven lower bound 7.07/2.59 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 7.07/2.59 (7) BOUNDS(n^1, INF) 7.07/2.59 (8) TRS for Loop Detection 7.07/2.59 (9) DecreasingLoopProof [FINISHED, 796 ms] 7.07/2.59 (10) BOUNDS(EXP, INF) 7.07/2.59 7.07/2.59 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (0) 7.07/2.59 Obligation: 7.07/2.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.07/2.59 7.07/2.59 7.07/2.59 The TRS R consists of the following rules: 7.07/2.59 7.07/2.59 zeros -> cons(0, n__zeros) 7.07/2.59 U11(tt, L) -> s(length(activate(L))) 7.07/2.59 U21(tt) -> nil 7.07/2.59 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 7.07/2.59 and(tt, X) -> activate(X) 7.07/2.59 isNat(n__0) -> tt 7.07/2.59 isNat(n__length(V1)) -> isNatList(activate(V1)) 7.07/2.59 isNat(n__s(V1)) -> isNat(activate(V1)) 7.07/2.59 isNatIList(V) -> isNatList(activate(V)) 7.07/2.59 isNatIList(n__zeros) -> tt 7.07/2.59 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 isNatList(n__nil) -> tt 7.07/2.59 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 7.07/2.59 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 length(nil) -> 0 7.07/2.59 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 7.07/2.59 take(0, IL) -> U21(isNatIList(IL)) 7.07/2.59 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) 7.07/2.59 zeros -> n__zeros 7.07/2.59 take(X1, X2) -> n__take(X1, X2) 7.07/2.59 0 -> n__0 7.07/2.59 length(X) -> n__length(X) 7.07/2.59 s(X) -> n__s(X) 7.07/2.59 cons(X1, X2) -> n__cons(X1, X2) 7.07/2.59 isNatIList(X) -> n__isNatIList(X) 7.07/2.59 nil -> n__nil 7.07/2.59 isNatList(X) -> n__isNatList(X) 7.07/2.59 isNat(X) -> n__isNat(X) 7.07/2.59 and(X1, X2) -> n__and(X1, X2) 7.07/2.59 activate(n__zeros) -> zeros 7.07/2.59 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.07/2.59 activate(n__0) -> 0 7.07/2.59 activate(n__length(X)) -> length(activate(X)) 7.07/2.59 activate(n__s(X)) -> s(activate(X)) 7.07/2.59 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.07/2.59 activate(n__isNatIList(X)) -> isNatIList(X) 7.07/2.59 activate(n__nil) -> nil 7.07/2.59 activate(n__isNatList(X)) -> isNatList(X) 7.07/2.59 activate(n__isNat(X)) -> isNat(X) 7.07/2.59 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.07/2.59 activate(X) -> X 7.07/2.59 7.07/2.59 S is empty. 7.07/2.59 Rewrite Strategy: FULL 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 7.07/2.59 Transformed a relative TRS into a decreasing-loop problem. 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (2) 7.07/2.59 Obligation: 7.07/2.59 Analyzing the following TRS for decreasing loops: 7.07/2.59 7.07/2.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.07/2.59 7.07/2.59 7.07/2.59 The TRS R consists of the following rules: 7.07/2.59 7.07/2.59 zeros -> cons(0, n__zeros) 7.07/2.59 U11(tt, L) -> s(length(activate(L))) 7.07/2.59 U21(tt) -> nil 7.07/2.59 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 7.07/2.59 and(tt, X) -> activate(X) 7.07/2.59 isNat(n__0) -> tt 7.07/2.59 isNat(n__length(V1)) -> isNatList(activate(V1)) 7.07/2.59 isNat(n__s(V1)) -> isNat(activate(V1)) 7.07/2.59 isNatIList(V) -> isNatList(activate(V)) 7.07/2.59 isNatIList(n__zeros) -> tt 7.07/2.59 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 isNatList(n__nil) -> tt 7.07/2.59 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 7.07/2.59 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 length(nil) -> 0 7.07/2.59 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 7.07/2.59 take(0, IL) -> U21(isNatIList(IL)) 7.07/2.59 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) 7.07/2.59 zeros -> n__zeros 7.07/2.59 