305.76/291.55 WORST_CASE(Omega(n^1), ?) 305.76/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 305.76/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 305.76/291.56 305.76/291.56 305.76/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.76/291.56 305.76/291.56 (0) CpxTRS 305.76/291.56 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 305.76/291.56 (2) TRS for Loop Detection 305.76/291.56 (3) DecreasingLoopProof [LOWER BOUND(ID), 200 ms] 305.76/291.56 (4) BEST 305.76/291.56 (5) proven lower bound 305.76/291.56 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 305.76/291.56 (7) BOUNDS(n^1, INF) 305.76/291.56 (8) TRS for Loop Detection 305.76/291.56 305.76/291.56 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (0) 305.76/291.56 Obligation: 305.76/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.76/291.56 305.76/291.56 305.76/291.56 The TRS R consists of the following rules: 305.76/291.56 305.76/291.56 zeros -> cons(0, n__zeros) 305.76/291.56 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 U12(tt, V1) -> U13(isNatList(activate(V1))) 305.76/291.56 U13(tt) -> tt 305.76/291.56 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 U22(tt, V1) -> U23(isNat(activate(V1))) 305.76/291.56 U23(tt) -> tt 305.76/291.56 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 U32(tt, V) -> U33(isNatList(activate(V))) 305.76/291.56 U33(tt) -> tt 305.76/291.56 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 305.76/291.56 U45(tt, V2) -> U46(isNatIList(activate(V2))) 305.76/291.56 U46(tt) -> tt 305.76/291.56 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 305.76/291.56 U52(tt) -> tt 305.76/291.56 U61(tt) -> tt 305.76/291.56 U71(tt) -> tt 305.76/291.56 U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) 305.76/291.56 U85(tt, V2) -> U86(isNatList(activate(V2))) 305.76/291.56 U86(tt) -> tt 305.76/291.56 U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) 305.76/291.56 U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) 305.76/291.56 U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) 305.76/291.56 U94(tt, L) -> s(length(activate(L))) 305.76/291.56 isNat(n__0) -> tt 305.76/291.56 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 isNatIList(n__zeros) -> tt 305.76/291.56 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 isNatIListKind(n__nil) -> tt 305.76/291.56 isNatIListKind(n__zeros) -> tt 305.76/291.56 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 305.76/291.56 isNatKind(n__0) -> tt 305.76/291.56 isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) 305.76/291.56 isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) 305.76/291.56 isNatList(n__nil) -> tt 305.76/291.56 isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 length(nil) -> 0 305.76/291.56 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) 305.76/291.56 zeros -> n__zeros 305.76/291.56 0 -> n__0 305.76/291.56 length(X) -> n__length(X) 305.76/291.56 s(X) -> n__s(X) 305.76/291.56 cons(X1, X2) -> n__cons(X1, X2) 305.76/291.56 nil -> n__nil 305.76/291.56 activate(n__zeros) -> zeros 305.76/291.56 activate(n__0) -> 0 305.76/291.56 activate(n__length(X)) -> length(X) 305.76/291.56 activate(n__s(X)) -> s(X) 305.76/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 305.76/291.56 activate(n__nil) -> nil 305.76/291.56 activate(X) -> X 305.76/291.56 305.76/291.56 S is empty. 305.76/291.56 Rewrite Strategy: FULL 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 305.76/291.56 Transformed a relative TRS into a decreasing-loop problem. 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (2) 305.76/291.56 Obligation: 305.76/291.56 Analyzing the following TRS for decreasing loops: 305.76/291.56 305.76/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.76/291.56 305.76/291.56 305.76/291.56 The TRS R consists of the following rules: 305.76/291.56 305.76/291.56 zeros -> cons(0, n__zeros) 305.76/291.56 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 U12(tt, V1) -> U13(isNatList(activate(V1))) 305.76/291.56 U13(tt) -> tt 305.76/291.56 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 U22(tt, V1) -> U23(isNat(activate(V1))) 305.76/291.56 U23(tt) -> tt 305.76/291.56 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 U32(tt, V) -> U33(isNatList(activate(V))) 305.76/291.56 U33(tt) -> tt 305.76/291.56 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 305.76/291.56 U45(tt, V2) -> U46(isNatIList(activate(V2))) 305.76/291.56 U46(tt) -> tt 305.76/291.56 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 305.76/291.56 U52(tt) -> tt 305.76/291.56 