317.25/291.50 WORST_CASE(Omega(n^1), ?) 317.25/291.50 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 317.25/291.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 317.25/291.50 317.25/291.50 317.25/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.25/291.50 317.25/291.50 (0) CpxTRS 317.25/291.50 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 317.25/291.50 (2) TRS for Loop Detection 317.25/291.50 (3) DecreasingLoopProof [LOWER BOUND(ID), 166 ms] 317.25/291.50 (4) BEST 317.25/291.50 (5) proven lower bound 317.25/291.50 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 317.25/291.50 (7) BOUNDS(n^1, INF) 317.25/291.50 (8) TRS for Loop Detection 317.25/291.50 317.25/291.50 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (0) 317.25/291.50 Obligation: 317.25/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.25/291.50 317.25/291.50 317.25/291.50 The TRS R consists of the following rules: 317.25/291.50 317.25/291.50 zeros -> cons(0, n__zeros) 317.25/291.50 U11(tt, V1) -> U12(isNatList(activate(V1))) 317.25/291.50 U12(tt) -> tt 317.25/291.50 U21(tt, V1) -> U22(isNat(activate(V1))) 317.25/291.50 U22(tt) -> tt 317.25/291.50 U31(tt, V) -> U32(isNatList(activate(V))) 317.25/291.50 U32(tt) -> tt 317.25/291.50 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 317.25/291.50 U42(tt, V2) -> U43(isNatIList(activate(V2))) 317.25/291.50 U43(tt) -> tt 317.25/291.50 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 317.25/291.50 U52(tt, V2) -> U53(isNatList(activate(V2))) 317.25/291.50 U53(tt) -> tt 317.25/291.50 U61(tt, L) -> s(length(activate(L))) 317.25/291.50 and(tt, X) -> activate(X) 317.25/291.50 isNat(n__0) -> tt 317.25/291.50 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 317.25/291.50 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.25/291.50 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 317.25/291.50 isNatIList(n__zeros) -> tt 317.25/291.50 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 isNatIListKind(n__nil) -> tt 317.25/291.50 isNatIListKind(n__zeros) -> tt 317.25/291.50 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 317.25/291.50 isNatKind(n__0) -> tt 317.25/291.50 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 317.25/291.50 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 317.25/291.50 isNatList(n__nil) -> tt 317.25/291.50 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 length(nil) -> 0 317.25/291.50 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) 317.25/291.50 zeros -> n__zeros 317.25/291.50 0 -> n__0 317.25/291.50 length(X) -> n__length(X) 317.25/291.50 s(X) -> n__s(X) 317.25/291.50 cons(X1, X2) -> n__cons(X1, X2) 317.25/291.50 isNatIListKind(X) -> n__isNatIListKind(X) 317.25/291.50 nil -> n__nil 317.25/291.50 and(X1, X2) -> n__and(X1, X2) 317.25/291.50 isNatKind(X) -> n__isNatKind(X) 317.25/291.50 activate(n__zeros) -> zeros 317.25/291.50 activate(n__0) -> 0 317.25/291.50 activate(n__length(X)) -> length(X) 317.25/291.50 activate(n__s(X)) -> s(X) 317.25/291.50 activate(n__cons(X1, X2)) -> cons(X1, X2) 317.25/291.50 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 317.25/291.50 activate(n__nil) -> nil 317.25/291.50 activate(n__and(X1, X2)) -> and(X1, X2) 317.25/291.50 activate(n__isNatKind(X)) -> isNatKind(X) 317.25/291.50 activate(X) -> X 317.25/291.50 317.25/291.50 S is empty. 317.25/291.50 Rewrite Strategy: FULL 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 317.25/291.50 Transformed a relative TRS into a decreasing-loop problem. 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (2) 317.25/291.50 Obligation: 317.25/291.50 Analyzing the following TRS for decreasing loops: 317.25/291.50 317.25/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.25/291.50 317.25/291.50 317.25/291.50 The TRS R consists of the following rules: 317.25/291.50 317.25/291.50 zeros -> cons(0, n__zeros) 317.25/291.50 U11(tt, V1) -> U12(isNatList(activate(V1))) 317.25/291.50 U12(tt) -> tt 317.25/291.50 U21(tt, V1) -> U22(isNat(activate(V1))) 317.25/291.50 U22(tt) -> tt 317.25/291.50 U31(tt, V) -> U32(isNatList(activate(V))) 317.25/291.50 U32(tt) -> tt 317.25/291.50 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 317.25/291.50 U42(tt, V2) -> U43(isNatIList(activate(V2))) 317.25/291.50 U43(tt) -> tt 317.25/291.50 