446.63/291.59 WORST_CASE(Omega(n^1), ?) 446.63/291.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 446.63/291.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 446.63/291.60 446.63/291.60 446.63/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 446.63/291.60 446.63/291.60 (0) CpxTRS 446.63/291.60 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 446.63/291.60 (2) TRS for Loop Detection 446.63/291.60 (3) DecreasingLoopProof [LOWER BOUND(ID), 363 ms] 446.63/291.60 (4) BEST 446.63/291.60 (5) proven lower bound 446.63/291.60 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 446.63/291.60 (7) BOUNDS(n^1, INF) 446.63/291.60 (8) TRS for Loop Detection 446.63/291.60 446.63/291.60 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (0) 446.63/291.60 Obligation: 446.63/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 446.63/291.60 446.63/291.60 446.63/291.60 The TRS R consists of the following rules: 446.63/291.60 446.63/291.60 zeros -> cons(0, n__zeros) 446.63/291.60 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 446.63/291.60 U105(tt, V2) -> U106(isNatIList(activate(V2))) 446.63/291.60 U106(tt) -> tt 446.63/291.60 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 446.63/291.60 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 446.63/291.60 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 446.63/291.60 U114(tt, L) -> s(length(activate(L))) 446.63/291.60 U12(tt, V1) -> U13(isNatList(activate(V1))) 446.63/291.60 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 446.63/291.60 U122(tt) -> nil 446.63/291.60 U13(tt) -> tt 446.63/291.60 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 446.63/291.60 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 446.63/291.60 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 U22(tt, V1) -> U23(isNat(activate(V1))) 446.63/291.60 U23(tt) -> tt 446.63/291.60 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 U32(tt, V) -> U33(isNatList(activate(V))) 446.63/291.60 U33(tt) -> tt 446.63/291.60 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 446.63/291.60 U45(tt, V2) -> U46(isNatIList(activate(V2))) 446.63/291.60 U46(tt) -> tt 446.63/291.60 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 446.63/291.60 U52(tt) -> tt 446.63/291.60 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 446.63/291.60 U62(tt) -> tt 446.63/291.60 U71(tt) -> tt 446.63/291.60 U81(tt) -> tt 446.63/291.60 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 446.63/291.60 U95(tt, V2) -> U96(isNatList(activate(V2))) 446.63/291.60 U96(tt) -> tt 446.63/291.60 isNat(n__0) -> tt 446.63/291.60 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 isNatIList(n__zeros) -> tt 446.63/291.60 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatIListKind(n__nil) -> tt 446.63/291.60 isNatIListKind(n__zeros) -> tt 446.63/291.60 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatKind(n__0) -> tt 446.63/291.60 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 446.63/291.60 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 446.63/291.60 isNatList(n__nil) -> tt 446.63/291.60 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 length(nil) -> 0 446.63/291.60 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 446.63/291.60 take(0, IL) -> U121(isNatIList(IL), IL) 446.63/291.60 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 446.63/291.60 zeros -> n__zeros 446.63/291.60 take(X1, X2) -> n__take(X1, X2) 446.63/291.60 0 -> n__0 446.63/291.60 length(X) -> n__length(X) 446.63/291.60 s(X) -> n__s(X) 446.63/291.60 cons(X1, X2) -> n__cons(X1, X2) 446.63/291.60 nil -> n__nil 446.63/291.60 activate(n__zeros) -> zeros 446.63/291.60 activate(n__take(X1, X2)) -> take(X1, X2) 446.63/291.60 activate(n__0) -> 0 446.63/291.60 activate(n__length(X)) -> length(X) 446.63/291.60 activate(n__s(X)) -> s(X) 446.63/291.60 activate(n__cons(X1, X2)) -> cons(X1, X2) 446.63/291.60 activate(n__nil) -> nil 446.63/291.60 activate(X) -> X 446.63/291.60 446.63/291.60 S is empty. 446.63/291.60 Rewrite Strategy: FULL 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 446.63/291.60 Transformed a relative TRS into a decreasing-loop problem. 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (2) 446.63/291.60 Obligation: 446.63/291.60 Analyzing the following TRS for decreasing loops: 446.63/291.60 446.63/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 446.63/291.60 446.63/291.60 446.63/291.60 The TRS R consists of the following rules: 446.63/291.60 446.63/291.60 zeros -> cons(0, n__zeros) 446.63/291.60 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 446.63/291.60 U105(tt, V2) -> U106(isNatIList(activate(V2))) 446.63/291.60 U106(tt) -> tt 446.63/291.60 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 446.63/291.60 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 446.63/291.60 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 446.63/291.60 U114(tt, L) -> s(length(activate(L))) 446.63/291.60 U12(tt, V1) -> U13(isNatList(activate(V1))) 446.63/291.60 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 446.63/291.60 U122(tt) -> nil 446.63/291.60 U13(tt) -> tt 446.63/291.60 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 446.63/291.60 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 