take(X1, X2) -> n__take(X1, X2) 7.07/2.59 0 -> n__0 7.07/2.59 length(X) -> n__length(X) 7.07/2.59 s(X) -> n__s(X) 7.07/2.59 cons(X1, X2) -> n__cons(X1, X2) 7.07/2.59 isNatIList(X) -> n__isNatIList(X) 7.07/2.59 nil -> n__nil 7.07/2.59 isNatList(X) -> n__isNatList(X) 7.07/2.59 isNat(X) -> n__isNat(X) 7.07/2.59 and(X1, X2) -> n__and(X1, X2) 7.07/2.59 activate(n__zeros) -> zeros 7.07/2.59 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.07/2.59 activate(n__0) -> 0 7.07/2.59 activate(n__length(X)) -> length(activate(X)) 7.07/2.59 activate(n__s(X)) -> s(activate(X)) 7.07/2.59 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.07/2.59 activate(n__isNatIList(X)) -> isNatIList(X) 7.07/2.59 activate(n__nil) -> nil 7.07/2.59 activate(n__isNatList(X)) -> isNatList(X) 7.07/2.59 activate(n__isNat(X)) -> isNat(X) 7.07/2.59 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.07/2.59 activate(X) -> X 7.07/2.59 7.07/2.59 S is empty. 7.07/2.59 Rewrite Strategy: FULL 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (3) DecreasingLoopProof (LOWER BOUND(ID)) 7.07/2.59 The following loop(s) give(s) rise to the lower bound Omega(n^1): 7.07/2.59 7.07/2.59 The rewrite sequence 7.07/2.59 7.07/2.59 activate(n__s(X)) ->^+ s(activate(X)) 7.07/2.59 7.07/2.59 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 7.07/2.59 7.07/2.59 The pumping substitution is [X / n__s(X)]. 7.07/2.59 7.07/2.59 The result substitution is [ ]. 7.07/2.59 7.07/2.59 7.07/2.59 7.07/2.59 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (4) 7.07/2.59 Complex Obligation (BEST) 7.07/2.59 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (5) 7.07/2.59 Obligation: 7.07/2.59 Proved the lower bound n^1 for the following obligation: 7.07/2.59 7.07/2.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.07/2.59 7.07/2.59 7.07/2.59 The TRS R consists of the following rules: 7.07/2.59 7.07/2.59 zeros -> cons(0, n__zeros) 7.07/2.59 U11(tt, L) -> s(length(activate(L))) 7.07/2.59 U21(tt) -> nil 7.07/2.59 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 7.07/2.59 and(tt, X) -> activate(X) 7.07/2.59 isNat(n__0) -> tt 7.07/2.59 isNat(n__length(V1)) -> isNatList(activate(V1)) 7.07/2.59 isNat(n__s(V1)) -> isNat(activate(V1)) 7.07/2.59 isNatIList(V) -> isNatList(activate(V)) 7.07/2.59 isNatIList(n__zeros) -> tt 7.07/2.59 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 isNatList(n__nil) -> tt 7.07/2.59 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 7.07/2.59 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 length(nil) -> 0 7.07/2.59 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 7.07/2.59 take(0, IL) -> U21(isNatIList(IL)) 7.07/2.59 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) 7.07/2.59 zeros -> n__zeros 7.07/2.59 take(X1, X2) -> n__take(X1, X2) 7.07/2.59 0 -> n__0 7.07/2.59 length(X) -> n__length(X) 7.07/2.59 s(X) -> n__s(X) 7.07/2.59 cons(X1, X2) -> n__cons(X1, X2) 7.07/2.59 isNatIList(X) -> n__isNatIList(X) 7.07/2.59 nil -> n__nil 7.07/2.59 isNatList(X) -> n__isNatList(X) 7.07/2.59 isNat(X) -> n__isNat(X) 7.07/2.59 and(X1, X2) -> n__and(X1, X2) 7.07/2.59 activate(n__zeros) -> zeros 7.07/2.59 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.07/2.59 activate(n__0) -> 0 7.07/2.59 activate(n__length(X)) -> length(activate(X)) 7.07/2.59 activate(n__s(X)) -> s(activate(X)) 7.07/2.59 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.07/2.59 activate(n__isNatIList(X)) -> isNatIList(X) 7.07/2.59 activate(n__nil) -> nil 7.07/2.59 activate(n__isNatList(X)) -> isNatList(X) 7.07/2.59 activate(n__isNat(X)) -> isNat(X) 7.07/2.59 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.07/2.59 activate(X) -> X 7.07/2.59 7.07/2.59 S is empty. 