303.51/291.51 WORST_CASE(Omega(n^1), ?) 303.51/291.52 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 303.51/291.52 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 303.51/291.52 303.51/291.52 303.51/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.51/291.52 303.51/291.52 (0) CpxTRS 303.51/291.52 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 303.51/291.52 (2) TRS for Loop Detection 303.51/291.52 (3) DecreasingLoopProof [LOWER BOUND(ID), 51 ms] 303.51/291.52 (4) BEST 303.51/291.52 (5) proven lower bound 303.51/291.52 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 303.51/291.52 (7) BOUNDS(n^1, INF) 303.51/291.52 (8) TRS for Loop Detection 303.51/291.52 303.51/291.52 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (0) 303.51/291.52 Obligation: 303.51/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.51/291.52 303.51/291.52 303.51/291.52 The TRS R consists of the following rules: 303.51/291.52 303.51/291.52 zeros -> cons(0, n__zeros) 303.51/291.52 U11(tt) -> tt 303.51/291.52 U21(tt) -> tt 303.51/291.52 U31(tt) -> tt 303.51/291.52 U41(tt, V2) -> U42(isNatIList(activate(V2))) 303.51/291.52 U42(tt) -> tt 303.51/291.52 U51(tt, V2) -> U52(isNatList(activate(V2))) 303.51/291.52 U52(tt) -> tt 303.51/291.52 U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) 303.51/291.52 U62(tt, L) -> s(length(activate(L))) 303.51/291.52 isNat(n__0) -> tt 303.51/291.52 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 303.51/291.52 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 303.51/291.52 isNatIList(V) -> U31(isNatList(activate(V))) 303.51/291.52 isNatIList(n__zeros) -> tt 303.51/291.52 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 303.51/291.52 isNatList(n__nil) -> tt 303.51/291.52 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 303.51/291.52 length(nil) -> 0 303.51/291.52 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) 303.51/291.52 zeros -> n__zeros 303.51/291.52 0 -> n__0 303.51/291.52 length(X) -> n__length(X) 303.51/291.52 s(X) -> n__s(X) 303.51/291.52 cons(X1, X2) -> n__cons(X1, X2) 303.51/291.52 nil -> n__nil 303.51/291.52 activate(n__zeros) -> zeros 303.51/291.52 activate(n__0) -> 0 303.51/291.52 activate(n__length(X)) -> length(X) 303.51/291.52 activate(n__s(X)) -> s(X) 303.51/291.52 activate(n__cons(X1, X2)) -> cons(X1, X2) 303.51/291.52 activate(n__nil) -> nil 303.51/291.52 activate(X) -> X 303.51/291.52 303.51/291.52 S is empty. 303.51/291.52 Rewrite Strategy: FULL 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 303.51/291.52 Transformed a relative TRS into a decreasing-loop problem. 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (2) 303.51/291.52 Obligation: 303.51/291.52 Analyzing the following TRS for decreasing loops: 303.51/291.52 303.51/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.51/291.52 303.51/291.52 303.51/291.52 The TRS R consists of the following rules: 303.51/291.52 303.51/291.52 zeros -> cons(0, n__zeros) 303.51/291.52 U11(tt) -> tt 303.51/291.52 U21(tt) -> tt 303.51/291.52 U31(tt) -> tt 303.51/291.52 U41(tt, V2) -> U42(isNatIList(activate(V2))) 303.51/291.52 U42(tt) -> tt 303.51/291.52 U51(tt, V2) -> U52(isNatList(activate(V2))) 303.51/291.52 U52(tt) -> tt 303.51/291.52 U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) 303.51/291.52 U62(tt, L) -> s(length(activate(L))) 303.51/291.52 isNat(n__0) -> tt 303.51/291.52 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 303.51/291.52 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 303.51/291.52 isNatIList(V) -> U31(isNatList(activate(V))) 303.51/291.52 isNatIList(n__zeros) -> tt 303.51/291.52 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 303.51/291.52 isNatList(n__nil) -> tt 303.51/291.52 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 303.51/291.52 length(nil) -> 0 303.51/291.52 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) 303.51/291.52 zeros -> n__zeros 303.51/291.52 0 -> n__0 303.51/291.52 length(X) -> n__length(X) 303.51/291.52 s(X) -> n__s(X) 303.51/291.52 cons(X1, X2) -> n__cons(X1, X2) 303.51/291.52 nil -> n__nil 303.51/291.52 activate(n__zeros) -> zeros 303.51/291.52 activate(n__0) -> 0 303.51/291.52 activate(n__length(X)) -> length(X) 303.51/291.52 activate(n__s(X)) -> s(X) 303.51/291.52 activate(n__cons(X1, X2)) -> cons(X1, X2) 303.51/291.52 activate(n__nil) -> nil 303.51/291.52 activate(X) -> X 303.51/291.52 303.51/291.52 S is empty. 