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 317.25/291.50 U52(tt, V2) -> U53(isNatList(activate(V2))) 317.25/291.50 U53(tt) -> tt 317.25/291.50 U61(tt, L) -> s(length(activate(L))) 317.25/291.50 and(tt, X) -> activate(X) 317.25/291.50 isNat(n__0) -> tt 317.25/291.50 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 317.25/291.50 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.25/291.50 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 317.25/291.50 isNatIList(n__zeros) -> tt 317.25/291.50 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 isNatIListKind(n__nil) -> tt 317.25/291.50 isNatIListKind(n__zeros) -> tt 317.25/291.50 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 317.25/291.50 isNatKind(n__0) -> tt 317.25/291.50 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 317.25/291.50 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 317.25/291.50 isNatList(n__nil) -> tt 317.25/291.50 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 length(nil) -> 0 317.25/291.50 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) 317.25/291.50 zeros -> n__zeros 317.25/291.50 0 -> n__0 317.25/291.50 length(X) -> n__length(X) 317.25/291.50 s(X) -> n__s(X) 317.25/291.50 cons(X1, X2) -> n__cons(X1, X2) 317.25/291.50 isNatIListKind(X) -> n__isNatIListKind(X) 317.25/291.50 nil -> n__nil 317.25/291.50 and(X1, X2) -> n__and(X1, X2) 317.25/291.50 isNatKind(X) -> n__isNatKind(X) 317.25/291.50 activate(n__zeros) -> zeros 317.25/291.50 activate(n__0) -> 0 317.25/291.50 activate(n__length(X)) -> length(X) 317.25/291.50 activate(n__s(X)) -> s(X) 317.25/291.50 activate(n__cons(X1, X2)) -> cons(X1, X2) 317.25/291.50 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 317.25/291.50 activate(n__nil) -> nil 317.25/291.50 activate(n__and(X1, X2)) -> and(X1, X2) 317.25/291.50 activate(n__isNatKind(X)) -> isNatKind(X) 317.25/291.50 activate(X) -> X 317.25/291.50 317.25/291.50 S is empty. 317.25/291.50 Rewrite Strategy: FULL 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (3) DecreasingLoopProof (LOWER BOUND(ID)) 317.25/291.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 317.25/291.50 317.25/291.50 The rewrite sequence 317.25/291.50 317.25/291.50 isNatKind(n__length(n__isNatKind(X1_0))) ->^+ isNatIListKind(isNatKind(X1_0)) 317.25/291.50 317.25/291.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 317.25/291.50 317.25/291.50 The pumping substitution is [X1_0 / n__length(n__isNatKind(X1_0))]. 317.25/291.50 317.25/291.50 The result substitution is [ ]. 317.25/291.50 317.25/291.50 317.25/291.50 317.25/291.50 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (4) 317.25/291.50 Complex Obligation (BEST) 317.25/291.50 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (5) 317.25/291.50 Obligation: 317.25/291.50 Proved the lower bound n^1 for the following obligation: 317.25/291.50 317.25/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.25/291.50 317.25/291.50 317.25/291.50 The TRS R consists of the following rules: 317.25/291.50 317.25/291.50 zeros -> cons(0, n__zeros) 317.25/291.50 U11(tt, V1) -> U12(isNatList(activate(V1))) 317.25/291.50 U12(tt) -> tt 317.25/291.50 U21(tt, V1) -> U22(isNat(activate(V1))) 317.25/291.50 U22(tt) -> tt 317.25/291.50 U31(tt, V) -> U32(isNatList(activate(V))) 317.25/291.50 U32(tt) -> tt 317.25/291.50 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 317.25/291.50 U42(tt, V2) -> U43(isNatIList(activate(V2))) 317.25/291.50 U43(tt) -> tt 317.25/291.50 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 317.25/291.50 U52(tt, V2) -> U53(isNatList(activate(V2))) 317.25/291.50 U53(tt) -> tt 317.25/291.50 U61(tt, L) -> s(length(activate(L))) 317.25/291.50 and(tt, X) -> activate(X) 317.25/291.50 isNat(n__0) -> tt 317.25/291.50 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 317.25/291.50 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.25/291.50 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 317.25/291.50 isNatIList(n__zeros) -> tt 317.25/291.50 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 isNatIListKind(n__nil) -> tt 317.25/291.50 isNatIListKind(n__zeros) -> tt 317.25/291.50 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 317.25/291.50 isNatKind(n__0) -> tt 317.25/291.50 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 317.25/291.50 