8.69/3.08 WORST_CASE(NON_POLY, ?) 8.69/3.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 8.69/3.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 8.69/3.08 8.69/3.08 8.69/3.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.69/3.08 8.69/3.08 (0) CpxTRS 8.69/3.08 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 8.69/3.08 (2) TRS for Loop Detection 8.69/3.08 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 8.69/3.08 (4) BEST 8.69/3.08 (5) proven lower bound 8.69/3.08 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 8.69/3.08 (7) BOUNDS(n^1, INF) 8.69/3.08 (8) TRS for Loop Detection 8.69/3.08 (9) DecreasingLoopProof [FINISHED, 947 ms] 8.69/3.08 (10) BOUNDS(EXP, INF) 8.69/3.08 8.69/3.08 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (0) 8.69/3.08 Obligation: 8.69/3.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.69/3.08 8.69/3.08 8.69/3.08 The TRS R consists of the following rules: 8.69/3.08 8.69/3.08 zeros -> cons(0, n__zeros) 8.69/3.08 U11(tt) -> tt 8.69/3.08 U21(tt) -> tt 8.69/3.08 U31(tt) -> tt 8.69/3.08 U41(tt, V2) -> U42(isNatIList(activate(V2))) 8.69/3.08 U42(tt) -> tt 8.69/3.08 U51(tt, V2) -> U52(isNatList(activate(V2))) 8.69/3.08 U52(tt) -> tt 8.69/3.08 U61(tt, V2) -> U62(isNatIList(activate(V2))) 8.69/3.08 U62(tt) -> tt 8.69/3.08 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 8.69/3.08 U72(tt, L) -> s(length(activate(L))) 8.69/3.08 U81(tt) -> nil 8.69/3.08 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 8.69/3.08 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 8.69/3.08 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 8.69/3.08 isNat(n__0) -> tt 8.69/3.08 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 8.69/3.08 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 8.69/3.08 isNatIList(V) -> U31(isNatList(activate(V))) 8.69/3.08 isNatIList(n__zeros) -> tt 8.69/3.08 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 8.69/3.08 isNatList(n__nil) -> tt 8.69/3.08 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 8.69/3.08 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 8.69/3.08 length(nil) -> 0 8.69/3.08 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 8.69/3.08 take(0, IL) -> U81(isNatIList(IL)) 8.69/3.08 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 8.69/3.08 zeros -> n__zeros 8.69/3.08 take(X1, X2) -> n__take(X1, X2) 8.69/3.08 0 -> n__0 8.69/3.08 length(X) -> n__length(X) 8.69/3.08 s(X) -> n__s(X) 8.69/3.08 cons(X1, X2) -> n__cons(X1, X2) 8.69/3.08 nil -> n__nil 8.69/3.08 activate(n__zeros) -> zeros 8.69/3.08 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 8.69/3.08 activate(n__0) -> 0 8.69/3.08 activate(n__length(X)) -> length(activate(X)) 8.69/3.08 activate(n__s(X)) -> s(activate(X)) 8.69/3.08 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 8.69/3.08 activate(n__nil) -> nil 8.69/3.08 activate(X) -> X 8.69/3.08 8.69/3.08 S is empty. 8.69/3.08 Rewrite Strategy: FULL 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 8.69/3.08 Transformed a relative TRS into a decreasing-loop problem. 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (2) 8.69/3.08 Obligation: 8.69/3.08 Analyzing the following TRS for decreasing loops: 8.69/3.08 8.69/3.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.69/3.08 8.69/3.08 8.69/3.08 The TRS R consists of the following rules: 8.69/3.08 8.69/3.08 zeros -> cons(0, n__zeros) 8.69/3.08 U11(tt) -> tt 8.69/3.08 U21(tt) -> tt 8.69/3.08 U31(tt) -> tt 8.69/3.08 U41(tt, V2) -> U42(isNatIList(activate(V2))) 8.69/3.08 U42(tt) -> tt 8.69/3.08 U51(tt, V2) -> U52(isNatList(activate(V2))) 8.69/3.08 U52(tt) -> tt 8.69/3.08 U61(tt, V2) -> U62(isNatIList(activate(V2))) 8.69/3.08 U62(tt) -> tt 8.69/3.08 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 8.69/3.08 U72(tt, L) -> s(length(activate(L))) 8.69/3.08 U81(tt) -> nil 8.69/3.08 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 8.69/3.08 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 8.69/3.08 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 