446.63/291.60 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 U22(tt, V1) -> U23(isNat(activate(V1))) 446.63/291.60 U23(tt) -> tt 446.63/291.60 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 U32(tt, V) -> U33(isNatList(activate(V))) 446.63/291.60 U33(tt) -> tt 446.63/291.60 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 446.63/291.60 U45(tt, V2) -> U46(isNatIList(activate(V2))) 446.63/291.60 U46(tt) -> tt 446.63/291.60 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 446.63/291.60 U52(tt) -> tt 446.63/291.60 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 446.63/291.60 U62(tt) -> tt 446.63/291.60 U71(tt) -> tt 446.63/291.60 U81(tt) -> tt 446.63/291.60 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 446.63/291.60 U95(tt, V2) -> U96(isNatList(activate(V2))) 446.63/291.60 U96(tt) -> tt 446.63/291.60 isNat(n__0) -> tt 446.63/291.60 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 isNatIList(n__zeros) -> tt 446.63/291.60 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatIListKind(n__nil) -> tt 446.63/291.60 isNatIListKind(n__zeros) -> tt 446.63/291.60 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatKind(n__0) -> tt 446.63/291.60 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 446.63/291.60 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 446.63/291.60 isNatList(n__nil) -> tt 446.63/291.60 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 length(nil) -> 0 446.63/291.60 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 446.63/291.60 take(0, IL) -> U121(isNatIList(IL), IL) 446.63/291.60 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 446.63/291.60 zeros -> n__zeros 446.63/291.60 take(X1, X2) -> n__take(X1, X2) 446.63/291.60 0 -> n__0 446.63/291.60 length(X) -> n__length(X) 446.63/291.60 s(X) -> n__s(X) 446.63/291.60 cons(X1, X2) -> n__cons(X1, X2) 446.63/291.60 nil -> n__nil 446.63/291.60 activate(n__zeros) -> zeros 446.63/291.60 activate(n__take(X1, X2)) -> take(X1, X2) 446.63/291.60 activate(n__0) -> 0 446.63/291.60 activate(n__length(X)) -> length(X) 446.63/291.60 activate(n__s(X)) -> s(X) 446.63/291.60 activate(n__cons(X1, X2)) -> cons(X1, X2) 446.63/291.60 activate(n__nil) -> nil 446.63/291.60 activate(X) -> X 446.63/291.60 446.63/291.60 S is empty. 446.63/291.60 Rewrite Strategy: FULL 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (3) DecreasingLoopProof (LOWER BOUND(ID)) 446.63/291.60 The following loop(s) give(s) rise to the lower bound Omega(n^1): 446.63/291.60 446.63/291.60 The rewrite sequence 446.63/291.60 446.63/291.60 isNatKind(n__s(V1)) ->^+ U81(isNatKind(V1)) 446.63/291.60 446.63/291.60 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 446.63/291.60 446.63/291.60 The pumping substitution is [V1 / n__s(V1)]. 446.63/291.60 446.63/291.60 The result substitution is [ ]. 446.63/291.60 446.63/291.60 446.63/291.60 446.63/291.60 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (4) 446.63/291.60 Complex Obligation (BEST) 446.63/291.60 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (5) 446.63/291.60 Obligation: 446.63/291.60 Proved the lower bound n^1 for the following obligation: 446.63/291.60 446.63/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 446.63/291.60 446.63/291.60 446.63/291.60 The TRS R consists of the following rules: 446.63/291.60 446.63/291.60 zeros -> cons(0, n__zeros) 446.63/291.60 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 446.63/291.60 U105(tt, V2) -> U106(isNatIList(activate(V2))) 446.63/291.60 U106(tt) -> tt 446.63/291.60 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 446.63/291.60 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 446.63/291.60 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 446.63/291.60 U114(tt, L) -> s(length(activate(L))) 446.63/291.60 U12(tt, V1) -> U13(isNatList(activate(V1))) 446.63/291.60 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 446.63/291.60 U122(tt) -> nil 446.63/291.60 U13(tt) -> tt 446.63/291.60 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 446.63/291.60 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 446.63/291.60 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 U22(tt, V1) -> U23(isNat(activate(V1))) 446.63/291.60 U23(tt) -> tt 446.63/291.60 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 U32(tt, V) -> U33(isNatList(activate(V))) 446.63/291.60 U33(tt) -> tt 446.63/291.60 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 446.63/291.60 U45(tt, V2) -> U46(isNatIList(activate(V2))) 446.63/291.60 U46(tt) -> tt 446.63/291.60 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 446.63/291.60 U52(tt) -> tt 446.63/291.60 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 446.63/291.60 U62(tt) -> tt 446.63/291.60 U71(tt) -> tt 446.63/291.60 U81(tt) -> tt 446.63/291.60 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 446.63/291.60 U95(tt, V2) -> U96(isNatList(activate(V2))) 446.63/291.60 U96(tt) -> tt 446.63/291.60 isNat(n__0) -> tt 446.63/291.60 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 isNatIList(n__zeros) -> tt 446.63/291.60 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatIListKind(n__nil) -> tt 446.63/291.60 isNatIListKind(n__zeros) -> tt 446.63/291.60 