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 317.25/291.50 isNatList(n__nil) -> tt 317.25/291.50 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 length(nil) -> 0 317.25/291.50 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) 317.25/291.50 zeros -> n__zeros 317.25/291.50 0 -> n__0 317.25/291.50 length(X) -> n__length(X) 317.25/291.50 s(X) -> n__s(X) 317.25/291.50 cons(X1, X2) -> n__cons(X1, X2) 317.25/291.50 isNatIListKind(X) -> n__isNatIListKind(X) 317.25/291.50 nil -> n__nil 317.25/291.50 and(X1, X2) -> n__and(X1, X2) 317.25/291.50 isNatKind(X) -> n__isNatKind(X) 317.25/291.50 activate(n__zeros) -> zeros 317.25/291.50 activate(n__0) -> 0 317.25/291.50 activate(n__length(X)) -> length(X) 317.25/291.50 activate(n__s(X)) -> s(X) 317.25/291.50 activate(n__cons(X1, X2)) -> cons(X1, X2) 317.25/291.50 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 317.25/291.50 activate(n__nil) -> nil 317.25/291.50 activate(n__and(X1, X2)) -> and(X1, X2) 317.25/291.50 activate(n__isNatKind(X)) -> isNatKind(X) 317.25/291.50 activate(X) -> X 317.25/291.50 317.25/291.50 S is empty. 317.25/291.50 Rewrite Strategy: FULL 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (6) LowerBoundPropagationProof (FINISHED) 317.25/291.50 Propagated lower bound. 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (7) 317.25/291.50 BOUNDS(n^1, INF) 317.25/291.50 317.25/291.50 ---------------------------------------- 317.25/291.50 317.25/291.50 (8) 317.25/291.50 Obligation: 317.25/291.50 Analyzing the following TRS for decreasing loops: 317.25/291.50 317.25/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 317.25/291.50 317.25/291.50 317.25/291.50 The TRS R consists of the following rules: 317.25/291.50 317.25/291.50 zeros -> cons(0, n__zeros) 317.25/291.50 U11(tt, V1) -> U12(isNatList(activate(V1))) 317.25/291.50 U12(tt) -> tt 317.25/291.50 U21(tt, V1) -> U22(isNat(activate(V1))) 317.25/291.50 U22(tt) -> tt 317.25/291.50 U31(tt, V) -> U32(isNatList(activate(V))) 317.25/291.50 U32(tt) -> tt 317.25/291.50 U41(tt, V1, V2) -> U42(isNat(activate(V1)), activate(V2)) 317.25/291.50 U42(tt, V2) -> U43(isNatIList(activate(V2))) 317.25/291.50 U43(tt) -> tt 317.25/291.50 U51(tt, V1, V2) -> U52(isNat(activate(V1)), activate(V2)) 317.25/291.50 U52(tt, V2) -> U53(isNatList(activate(V2))) 317.25/291.50 U53(tt) -> tt 317.25/291.50 U61(tt, L) -> s(length(activate(L))) 317.25/291.50 and(tt, X) -> activate(X) 317.25/291.50 isNat(n__0) -> tt 317.25/291.50 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 317.25/291.50 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 317.25/291.50 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 317.25/291.50 isNatIList(n__zeros) -> tt 317.25/291.50 isNatIList(n__cons(V1, V2)) -> U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 isNatIListKind(n__nil) -> tt 317.25/291.50 isNatIListKind(n__zeros) -> tt 317.25/291.50 isNatIListKind(n__cons(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))) 317.25/291.50 isNatKind(n__0) -> tt 317.25/291.50 isNatKind(n__length(V1)) -> isNatIListKind(activate(V1)) 317.25/291.50 isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 317.25/291.50 isNatList(n__nil) -> tt 317.25/291.50 isNatList(n__cons(V1, V2)) -> U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2)) 317.25/291.50 length(nil) -> 0 317.25/291.50 length(cons(N, L)) -> U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L)) 317.25/291.50 zeros -> n__zeros 317.25/291.50 0 -> n__0 317.25/291.50 length(X) -> n__length(X) 317.25/291.50 s(X) -> n__s(X) 317.25/291.50 cons(X1, X2) -> n__cons(X1, X2) 317.25/291.50 isNatIListKind(X) -> n__isNatIListKind(X) 317.25/291.50 nil -> n__nil 317.25/291.50 and(X1, X2) -> n__and(X1, X2) 317.25/291.50 isNatKind(X) -> n__isNatKind(X) 317.25/291.50 activate(n__zeros) -> zeros 317.25/291.50 activate(n__0) -> 0 317.25/291.50 activate(n__length(X)) -> length(X) 317.25/291.50 activate(n__s(X)) -> s(X) 317.25/291.50 activate(n__cons(X1, X2)) -> cons(X1, X2) 317.25/291.50 activate(n__isNatIListKind(X)) -> isNatIListKind(X) 317.25/291.50 activate(n__nil) -> nil 317.25/291.50 activate(n__and(X1, X2)) -> and(X1, X2) 317.25/291.50 activate(n__isNatKind(X)) -> isNatKind(X) 317.25/291.50 activate(X) -> X 317.25/291.50 317.25/291.50 S is empty. 317.25/291.50 Rewrite Strategy: FULL 317.25/291.54 EOF