7.07/2.59 Rewrite Strategy: FULL 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (6) LowerBoundPropagationProof (FINISHED) 7.07/2.59 Propagated lower bound. 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (7) 7.07/2.59 BOUNDS(n^1, INF) 7.07/2.59 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (8) 7.07/2.59 Obligation: 7.07/2.59 Analyzing the following TRS for decreasing loops: 7.07/2.59 7.07/2.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.07/2.59 7.07/2.59 7.07/2.59 The TRS R consists of the following rules: 7.07/2.59 7.07/2.59 zeros -> cons(0, n__zeros) 7.07/2.59 U11(tt, L) -> s(length(activate(L))) 7.07/2.59 U21(tt) -> nil 7.07/2.59 U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 7.07/2.59 and(tt, X) -> activate(X) 7.07/2.59 isNat(n__0) -> tt 7.07/2.59 isNat(n__length(V1)) -> isNatList(activate(V1)) 7.07/2.59 isNat(n__s(V1)) -> isNat(activate(V1)) 7.07/2.59 isNatIList(V) -> isNatList(activate(V)) 7.07/2.59 isNatIList(n__zeros) -> tt 7.07/2.59 isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 isNatList(n__nil) -> tt 7.07/2.59 isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) 7.07/2.59 isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 7.07/2.59 length(nil) -> 0 7.07/2.59 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) 7.07/2.59 take(0, IL) -> U21(isNatIList(IL)) 7.07/2.59 take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) 7.07/2.59 zeros -> n__zeros 7.07/2.59 take(X1, X2) -> n__take(X1, X2) 7.07/2.59 0 -> n__0 7.07/2.59 length(X) -> n__length(X) 7.07/2.59 s(X) -> n__s(X) 7.07/2.59 cons(X1, X2) -> n__cons(X1, X2) 7.07/2.59 isNatIList(X) -> n__isNatIList(X) 7.07/2.59 nil -> n__nil 7.07/2.59 isNatList(X) -> n__isNatList(X) 7.07/2.59 isNat(X) -> n__isNat(X) 7.07/2.59 and(X1, X2) -> n__and(X1, X2) 7.07/2.59 activate(n__zeros) -> zeros 7.07/2.59 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.07/2.59 activate(n__0) -> 0 7.07/2.59 activate(n__length(X)) -> length(activate(X)) 7.07/2.59 activate(n__s(X)) -> s(activate(X)) 7.07/2.59 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.07/2.59 activate(n__isNatIList(X)) -> isNatIList(X) 7.07/2.59 activate(n__nil) -> nil 7.07/2.59 activate(n__isNatList(X)) -> isNatList(X) 7.07/2.59 activate(n__isNat(X)) -> isNat(X) 7.07/2.59 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.07/2.59 activate(X) -> X 7.07/2.59 7.07/2.59 S is empty. 7.07/2.59 Rewrite Strategy: FULL 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (9) DecreasingLoopProof (FINISHED) 7.07/2.59 The following loop(s) give(s) rise to the lower bound EXP: 7.07/2.59 7.07/2.59 The rewrite sequence 7.07/2.59 7.07/2.59 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U11(and(isNatList(activate(X22_0)), n__isNat(activate(X11_0))), activate(X22_0)) 7.07/2.59 7.07/2.59 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0]. 7.07/2.59 7.07/2.59 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 7.07/2.59 7.07/2.59 The result substitution is [ ]. 7.07/2.59 7.07/2.59 7.07/2.59 7.07/2.59 The rewrite sequence 7.07/2.59 7.07/2.59 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U11(and(isNatList(activate(X22_0)), n__isNat(activate(X11_0))), activate(X22_0)) 7.07/2.59 7.07/2.59 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 7.07/2.59 7.07/2.59 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 7.07/2.59 7.07/2.59 The result substitution is [ ]. 7.07/2.59 7.07/2.59 7.07/2.59 7.07/2.59 7.07/2.59 ---------------------------------------- 7.07/2.59 7.07/2.59 (10) 7.07/2.59 BOUNDS(EXP, INF) 7.11/2.85 EOF