17.52/5.89 WORST_CASE(NON_POLY, ?) 17.52/5.90 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 17.52/5.90 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 17.52/5.90 17.52/5.90 17.52/5.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.52/5.90 17.52/5.90 (0) CpxTRS 17.52/5.90 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 17.52/5.90 (2) TRS for Loop Detection 17.52/5.90 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 17.52/5.90 (4) BEST 17.52/5.90 (5) proven lower bound 17.52/5.90 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 17.52/5.90 (7) BOUNDS(n^1, INF) 17.52/5.90 (8) TRS for Loop Detection 17.52/5.90 (9) DecreasingLoopProof [FINISHED, 3252 ms] 17.52/5.90 (10) BOUNDS(EXP, INF) 17.52/5.90 17.52/5.90 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (0) 17.52/5.90 Obligation: 17.52/5.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.52/5.90 17.52/5.90 17.52/5.90 The TRS R consists of the following rules: 17.52/5.90 17.52/5.90 zeros -> cons(0, n__zeros) 17.52/5.90 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 17.52/5.90 U105(tt, V2) -> U106(isNatIList(activate(V2))) 17.52/5.90 U106(tt) -> tt 17.52/5.90 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 17.52/5.90 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 17.52/5.90 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 17.52/5.90 U114(tt, L) -> s(length(activate(L))) 17.52/5.90 U12(tt, V1) -> U13(isNatList(activate(V1))) 17.52/5.90 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 17.52/5.90 U122(tt) -> nil 17.52/5.90 U13(tt) -> tt 17.52/5.90 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 17.52/5.90 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 17.52/5.90 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 U22(tt, V1) -> U23(isNat(activate(V1))) 17.52/5.90 U23(tt) -> tt 17.52/5.90 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 U32(tt, V) -> U33(isNatList(activate(V))) 17.52/5.90 U33(tt) -> tt 17.52/5.90 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 17.52/5.90 U45(tt, V2) -> U46(isNatIList(activate(V2))) 17.52/5.90 U46(tt) -> tt 17.52/5.90 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 17.52/5.90 U52(tt) -> tt 17.52/5.90 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 17.52/5.90 U62(tt) -> tt 17.52/5.90 U71(tt) -> tt 17.52/5.90 U81(tt) -> tt 17.52/5.90 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 17.52/5.90 U95(tt, V2) -> U96(isNatList(activate(V2))) 17.52/5.90 U96(tt) -> tt 17.52/5.90 isNat(n__0) -> tt 17.52/5.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 isNatIList(n__zeros) -> tt 17.52/5.90 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatIListKind(n__nil) -> tt 17.52/5.90 isNatIListKind(n__zeros) -> tt 17.52/5.90 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatKind(n__0) -> tt 17.52/5.90 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 17.52/5.90 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 17.52/5.90 isNatList(n__nil) -> tt 17.52/5.90 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 length(nil) -> 0 17.52/5.90 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 17.52/5.90 take(0, IL) -> U121(isNatIList(IL), IL) 17.52/5.90 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 17.52/5.90 zeros -> n__zeros 17.52/5.90 take(X1, X2) -> n__take(X1, X2) 17.52/5.90 0 -> n__0 17.52/5.90 length(X) -> n__length(X) 17.52/5.90 s(X) -> n__s(X) 17.52/5.90 cons(X1, X2) -> n__cons(X1, X2) 17.52/5.90 nil -> n__nil 17.52/5.90 activate(n__zeros) -> zeros 17.52/5.90 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 17.52/5.90 activate(n__0) -> 0 17.52/5.90 activate(n__length(X)) -> length(activate(X)) 17.52/5.90 activate(n__s(X)) -> s(activate(X)) 17.52/5.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 17.52/5.90 activate(n__nil) -> nil 17.52/5.90 activate(X) -> X 17.52/5.90 17.52/5.90 S is empty. 17.52/5.90 Rewrite Strategy: FULL 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 17.52/5.90 Transformed a relative TRS into a decreasing-loop problem. 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (2) 17.52/5.90 Obligation: 17.52/5.90 Analyzing the following TRS for decreasing loops: 17.52/5.90 17.52/5.