U61(tt) -> tt 305.76/291.56 U71(tt) -> tt 305.76/291.56 U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) 305.76/291.56 U85(tt, V2) -> U86(isNatList(activate(V2))) 305.76/291.56 U86(tt) -> tt 305.76/291.56 U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) 305.76/291.56 U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) 305.76/291.56 U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) 305.76/291.56 U94(tt, L) -> s(length(activate(L))) 305.76/291.56 isNat(n__0) -> tt 305.76/291.56 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 isNatIList(n__zeros) -> tt 305.76/291.56 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 isNatIListKind(n__nil) -> tt 305.76/291.56 isNatIListKind(n__zeros) -> tt 305.76/291.56 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 305.76/291.56 isNatKind(n__0) -> tt 305.76/291.56 isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) 305.76/291.56 isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) 305.76/291.56 isNatList(n__nil) -> tt 305.76/291.56 isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 length(nil) -> 0 305.76/291.56 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) 305.76/291.56 zeros -> n__zeros 305.76/291.56 0 -> n__0 305.76/291.56 length(X) -> n__length(X) 305.76/291.56 s(X) -> n__s(X) 305.76/291.56 cons(X1, X2) -> n__cons(X1, X2) 305.76/291.56 nil -> n__nil 305.76/291.56 activate(n__zeros) -> zeros 305.76/291.56 activate(n__0) -> 0 305.76/291.56 activate(n__length(X)) -> length(X) 305.76/291.56 activate(n__s(X)) -> s(X) 305.76/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 305.76/291.56 activate(n__nil) -> nil 305.76/291.56 activate(X) -> X 305.76/291.56 305.76/291.56 S is empty. 305.76/291.56 Rewrite Strategy: FULL 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (3) DecreasingLoopProof (LOWER BOUND(ID)) 305.76/291.56 The following loop(s) give(s) rise to the lower bound Omega(n^1): 305.76/291.56 305.76/291.56 The rewrite sequence 305.76/291.56 305.76/291.56 isNatKind(n__s(V1)) ->^+ U71(isNatKind(V1)) 305.76/291.56 305.76/291.56 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 305.76/291.56 305.76/291.56 The pumping substitution is [V1 / n__s(V1)]. 305.76/291.56 305.76/291.56 The result substitution is [ ]. 305.76/291.56 305.76/291.56 305.76/291.56 305.76/291.56 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (4) 305.76/291.56 Complex Obligation (BEST) 305.76/291.56 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (5) 305.76/291.56 Obligation: 305.76/291.56 Proved the lower bound n^1 for the following obligation: 305.76/291.56 305.76/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.76/291.56 305.76/291.56 305.76/291.56 The TRS R consists of the following rules: 305.76/291.56 305.76/291.56 zeros -> cons(0, n__zeros) 305.76/291.56 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 U12(tt, V1) -> U13(isNatList(activate(V1))) 305.76/291.56 U13(tt) -> tt 305.76/291.56 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 U22(tt, V1) -> U23(isNat(activate(V1))) 305.76/291.56 U23(tt) -> tt 305.76/291.56 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 U32(tt, V) -> U33(isNatList(activate(V))) 305.76/291.56 U33(tt) -> tt 305.76/291.56 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 305.76/291.56 U45(tt, V2) -> U46(isNatIList(activate(V2))) 305.76/291.56 U46(tt) -> tt 305.76/291.56 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 305.76/291.56 U52(tt) -> tt 305.76/291.56 U61(tt) -> tt 305.76/291.56 U71(tt) -> tt 305.76/291.56 U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) 305.76/291.56 U85(tt, V2) -> U86(isNatList(activate(V2))) 305.76/291.56 U86(tt) -> tt 305.76/291.56 U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) 305.76/291.56 U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) 305.76/291.56 U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) 305.76/291.56 U94(tt, L) -> s(length(activate(L))) 305.76/291.56 isNat(n__0) -> tt 305.76/291.56 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 isNatIList(n__zeros) -> tt 305.76/291.56 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 isNatIListKind(n__nil) -> tt 305.76/291.56 isNatIListKind(n__zeros) -> tt 305.76/291.56 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 305.76/291.56 isNatKind(n__0) -> tt 305.76/291.56 isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) 305.76/291.56 isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) 305.76/291.56 isNatList(n__nil) -> tt 305.76/291.56 isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 length(nil) -> 0 305.76/291.56 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) 305.76/291.56 zeros -> n__zeros 305.76/291.56 0 -> n__0 305.76/291.56 length(X) -> n__length(X) 305.76/291.56 s(X) -> n__s(X) 305.76/291.56 cons(X1, X2) -> n__cons(X1, X2) 305.76/291.56 nil -> n__nil 305.76/291.56 activate(n__zeros) -> zeros 305.76/291.56 activate(n__0) -> 0 305.76/291.56 activate(n__length(X)) -> length(X) 305.76/291.56 activate(n__s(X)) -> s(X) 305.76/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 305.76/291.56 activate(n__nil) -> nil 305.76/291.56 activate(X) -> X 305.76/291.56 305.76/291.56 S is empty. 