303.51/291.52 Rewrite Strategy: FULL 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (3) DecreasingLoopProof (LOWER BOUND(ID)) 303.51/291.52 The following loop(s) give(s) rise to the lower bound Omega(n^1): 303.51/291.52 303.51/291.52 The rewrite sequence 303.51/291.52 303.51/291.52 isNat(n__s(V1)) ->^+ U21(isNat(V1)) 303.51/291.52 303.51/291.52 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 303.51/291.52 303.51/291.52 The pumping substitution is [V1 / n__s(V1)]. 303.51/291.52 303.51/291.52 The result substitution is [ ]. 303.51/291.52 303.51/291.52 303.51/291.52 303.51/291.52 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (4) 303.51/291.52 Complex Obligation (BEST) 303.51/291.52 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (5) 303.51/291.52 Obligation: 303.51/291.52 Proved the lower bound n^1 for the following obligation: 303.51/291.52 303.51/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.51/291.52 303.51/291.52 303.51/291.52 The TRS R consists of the following rules: 303.51/291.52 303.51/291.52 zeros -> cons(0, n__zeros) 303.51/291.52 U11(tt) -> tt 303.51/291.52 U21(tt) -> tt 303.51/291.52 U31(tt) -> tt 303.51/291.52 U41(tt, V2) -> U42(isNatIList(activate(V2))) 303.51/291.52 U42(tt) -> tt 303.51/291.52 U51(tt, V2) -> U52(isNatList(activate(V2))) 303.51/291.52 U52(tt) -> tt 303.51/291.52 U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) 303.51/291.52 U62(tt, L) -> s(length(activate(L))) 303.51/291.52 isNat(n__0) -> tt 303.51/291.52 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 303.51/291.52 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 303.51/291.52 isNatIList(V) -> U31(isNatList(activate(V))) 303.51/291.52 isNatIList(n__zeros) -> tt 303.51/291.52 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 303.51/291.52 isNatList(n__nil) -> tt 303.51/291.52 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 303.51/291.52 length(nil) -> 0 303.51/291.52 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) 303.51/291.52 zeros -> n__zeros 303.51/291.52 0 -> n__0 303.51/291.52 length(X) -> n__length(X) 303.51/291.52 s(X) -> n__s(X) 303.51/291.52 cons(X1, X2) -> n__cons(X1, X2) 303.51/291.52 nil -> n__nil 303.51/291.52 activate(n__zeros) -> zeros 303.51/291.52 activate(n__0) -> 0 303.51/291.52 activate(n__length(X)) -> length(X) 303.51/291.52 activate(n__s(X)) -> s(X) 303.51/291.52 activate(n__cons(X1, X2)) -> cons(X1, X2) 303.51/291.52 activate(n__nil) -> nil 303.51/291.52 activate(X) -> X 303.51/291.52 303.51/291.52 S is empty. 303.51/291.52 Rewrite Strategy: FULL 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (6) LowerBoundPropagationProof (FINISHED) 303.51/291.52 Propagated lower bound. 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (7) 303.51/291.52 BOUNDS(n^1, INF) 303.51/291.52 303.51/291.52 ---------------------------------------- 303.51/291.52 303.51/291.52 (8) 303.51/291.52 Obligation: 303.51/291.52 Analyzing the following TRS for decreasing loops: 303.51/291.52 303.51/291.52 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 303.51/291.52 303.51/291.52 303.51/291.52 The TRS R consists of the following rules: 303.51/291.52 303.51/291.52 zeros -> cons(0, n__zeros) 303.51/291.52 U11(tt) -> tt 303.51/291.52 U21(tt) -> tt 303.51/291.52 U31(tt) -> tt 303.51/291.52 U41(tt, V2) -> U42(isNatIList(activate(V2))) 303.51/291.52 U42(tt) -> tt 303.51/291.52 U51(tt, V2) -> U52(isNatList(activate(V2))) 303.51/291.52 U52(tt) -> tt 303.51/291.52 U61(tt, L, N) -> U62(isNat(activate(N)), activate(L)) 303.51/291.52 U62(tt, L) -> s(length(activate(L))) 303.51/291.52 isNat(n__0) -> tt 303.51/291.52 isNat(n__length(V1)) -> U11(isNatList(activate(V1))) 303.51/291.52 isNat(n__s(V1)) -> U21(isNat(activate(V1))) 303.51/291.52 isNatIList(V) -> U31(isNatList(activate(V))) 303.51/291.52 isNatIList(n__zeros) -> tt 303.51/291.52 isNatIList(n__cons(V1, V2)) -> U41(isNat(activate(V1)), activate(V2)) 303.51/291.52 isNatList(n__nil) -> tt 303.51/291.52 isNatList(n__cons(V1, V2)) -> U51(isNat(activate(V1)), activate(V2)) 303.51/291.52 length(nil) -> 0 303.51/291.52 length(cons(N, L)) -> U61(isNatList(activate(L)), activate(L), N) 303.51/291.52 zeros -> n__zeros 303.51/291.52 0 -> n__0 303.51/291.52 length(X) -> n__length(X) 303.51/291.52 s(X) -> n__s(X) 303.51/291.52 cons(X1, X2) -> n__cons(X1, X2) 303.51/291.52 nil -> n__nil 303.51/291.52 activate(n__zeros) -> zeros 303.51/291.52 activate(n__0) -> 0 303.51/291.52 activate(n__length(X)) -> length(X) 303.51/291.52 activate(n__s(X)) -> s(X) 303.51/291.52 activate(n__cons(X1, X2)) -> cons(X1, X2) 303.51/291.52 activate(n__nil) -> nil 303.51/291.52 activate(X) -> X 303.51/291.52 303.51/291.52 S is empty. 303.51/291.52 Rewrite Strategy: FULL 303.59/291.56 EOF