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatKind(n__0) -> tt 446.63/291.60 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 446.63/291.60 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 446.63/291.60 isNatList(n__nil) -> tt 446.63/291.60 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 length(nil) -> 0 446.63/291.60 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 446.63/291.60 take(0, IL) -> U121(isNatIList(IL), IL) 446.63/291.60 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 446.63/291.60 zeros -> n__zeros 446.63/291.60 take(X1, X2) -> n__take(X1, X2) 446.63/291.60 0 -> n__0 446.63/291.60 length(X) -> n__length(X) 446.63/291.60 s(X) -> n__s(X) 446.63/291.60 cons(X1, X2) -> n__cons(X1, X2) 446.63/291.60 nil -> n__nil 446.63/291.60 activate(n__zeros) -> zeros 446.63/291.60 activate(n__take(X1, X2)) -> take(X1, X2) 446.63/291.60 activate(n__0) -> 0 446.63/291.60 activate(n__length(X)) -> length(X) 446.63/291.60 activate(n__s(X)) -> s(X) 446.63/291.60 activate(n__cons(X1, X2)) -> cons(X1, X2) 446.63/291.60 activate(n__nil) -> nil 446.63/291.60 activate(X) -> X 446.63/291.60 446.63/291.60 S is empty. 446.63/291.60 Rewrite Strategy: FULL 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (6) LowerBoundPropagationProof (FINISHED) 446.63/291.60 Propagated lower bound. 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (7) 446.63/291.60 BOUNDS(n^1, INF) 446.63/291.60 446.63/291.60 ---------------------------------------- 446.63/291.60 446.63/291.60 (8) 446.63/291.60 Obligation: 446.63/291.60 Analyzing the following TRS for decreasing loops: 446.63/291.60 446.63/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 446.63/291.60 446.63/291.60 446.63/291.60 The TRS R consists of the following rules: 446.63/291.60 446.63/291.60 zeros -> cons(0, n__zeros) 446.63/291.60 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 446.63/291.60 U105(tt, V2) -> U106(isNatIList(activate(V2))) 446.63/291.60 U106(tt) -> tt 446.63/291.60 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 446.63/291.60 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 446.63/291.60 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 446.63/291.60 U114(tt, L) -> s(length(activate(L))) 446.63/291.60 U12(tt, V1) -> U13(isNatList(activate(V1))) 446.63/291.60 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 446.63/291.60 U122(tt) -> nil 446.63/291.60 U13(tt) -> tt 446.63/291.60 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 446.63/291.60 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 446.63/291.60 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 446.63/291.60 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 446.63/291.60 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 U22(tt, V1) -> U23(isNat(activate(V1))) 446.63/291.60 U23(tt) -> tt 446.63/291.60 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 U32(tt, V) -> U33(isNatList(activate(V))) 446.63/291.60 U33(tt) -> tt 446.63/291.60 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 446.63/291.60 U45(tt, V2) -> U46(isNatIList(activate(V2))) 446.63/291.60 U46(tt) -> tt 446.63/291.60 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 446.63/291.60 U52(tt) -> tt 446.63/291.60 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 446.63/291.60 U62(tt) -> tt 446.63/291.60 U71(tt) -> tt 446.63/291.60 U81(tt) -> tt 446.63/291.60 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 446.63/291.60 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 446.63/291.60 U95(tt, V2) -> U96(isNatList(activate(V2))) 446.63/291.60 U96(tt) -> tt 446.63/291.60 isNat(n__0) -> tt 446.63/291.60 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 446.63/291.60 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 446.63/291.60 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 446.63/291.60 isNatIList(n__zeros) -> tt 446.63/291.60 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatIListKind(n__nil) -> tt 446.63/291.60 isNatIListKind(n__zeros) -> tt 446.63/291.60 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 446.63/291.60 isNatKind(n__0) -> tt 446.63/291.60 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 446.63/291.60 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 446.63/291.60 isNatList(n__nil) -> tt 446.63/291.60 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 446.63/291.60 length(nil) -> 0 446.63/291.60 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 446.63/291.60 take(0, IL) -> U121(isNatIList(IL), IL) 446.63/291.60 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 446.63/291.60 zeros -> n__zeros 446.63/291.60 take(X1, X2) -> n__take(X1, X2) 446.63/291.60 0 -> n__0 446.63/291.60 length(X) -> n__length(X) 446.63/291.60 s(X) -> n__s(X) 446.63/291.60 cons(X1, X2) -> n__cons(X1, X2) 446.63/291.60 nil -> n__nil 446.63/291.60 activate(n__zeros) -> zeros 446.63/291.60 activate(n__take(X1, X2)) -> take(X1, X2) 446.63/291.60 activate(n__0) -> 0 446.63/291.60 activate(n__length(X)) -> length(X) 446.63/291.60 activate(n__s(X)) -> s(X) 446.63/291.60 activate(n__cons(X1, X2)) -> cons(X1, X2) 446.63/291.60 activate(n__nil) -> nil 446.63/291.60 activate(X) -> X 446.63/291.60 446.63/291.60 S is empty. 446.63/291.60 Rewrite Strategy: FULL 446.70/291.64 EOF