316.56/291.53 WORST_CASE(Omega(n^1), ?) 316.56/291.54 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 316.56/291.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 316.56/291.54 316.56/291.54 316.56/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.56/291.54 316.56/291.54 (0) CpxTRS 316.56/291.54 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 316.56/291.54 (2) TRS for Loop Detection 316.56/291.54 (3) DecreasingLoopProof [LOWER BOUND(ID), 126 ms] 316.56/291.54 (4) BEST 316.56/291.54 (5) proven lower bound 316.56/291.54 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 316.56/291.54 (7) BOUNDS(n^1, INF) 316.56/291.54 (8) TRS for Loop Detection 316.56/291.54 316.56/291.54 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (0) 316.56/291.54 Obligation: 316.56/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.56/291.54 316.56/291.54 316.56/291.54 The TRS R consists of the following rules: 316.56/291.54 316.56/291.54 zeros -> cons(0, n__zeros) 316.56/291.54 U11(tt) -> tt 316.56/291.54 U21(tt) -> tt 316.56/291.54 U31(tt) -> tt 316.56/291.54 U41(tt, V2) -> U42(isNatIList(activate(V2))) 316.56/291.54 U42(tt) -> tt 316.56/291.54 U51(tt, V2) -> U52(isNatList(activate(V2))) 316.56/291.54 U52(tt) -> tt 316.56/291.54 U61(tt, V2) -> U62(isNatIList(activate(V2))) 316.56/291.54 U62(tt) -> tt 316.56/291.54 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 316.56/291.54 U72(tt, L) -> s(length(activate(L))) 316.56/291.54 U81(tt) -> nil 316.56/291.54 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 316.56/291.54 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 316.56/291.54 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 316.56/291.54 isNat(n__0) -> tt 316.56/291.54 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 316.56/291.54 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 316.56/291.54 isNatIList(V) -> U31(isNatList(activate(V))) 316.56/291.54 isNatIList(n__zeros) -> tt 316.56/291.54 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__nil) -> tt 316.56/291.54 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 316.56/291.54 length(nil) -> 0 316.56/291.54 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 316.56/291.54 take(0, IL) -> U81(isNatIList(IL)) 316.56/291.54 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 316.56/291.54 zeros -> n__zeros 316.56/291.54 take(X1, X2) -> n__take(X1, X2) 316.56/291.54 0 -> n__0 316.56/291.54 length(X) -> n__length(X) 316.56/291.54 s(X) -> n__s(X) 316.56/291.54 cons(X1, X2) -> n__cons(X1, X2) 316.56/291.54 nil -> n__nil 316.56/291.54 activate(n__zeros) -> zeros 316.56/291.54 activate(n__take(X1, X2)) -> take(X1, X2) 316.56/291.54 activate(n__0) -> 0 316.56/291.54 activate(n__length(X)) -> length(X) 316.56/291.54 activate(n__s(X)) -> s(X) 316.56/291.54 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.56/291.54 activate(n__nil) -> nil 316.56/291.54 activate(X) -> X 316.56/291.54 316.56/291.54 S is empty. 316.56/291.54 Rewrite Strategy: FULL 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 316.56/291.54 Transformed a relative TRS into a decreasing-loop problem. 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (2) 316.56/291.54 Obligation: 316.56/291.54 Analyzing the following TRS for decreasing loops: 316.56/291.54 316.56/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.56/291.54 316.56/291.54 316.56/291.54 The TRS R consists of the following rules: 316.56/291.54 316.56/291.54 zeros -> cons(0, n__zeros) 316.56/291.54 U11(tt) -> tt 316.56/291.54 U21(tt) -> tt 316.56/291.54 U31(tt) -> tt 316.56/291.54 U41(tt, V2) -> U42(isNatIList(activate(V2))) 316.56/291.54 U42(tt) -> tt 316.56/291.54 U51(tt, V2) -> U52(isNatList(activate(V2))) 316.56/291.54 U52(tt) -> tt 316.56/291.54 