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.52/5.90 17.52/5.90 17.52/5.90 The TRS R consists of the following rules: 17.52/5.90 17.52/5.90 zeros -> cons(0, n__zeros) 17.52/5.90 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 17.52/5.90 U105(tt, V2) -> U106(isNatIList(activate(V2))) 17.52/5.90 U106(tt) -> tt 17.52/5.90 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 17.52/5.90 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 17.52/5.90 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 17.52/5.90 U114(tt, L) -> s(length(activate(L))) 17.52/5.90 U12(tt, V1) -> U13(isNatList(activate(V1))) 17.52/5.90 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 17.52/5.90 U122(tt) -> nil 17.52/5.90 U13(tt) -> tt 17.52/5.90 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 17.52/5.90 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 17.52/5.90 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 U22(tt, V1) -> U23(isNat(activate(V1))) 17.52/5.90 U23(tt) -> tt 17.52/5.90 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 U32(tt, V) -> U33(isNatList(activate(V))) 17.52/5.90 U33(tt) -> tt 17.52/5.90 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 17.52/5.90 U45(tt, V2) -> U46(isNatIList(activate(V2))) 17.52/5.90 U46(tt) -> tt 17.52/5.90 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 17.52/5.90 U52(tt) -> tt 17.52/5.90 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 17.52/5.90 U62(tt) -> tt 17.52/5.90 U71(tt) -> tt 17.52/5.90 U81(tt) -> tt 17.52/5.90 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 17.52/5.90 U95(tt, V2) -> U96(isNatList(activate(V2))) 17.52/5.90 U96(tt) -> tt 17.52/5.90 isNat(n__0) -> tt 17.52/5.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 isNatIList(n__zeros) -> tt 17.52/5.90 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatIListKind(n__nil) -> tt 17.52/5.90 isNatIListKind(n__zeros) -> tt 17.52/5.90 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatKind(n__0) -> tt 17.52/5.90 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 17.52/5.90 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 17.52/5.90 isNatList(n__nil) -> tt 17.52/5.90 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 length(nil) -> 0 17.52/5.90 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 17.52/5.90 take(0, IL) -> U121(isNatIList(IL), IL) 17.52/5.90 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 17.52/5.90 zeros -> n__zeros 17.52/5.90 take(X1, X2) -> n__take(X1, X2) 17.52/5.90 0 -> n__0 17.52/5.90 length(X) -> n__length(X) 17.52/5.90 s(X) -> n__s(X) 17.52/5.90 cons(X1, X2) -> n__cons(X1, X2) 17.52/5.90 nil -> n__nil 17.52/5.90 activate(n__zeros) -> zeros 17.52/5.90 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 17.52/5.90 activate(n__0) -> 0 17.52/5.90 activate(n__length(X)) -> length(activate(X)) 17.52/5.90 activate(n__s(X)) -> s(activate(X)) 17.52/5.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 17.52/5.90 activate(n__nil) -> nil 17.52/5.90 activate(X) -> X 17.52/5.90 17.52/5.90 S is empty. 17.52/5.90 Rewrite Strategy: FULL 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (3) DecreasingLoopProof (LOWER BOUND(ID)) 17.52/5.90 The following loop(s) give(s) rise to the lower bound Omega(n^1): 17.52/5.90 17.52/5.90 The rewrite sequence 17.52/5.90 17.52/5.90 activate(n__s(X)) ->^+ s(activate(X)) 17.52/5.90 17.52/5.90 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 17.52/5.90 17.52/5.90 The pumping substitution is [X / n__s(X)]. 17.52/5.90 17.52/5.90 The result substitution is [ ]. 17.52/5.90 17.52/5.90 17.52/5.90 17.52/5.90 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (4) 17.52/5.90 Complex Obligation (BEST) 17.52/5.90 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (5) 17.52/5.90 Obligation: 17.52/5.90 Proved the lower bound n^1 for the following obligation: 17.52/5.90 17.52/5.