8.69/3.08 isNat(n__0) -> tt 8.69/3.08 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 8.69/3.08 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 8.69/3.08 isNatIList(V) -> U31(isNatList(activate(V))) 8.69/3.08 isNatIList(n__zeros) -> tt 8.69/3.08 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 8.69/3.08 isNatList(n__nil) -> tt 8.69/3.08 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 8.69/3.08 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 8.69/3.08 length(nil) -> 0 8.69/3.08 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 8.69/3.08 take(0, IL) -> U81(isNatIList(IL)) 8.69/3.08 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 8.69/3.08 zeros -> n__zeros 8.69/3.08 take(X1, X2) -> n__take(X1, X2) 8.69/3.08 0 -> n__0 8.69/3.08 length(X) -> n__length(X) 8.69/3.08 s(X) -> n__s(X) 8.69/3.08 cons(X1, X2) -> n__cons(X1, X2) 8.69/3.08 nil -> n__nil 8.69/3.08 activate(n__zeros) -> zeros 8.69/3.08 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 8.69/3.08 activate(n__0) -> 0 8.69/3.08 activate(n__length(X)) -> length(activate(X)) 8.69/3.08 activate(n__s(X)) -> s(activate(X)) 8.69/3.08 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 8.69/3.08 activate(n__nil) -> nil 8.69/3.08 activate(X) -> X 8.69/3.08 8.69/3.08 S is empty. 8.69/3.08 Rewrite Strategy: FULL 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (3) DecreasingLoopProof (LOWER BOUND(ID)) 8.69/3.08 The following loop(s) give(s) rise to the lower bound Omega(n^1): 8.69/3.08 8.69/3.08 The rewrite sequence 8.69/3.08 8.69/3.08 activate(n__s(X)) ->^+ s(activate(X)) 8.69/3.08 8.69/3.08 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 8.69/3.08 8.69/3.08 The pumping substitution is [X / n__s(X)]. 8.69/3.08 8.69/3.08 The result substitution is [ ]. 8.69/3.08 8.69/3.08 8.69/3.08 8.69/3.08 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (4) 8.69/3.08 Complex Obligation (BEST) 8.69/3.08 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (5) 8.69/3.08 Obligation: 8.69/3.08 Proved the lower bound n^1 for the following obligation: 8.69/3.08 8.69/3.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.69/3.08 8.69/3.08 8.69/3.08 The TRS R consists of the following rules: 8.69/3.08 8.69/3.08 zeros -> cons(0, n__zeros) 8.69/3.08 U11(tt) -> tt 8.69/3.08 U21(tt) -> tt 8.69/3.08 U31(tt) -> tt 8.69/3.08 U41(tt, V2) -> U42(isNatIList(activate(V2))) 8.69/3.08 U42(tt) -> tt 8.69/3.08 U51(tt, V2) -> U52(isNatList(activate(V2))) 8.69/3.08 U52(tt) -> tt 8.69/3.08 U61(tt, V2) -> U62(isNatIList(activate(V2))) 8.69/3.08 U62(tt) -> tt 8.69/3.08 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 8.69/3.08 U72(tt, L) -> s(length(activate(L))) 8.69/3.08 U81(tt) -> nil 8.69/3.08 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 8.69/3.08 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 8.69/3.08 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 8.69/3.08 isNat(n__0) -> tt 8.69/3.08 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 8.69/3.08 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 8.69/3.08 isNatIList(V) -> U31(isNatList(activate(V))) 8.69/3.08 isNatIList(n__zeros) -> tt 8.69/3.08 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 8.69/3.08 isNatList(n__nil) -> tt 8.69/3.08 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 8.69/3.08 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 8.69/3.08 length(nil) -> 0 8.69/3.08 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 8.69/3.08 take(0, IL) -> U81(isNatIList(IL)) 8.69/3.08 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 8.69/3.08 zeros -> n__zeros 8.69/3.08 take(X1, X2) -> n__take(X1, X2) 8.69/3.08 0 -> n__0 8.69/3.08 length(X) -> n__length(X) 8.69/3.08 s(X) -> n__s(X) 8.69/3.08 cons(X1, X2) -> n__cons(X1, X2) 8.69/3.08 nil -> n__nil 8.69/3.08 activate(n__zeros) -> zeros 8.69/3.08 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 8.69/3.08 activate(n__0) -> 0 8.69/3.08 activate(n__length(X)) -> length(activate(X)) 8.69/3.08 activate(n__s(X)) -> s(activate(X)) 8.69/3.08 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 8.69/3.08 activate(n__nil) -> nil 8.69/3.08 activate(X) -> X 8.69/3.08 8.69/3.08 S is empty. 