U61(tt, V2) -> U62(isNatIList(activate(V2))) 316.56/291.54 U62(tt) -> tt 316.56/291.54 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 316.56/291.54 U72(tt, L) -> s(length(activate(L))) 316.56/291.54 U81(tt) -> nil 316.56/291.54 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 316.56/291.54 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 316.56/291.54 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 316.56/291.54 isNat(n__0) -> tt 316.56/291.54 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 316.56/291.54 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 316.56/291.54 isNatIList(V) -> U31(isNatList(activate(V))) 316.56/291.54 isNatIList(n__zeros) -> tt 316.56/291.54 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__nil) -> tt 316.56/291.54 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 316.56/291.54 length(nil) -> 0 316.56/291.54 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 316.56/291.54 take(0, IL) -> U81(isNatIList(IL)) 316.56/291.54 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 316.56/291.54 zeros -> n__zeros 316.56/291.54 take(X1, X2) -> n__take(X1, X2) 316.56/291.54 0 -> n__0 316.56/291.54 length(X) -> n__length(X) 316.56/291.54 s(X) -> n__s(X) 316.56/291.54 cons(X1, X2) -> n__cons(X1, X2) 316.56/291.54 nil -> n__nil 316.56/291.54 activate(n__zeros) -> zeros 316.56/291.54 activate(n__take(X1, X2)) -> take(X1, X2) 316.56/291.54 activate(n__0) -> 0 316.56/291.54 activate(n__length(X)) -> length(X) 316.56/291.54 activate(n__s(X)) -> s(X) 316.56/291.54 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.56/291.54 activate(n__nil) -> nil 316.56/291.54 activate(X) -> X 316.56/291.54 316.56/291.54 S is empty. 316.56/291.54 Rewrite Strategy: FULL 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (3) DecreasingLoopProof (LOWER BOUND(ID)) 316.56/291.54 The following loop(s) give(s) rise to the lower bound Omega(n^1): 316.56/291.54 316.56/291.54 The rewrite sequence 316.56/291.54 316.56/291.54 isNat(n__s(V1)) ->^+ U21(isNat(V1)) 316.56/291.54 316.56/291.54 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 316.56/291.54 316.56/291.54 The pumping substitution is [V1 / n__s(V1)]. 316.56/291.54 316.56/291.54 The result substitution is [ ]. 316.56/291.54 316.56/291.54 316.56/291.54 316.56/291.54 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (4) 316.56/291.54 Complex Obligation (BEST) 316.56/291.54 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (5) 316.56/291.54 Obligation: 316.56/291.54 Proved the lower bound n^1 for the following obligation: 316.56/291.54 316.56/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.56/291.54 316.56/291.54 316.56/291.54 The TRS R consists of the following rules: 316.56/291.54 316.56/291.54 zeros -> cons(0, n__zeros) 316.56/291.54 U11(tt) -> tt 316.56/291.54 U21(tt) -> tt 316.56/291.54 U31(tt) -> tt 316.56/291.54 U41(tt, V2) -> U42(isNatIList(activate(V2))) 316.56/291.54 U42(tt) -> tt 316.56/291.54 U51(tt, V2) -> U52(isNatList(activate(V2))) 316.56/291.54 U52(tt) -> tt 316.56/291.54 U61(tt, V2) -> U62(isNatIList(activate(V2))) 316.56/291.54 U62(tt) -> tt 316.56/291.54 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 316.56/291.54 U72(tt, L) -> s(length(activate(L))) 316.56/291.54 U81(tt) -> nil 316.56/291.54 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 316.56/291.54 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 316.56/291.54 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 316.56/291.54 isNat(n__0) -> tt 316.56/291.54 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 316.56/291.54 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 316.56/291.54 isNatIList(V) -> U31(isNatList(activate(V))) 316.56/291.54 isNatIList(n__zeros) -> tt 316.56/291.54 