305.76/291.56 Rewrite Strategy: FULL 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (6) LowerBoundPropagationProof (FINISHED) 305.76/291.56 Propagated lower bound. 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (7) 305.76/291.56 BOUNDS(n^1, INF) 305.76/291.56 305.76/291.56 ---------------------------------------- 305.76/291.56 305.76/291.56 (8) 305.76/291.56 Obligation: 305.76/291.56 Analyzing the following TRS for decreasing loops: 305.76/291.56 305.76/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 305.76/291.56 305.76/291.56 305.76/291.56 The TRS R consists of the following rules: 305.76/291.56 305.76/291.56 zeros -> cons(0, n__zeros) 305.76/291.56 U11(tt, V1) -> U12(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 U12(tt, V1) -> U13(isNatList(activate(V1))) 305.76/291.56 U13(tt) -> tt 305.76/291.56 U21(tt, V1) -> U22(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 U22(tt, V1) -> U23(isNat(activate(V1))) 305.76/291.56 U23(tt) -> tt 305.76/291.56 U31(tt, V) -> U32(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 U32(tt, V) -> U33(isNatList(activate(V))) 305.76/291.56 U33(tt) -> tt 305.76/291.56 U41(tt, V1, V2) -> U42(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U42(tt, V1, V2) -> U43(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U43(tt, V1, V2) -> U44(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U44(tt, V1, V2) -> U45(isNat(activate(V1)), activate(V2)) 305.76/291.56 U45(tt, V2) -> U46(isNatIList(activate(V2))) 305.76/291.56 U46(tt) -> tt 305.76/291.56 U51(tt, V2) -> U52(isNatIListKind(activate(V2))) 305.76/291.56 U52(tt) -> tt 305.76/291.56 U61(tt) -> tt 305.76/291.56 U71(tt) -> tt 305.76/291.56 U81(tt, V1, V2) -> U82(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 U82(tt, V1, V2) -> U83(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U83(tt, V1, V2) -> U84(isNatIListKind(activate(V2)), activate(V1), activate(V2)) 305.76/291.56 U84(tt, V1, V2) -> U85(isNat(activate(V1)), activate(V2)) 305.76/291.56 U85(tt, V2) -> U86(isNatList(activate(V2))) 305.76/291.56 U86(tt) -> tt 305.76/291.56 U91(tt, L, N) -> U92(isNatIListKind(activate(L)), activate(L), activate(N)) 305.76/291.56 U92(tt, L, N) -> U93(isNat(activate(N)), activate(L), activate(N)) 305.76/291.56 U93(tt, L, N) -> U94(isNatKind(activate(N)), activate(L)) 305.76/291.56 U94(tt, L) -> s(length(activate(L))) 305.76/291.56 isNat(n__0) -> tt 305.76/291.56 isNat(n__length(V1)) -> U11(isNatIListKind(activate(V1)), activate(V1)) 305.76/291.56 isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) 305.76/291.56 isNatIList(V) -> U31(isNatIListKind(activate(V)), activate(V)) 305.76/291.56 isNatIList(n__zeros) -> tt 305.76/291.56 isNatIList(n__cons(V1, V2)) -> U41(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 isNatIListKind(n__nil) -> tt 305.76/291.56 isNatIListKind(n__zeros) -> tt 305.76/291.56 isNatIListKind(n__cons(V1, V2)) -> U51(isNatKind(activate(V1)), activate(V2)) 305.76/291.56 isNatKind(n__0) -> tt 305.76/291.56 isNatKind(n__length(V1)) -> U61(isNatIListKind(activate(V1))) 305.76/291.56 isNatKind(n__s(V1)) -> U71(isNatKind(activate(V1))) 305.76/291.56 isNatList(n__nil) -> tt 305.76/291.56 isNatList(n__cons(V1, V2)) -> U81(isNatKind(activate(V1)), activate(V1), activate(V2)) 305.76/291.56 length(nil) -> 0 305.76/291.56 length(cons(N, L)) -> U91(isNatList(activate(L)), activate(L), N) 305.76/291.56 zeros -> n__zeros 305.76/291.56 0 -> n__0 305.76/291.56 length(X) -> n__length(X) 305.76/291.56 s(X) -> n__s(X) 305.76/291.56 cons(X1, X2) -> n__cons(X1, X2) 305.76/291.56 nil -> n__nil 305.76/291.56 activate(n__zeros) -> zeros 305.76/291.56 activate(n__0) -> 0 305.76/291.56 activate(n__length(X)) -> length(X) 305.76/291.56 activate(n__s(X)) -> s(X) 305.76/291.56 activate(n__cons(X1, X2)) -> cons(X1, X2) 305.76/291.56 activate(n__nil) -> nil 305.76/291.56 activate(X) -> X 305.76/291.56 305.76/291.56 S is empty. 305.76/291.56 Rewrite Strategy: FULL 305.85/291.59 EOF