8.69/3.08 Rewrite Strategy: FULL 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (6) LowerBoundPropagationProof (FINISHED) 8.69/3.08 Propagated lower bound. 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (7) 8.69/3.08 BOUNDS(n^1, INF) 8.69/3.08 8.69/3.08 ---------------------------------------- 8.69/3.08 8.69/3.08 (8) 8.69/3.08 Obligation: 8.69/3.08 Analyzing the following TRS for decreasing loops: 8.69/3.08 8.69/3.08 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 8.69/3.08 8.69/3.08 8.69/3.08 The TRS R consists of the following rules: 8.69/3.08 8.69/3.08 zeros -> cons(0, n__zeros) 8.69/3.08 U11(tt) -> tt 8.69/3.08 U21(tt) -> tt 8.69/3.08 U31(tt) -> tt 8.69/3.08 U41(tt, V2) -> U42(isNatIList(activate(V2))) 8.69/3.08 U42(tt) -> tt 8.69/3.08 U51(tt, V2) -> U52(isNatList(activate(V2))) 8.69/3.08 U52(tt) -> tt 8.69/3.08 U61(tt, V2) -> U62(isNatIList(activate(V2))) 8.69/3.08 U62(tt) -> tt 8.69/3.08 U71(tt, L, N) -> U72(isNat(activate(N)), activate(L)) 8.69/3.08 U72(tt, L) -> s(length(activate(L))) 8.69/3.08 U81(tt) -> nil 8.69/3.08 U91(tt, IL, M, N) -> U92(isNat(activate(M)), activate(IL), activate(M), activate(N)) 8.69/3.09 U92(tt, IL, M, N) -> U93(isNat(activate(N)), activate(IL), activate(M), activate(N)) 8.69/3.09 U93(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) 8.69/3.09 isNat(n__0) -> tt 8.69/3.09 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 8.69/3.09 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 8.69/3.09 isNatIList(V) -> U31(isNatList(activate(V))) 8.69/3.09 isNatIList(n__zeros) -> tt 8.69/3.09 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 8.69/3.09 isNatList(n__nil) -> tt 8.69/3.09 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 8.69/3.09 isNatList(n__take(V1, V2)) -> U61(isNat(activate(V1)), activate(V2)) 8.69/3.09 length(nil) -> 0 8.69/3.09 length(cons(N, L)) -> U71(isNatList(activate(L)), activate(L), N) 8.69/3.09 take(0, IL) -> U81(isNatIList(IL)) 8.69/3.09 take(s(M), cons(N, IL)) -> U91(isNatIList(activate(IL)), activate(IL), M, N) 8.69/3.09 zeros -> n__zeros 8.69/3.09 take(X1, X2) -> n__take(X1, X2) 8.69/3.09 0 -> n__0 8.69/3.09 length(X) -> n__length(X) 8.69/3.09 s(X) -> n__s(X) 8.69/3.09 cons(X1, X2) -> n__cons(X1, X2) 8.69/3.09 nil -> n__nil 8.69/3.09 activate(n__zeros) -> zeros 8.69/3.09 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 8.69/3.09 activate(n__0) -> 0 8.69/3.09 activate(n__length(X)) -> length(activate(X)) 8.69/3.09 activate(n__s(X)) -> s(activate(X)) 8.69/3.09 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 8.69/3.09 activate(n__nil) -> nil 8.69/3.09 activate(X) -> X 8.69/3.09 8.69/3.09 S is empty. 8.69/3.09 Rewrite Strategy: FULL 8.69/3.09 ---------------------------------------- 8.69/3.09 8.69/3.09 (9) DecreasingLoopProof (FINISHED) 8.69/3.09 The following loop(s) give(s) rise to the lower bound EXP: 8.69/3.09 8.69/3.09 The rewrite sequence 8.69/3.09 8.69/3.09 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U71(isNatList(activate(X22_0)), activate(X22_0), activate(X11_0)) 8.69/3.09 8.69/3.09 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 8.69/3.09 8.69/3.09 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 8.69/3.09 8.69/3.09 The result substitution is [ ]. 8.69/3.09 8.69/3.09 8.69/3.09 8.69/3.09 The rewrite sequence 8.69/3.09 8.69/3.09 activate(n__length(n__cons(X11_0, X22_0))) ->^+ U71(isNatList(activate(X22_0)), activate(X22_0), activate(X11_0)) 8.69/3.09 8.69/3.09 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 8.69/3.09 8.69/3.09 The pumping substitution is [X22_0 / n__length(n__cons(X11_0, X22_0))]. 8.69/3.09 8.69/3.09 The result substitution is [ ]. 8.69/3.09 8.69/3.09 8.69/3.09 8.69/3.09 8.69/3.09 ---------------------------------------- 8.69/3.09 8.69/3.09 (10) 8.69/3.09 BOUNDS(EXP, INF) 8.98/3.13 EOF