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__nil) -> tt 316.56/291.54 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 316.56/291.54 length(nil) -> 0 316.56/291.54 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 316.56/291.54 take(0, IL) -> U81(isNatIList(IL)) 316.56/291.54 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 316.56/291.54 zeros -> n__zeros 316.56/291.54 take(X1, X2) -> n__take(X1, X2) 316.56/291.54 0 -> n__0 316.56/291.54 length(X) -> n__length(X) 316.56/291.54 s(X) -> n__s(X) 316.56/291.54 cons(X1, X2) -> n__cons(X1, X2) 316.56/291.54 nil -> n__nil 316.56/291.54 activate(n__zeros) -> zeros 316.56/291.54 activate(n__take(X1, X2)) -> take(X1, X2) 316.56/291.54 activate(n__0) -> 0 316.56/291.54 activate(n__length(X)) -> length(X) 316.56/291.54 activate(n__s(X)) -> s(X) 316.56/291.54 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.56/291.54 activate(n__nil) -> nil 316.56/291.54 activate(X) -> X 316.56/291.54 316.56/291.54 S is empty. 316.56/291.54 Rewrite Strategy: FULL 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (6) LowerBoundPropagationProof (FINISHED) 316.56/291.54 Propagated lower bound. 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (7) 316.56/291.54 BOUNDS(n^1, INF) 316.56/291.54 316.56/291.54 ---------------------------------------- 316.56/291.54 316.56/291.54 (8) 316.56/291.54 Obligation: 316.56/291.54 Analyzing the following TRS for decreasing loops: 316.56/291.54 316.56/291.54 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 316.56/291.54 316.56/291.54 316.56/291.54 The TRS R consists of the following rules: 316.56/291.54 316.56/291.54 zeros -> cons(0, n__zeros) 316.56/291.54 U11(tt) -> tt 316.56/291.54 U21(tt) -> tt 316.56/291.54 U31(tt) -> tt 316.56/291.54 U41(tt, V2) -> U42(isNatIList(activate(V2))) 316.56/291.54 U42(tt) -> tt 316.56/291.54 U51(tt, V2) -> U52(isNatList(activate(V2))) 316.56/291.54 U52(tt) -> tt 316.56/291.54 U61(tt, V2) -> U62(isNatIList(activate(V2))) 316.56/291.54 U62(tt) -> tt 316.56/291.54 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 316.56/291.54 U72(tt, L) -> s(length(activate(L))) 316.56/291.54 U81(tt) -> nil 316.56/291.54 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 316.56/291.54 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 316.56/291.54 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 316.56/291.54 isNat(n__0) -> tt 316.56/291.54 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 316.56/291.54 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 316.56/291.54 isNatIList(V) -> U31(isNatList(activate(V))) 316.56/291.54 isNatIList(n__zeros) -> tt 316.56/291.54 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__nil) -> tt 316.56/291.54 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 316.56/291.54 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 316.56/291.54 length(nil) -> 0 316.56/291.54 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 316.56/291.54 take(0, IL) -> U81(isNatIList(IL)) 316.56/291.54 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 316.56/291.54 zeros -> n__zeros 316.56/291.54 take(X1, X2) -> n__take(X1, X2) 316.56/291.54 0 -> n__0 316.56/291.54 length(X) -> n__length(X) 316.56/291.54 s(X) -> n__s(X) 316.56/291.54 cons(X1, X2) -> n__cons(X1, X2) 316.56/291.54 nil -> n__nil 316.56/291.54 activate(n__zeros) -> zeros 316.56/291.54 activate(n__take(X1, X2)) -> take(X1, X2) 316.56/291.54 activate(n__0) -> 0 316.56/291.54 activate(n__length(X)) -> length(X) 316.56/291.54 activate(n__s(X)) -> s(X) 316.56/291.54 activate(n__cons(X1, X2)) -> cons(X1, X2) 316.56/291.54 activate(n__nil) -> nil 316.56/291.54 activate(X) -> X 316.56/291.54 316.56/291.54 S is empty. 316.56/291.54 Rewrite Strategy: FULL 316.63/291.57 EOF