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.52/5.90 17.52/5.90 17.52/5.90 The TRS R consists of the following rules: 17.52/5.90 17.52/5.90 zeros -> cons(0, n__zeros) 17.52/5.90 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 17.52/5.90 U105(tt, V2) -> U106(isNatIList(activate(V2))) 17.52/5.90 U106(tt) -> tt 17.52/5.90 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 17.52/5.90 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 17.52/5.90 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 17.52/5.90 U114(tt, L) -> s(length(activate(L))) 17.52/5.90 U12(tt, V1) -> U13(isNatList(activate(V1))) 17.52/5.90 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 17.52/5.90 U122(tt) -> nil 17.52/5.90 U13(tt) -> tt 17.52/5.90 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 17.52/5.90 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 17.52/5.90 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 U22(tt, V1) -> U23(isNat(activate(V1))) 17.52/5.90 U23(tt) -> tt 17.52/5.90 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 U32(tt, V) -> U33(isNatList(activate(V))) 17.52/5.90 U33(tt) -> tt 17.52/5.90 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 17.52/5.90 U45(tt, V2) -> U46(isNatIList(activate(V2))) 17.52/5.90 U46(tt) -> tt 17.52/5.90 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 17.52/5.90 U52(tt) -> tt 17.52/5.90 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 17.52/5.90 U62(tt) -> tt 17.52/5.90 U71(tt) -> tt 17.52/5.90 U81(tt) -> tt 17.52/5.90 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 17.52/5.90 U95(tt, V2) -> U96(isNatList(activate(V2))) 17.52/5.90 U96(tt) -> tt 17.52/5.90 isNat(n__0) -> tt 17.52/5.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 isNatIList(n__zeros) -> tt 17.52/5.90 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatIListKind(n__nil) -> tt 17.52/5.90 isNatIListKind(n__zeros) -> tt 17.52/5.90 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatKind(n__0) -> tt 17.52/5.90 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 17.52/5.90 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 17.52/5.90 isNatList(n__nil) -> tt 17.52/5.90 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 length(nil) -> 0 17.52/5.90 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 17.52/5.90 take(0, IL) -> U121(isNatIList(IL), IL) 17.52/5.90 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 17.52/5.90 zeros -> n__zeros 17.52/5.90 take(X1, X2) -> n__take(X1, X2) 17.52/5.90 0 -> n__0 17.52/5.90 length(X) -> n__length(X) 17.52/5.90 s(X) -> n__s(X) 17.52/5.90 cons(X1, X2) -> n__cons(X1, X2) 17.52/5.90 nil -> n__nil 17.52/5.90 activate(n__zeros) -> zeros 17.52/5.90 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 17.52/5.90 activate(n__0) -> 0 17.52/5.90 activate(n__length(X)) -> length(activate(X)) 17.52/5.90 activate(n__s(X)) -> s(activate(X)) 17.52/5.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 17.52/5.90 activate(n__nil) -> nil 17.52/5.90 activate(X) -> X 17.52/5.90 17.52/5.90 S is empty. 17.52/5.90 Rewrite Strategy: FULL 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (6) LowerBoundPropagationProof (FINISHED) 17.52/5.90 Propagated lower bound. 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (7) 17.52/5.90 BOUNDS(n^1, INF) 17.52/5.90 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (8) 17.52/5.90 Obligation: 17.52/5.90 Analyzing the following TRS for decreasing loops: 17.52/5.90 17.52/5.90 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 17.52/5.90 17.52/5.90 17.52/5.90 The TRS R consists of the following rules: 17.52/5.90 17.52/5.90 zeros -> cons(0, n__zeros) 17.52/5.90 U101(tt, V1, V2) -> U102(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U102(tt, V1, V2) -> U103(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U103(tt, V1, V2) -> U104(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U104(tt, V1, V2) -> U105(isNat(activate(V1)), activate(V2)) 17.52/5.90 U105(tt, V2) -> U106(isNatIList(activate(V2))) 17.52/5.90 U106(tt) -> tt 17.52/5.90 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 U111(tt, L, N) -> U112(isNatIListKind(activate(L)), activate(L), activate(N)) 17.52/5.90 U112(tt, L, N) -> U113(isNat(activate(N)), activate(L), activate(N)) 17.52/5.90 U113(tt, L, N) -> U114(isNatKind(activate(N)), activate(L)) 17.52/5.90 U114(tt, L) -> s(length(activate(L))) 17.52/5.90 U12(tt, V1) -> U13(isNatList(activate(V1))) 17.52/5.90 U121(tt, IL) -> U122(isNatIListKind(activate(IL))) 17.52/5.90 U122(tt) -> nil 17.52/5.90 U13(tt) -> tt 17.52/5.90 U131(tt, IL, M, N) -> U132(isNatIListKind(activate(IL)), activate(IL), activate(M), activate(N)) 17.52/5.90 U132(tt, IL, M, N) -> U133(isNat(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U133(tt, IL, M, N) -> U134(isNatKind(activate(M)), activate(IL), activate(M), activate(N)) 17.52/5.90 U134(tt, IL, M, N) -> U135(isNat(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U135(tt, IL, M, N) -> U136(isNatKind(activate(N)), activate(IL), activate(M), activate(N)) 17.52/5.90 U136(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 17.52/5.90 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 U22(tt, V1) -> U23(isNat(activate(V1))) 17.52/5.90 U23(tt) -> tt 17.52/5.90 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 U32(tt, V) -> U33(isNatList(activate(V))) 17.52/5.90 U33(tt) -> tt 17.52/5.90 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 17.52/5.90 U45(tt, V2) -> U46(isNatIList(activate(V2))) 17.52/5.90 U46(tt) -> tt 17.52/5.90 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 17.52/5.90 U52(tt) -> tt 17.52/5.90 U61(tt, V2) -> U62(isNatIListKind(activate(V2))) 17.52/5.90 U62(tt) -> tt 17.52/5.90 U71(tt) -> tt 17.52/5.90 U81(tt) -> tt 17.52/5.90 U91(tt, V1, V2) -> U92(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 U92(tt, V1, V2) -> U93(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U93(tt, V1, V2) -> U94(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 17.52/5.90 U94(tt, V1, V2) -> U95(isNat(activate(V1)), activate(V2)) 17.52/5.90 U95(tt, V2) -> U96(isNatList(activate(V2))) 17.52/5.90 U96(tt) -> tt 17.52/5.90 isNat(n__0) -> tt 17.52/5.90 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 17.52/5.90 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 17.52/5.90 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 17.52/5.90 isNatIList(n__zeros) -> tt 17.52/5.90 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatIListKind(n__nil) -> tt 17.52/5.90 isNatIListKind(n__zeros) -> tt 17.52/5.90 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatIListKind(n__take(V1, V2)) -> U61(isNatKind(activate(V1)), activate(V2)) 17.52/5.90 isNatKind(n__0) -> tt 17.52/5.90 isNatKind(n__length(V1)) -> U71(isNatIListKind(activate(V1))) 17.52/5.90 isNatKind(n__s(V1)) -> U81(isNatKind(activate(V1))) 17.52/5.90 isNatList(n__nil) -> tt 17.52/5.90 isNatList(n__cons(V1, V2)) -> U91(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 isNatList(n__take(V1, V2)) -> U101(isNatKind(activate(V1)), activate(V1), activate(V2)) 17.52/5.90 length(nil) -> 0 17.52/5.90 length(cons(N, L)) -> U111(isNatList(activate(L)), activate(L), N) 17.52/5.90 take(0, IL) -> U121(isNatIList(IL), IL) 17.52/5.90 take(s(M), cons(N, IL)) -> U131(isNatIList(activate(IL)), activate(IL), M, N) 17.52/5.90 zeros -> n__zeros 17.52/5.90 take(X1, X2) -> n__take(X1, X2) 17.52/5.90 0 -> n__0 17.52/5.90 length(X) -> n__length(X) 17.52/5.90 s(X) -> n__s(X) 17.52/5.90 cons(X1, X2) -> n__cons(X1, X2) 17.52/5.90 nil -> n__nil 17.52/5.90 activate(n__zeros) -> zeros 17.52/5.90 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 17.52/5.90 activate(n__0) -> 0 17.52/5.90 activate(n__length(X)) -> length(activate(X)) 17.52/5.90 activate(n__s(X)) -> s(activate(X)) 17.52/5.90 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 17.52/5.90 activate(n__nil) -> nil 17.52/5.90 activate(X) -> X 17.52/5.90 17.52/5.90 S is empty. 17.52/5.90 Rewrite Strategy: FULL 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (9) DecreasingLoopProof (FINISHED) 17.52/5.90 The following loop(s) give(s) rise to the lower bound EXP: 17.52/5.90 17.52/5.90 The rewrite sequence 17.52/5.90 17.52/5.90 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U111(isNatList(activate(X22_0)), activate(X22_0), activate(X11_0)) 17.52/5.90 17.52/5.90 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 17.52/5.90 17.52/5.90 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 17.52/5.90 17.52/5.90 The result substitution is [ ]. 17.52/5.90 17.52/5.90 17.52/5.90 17.52/5.90 The rewrite sequence 17.52/5.90 17.52/5.90 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U111(isNatList(activate(X22_0)), activate(X22_0), activate(X11_0)) 17.52/5.90 17.52/5.90 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 17.52/5.90 17.52/5.90 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 17.52/5.90 17.52/5.90 The result substitution is [ ]. 17.52/5.90 17.52/5.90 17.52/5.90 17.52/5.90 17.52/5.90 ---------------------------------------- 17.52/5.90 17.52/5.90 (10) 17.52/5.90 BOUNDS(EXP, INF) 17.52/5.96 EOF