625.50/291.61 WORST_CASE(Omega(n^1), ?) 625.50/291.62 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 625.50/291.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 625.50/291.62 625.50/291.62 625.50/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 625.50/291.62 625.50/291.62 (0) CpxTRS 625.50/291.62 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 625.50/291.62 (2) TRS for Loop Detection 625.50/291.62 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 625.50/291.62 (4) BEST 625.50/291.62 (5) proven lower bound 625.50/291.62 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 625.50/291.62 (7) BOUNDS(n^1, INF) 625.50/291.62 (8) TRS for Loop Detection 625.50/291.62 625.50/291.62 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (0) 625.50/291.62 Obligation: 625.50/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 625.50/291.62 625.50/291.62 625.50/291.62 The TRS R consists of the following rules: 625.50/291.62 625.50/291.62 a__zeros -> cons(0, zeros) 625.50/291.62 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 625.50/291.62 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 625.50/291.62 a__U13(tt) -> tt 625.50/291.62 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 625.50/291.62 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 625.50/291.62 a__U23(tt) -> tt 625.50/291.62 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 625.50/291.62 a__U32(tt, V) -> a__U33(a__isNatList(V)) 625.50/291.62 a__U33(tt) -> tt 625.50/291.62 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 625.50/291.62 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 625.50/291.62 a__U46(tt) -> tt 625.50/291.62 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 625.50/291.62 a__U52(tt) -> tt 625.50/291.62 a__U61(tt) -> tt 625.50/291.62 a__U71(tt) -> tt 625.50/291.62 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 625.50/291.62 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 625.50/291.62 a__U86(tt) -> tt 625.50/291.62 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 625.50/291.62 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 625.50/291.62 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 625.50/291.62 a__U94(tt, L) -> s(a__length(mark(L))) 625.50/291.62 a__isNat(0) -> tt 625.50/291.62 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 625.50/291.62 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 625.50/291.62 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 625.50/291.62 a__isNatIList(zeros) -> tt 625.50/291.62 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 625.50/291.62 a__isNatIListKind(nil) -> tt 625.50/291.62 a__isNatIListKind(zeros) -> tt 625.50/291.62 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 625.50/291.62 a__isNatKind(0) -> tt 625.50/291.62 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 625.50/291.62 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 625.50/291.62 a__isNatList(nil) -> tt 625.50/291.62 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 625.50/291.62 a__length(nil) -> 0 625.50/291.62 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 625.50/291.62 mark(zeros) -> a__zeros 625.50/291.62 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 625.50/291.62 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 625.50/291.62 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 625.50/291.62 mark(U13(X)) -> a__U13(mark(X)) 625.50/291.62 mark(isNatList(X)) -> a__isNatList(X) 625.50/291.62 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 625.50/291.62 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 625.50/291.62 mark(isNatKind(X)) -> a__isNatKind(X) 625.50/291.62 mark(U23(X)) -> a__U23(mark(X)) 625.50/291.62 mark(isNat(X)) -> a__isNat(X) 625.50/291.62 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 625.50/291.62 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 625.50/291.62 mark(U33(X)) -> a__U33(mark(X)) 625.50/291.62 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 625.50/291.62 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 625.50/291.62 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 625.50/291.62 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 625.50/291.62 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 625.50/291.62 mark(U46(X)) -> a__U46(mark(X)) 625.50/291.62 mark(isNatIList(X)) -> a__isNatIList(X) 625.50/291.62 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 625.50/291.62 mark(U52(X)) -> a__U52(mark(X)) 625.50/291.62 mark(U61(X)) -> a__U61(mark(X)) 625.50/291.62 mark(U71(X)) -> a__U71(mark(X)) 625.50/291.62 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 625.50/291.62 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 625.50/291.62 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 625.50/291.62 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 625.50/291.62 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 625.50/291.62 mark(U86(X)) -> a__U86(mark(X)) 625.50/291.62 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 625.50/291.62 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 625.50/291.62 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 625.50/291.62 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 625.50/291.62 mark(length(X)) -> a__length(mark(X)) 625.50/291.62 mark(cons(X1, X2)) -> cons(mark(X1), X2) 625.50/291.62 mark(0) -> 0 625.50/291.62 mark(tt) -> tt 625.50/291.62 mark(s(X)) -> s(mark(X)) 625.50/291.62 mark(nil) -> nil 625.50/291.62 a__zeros -> zeros 625.50/291.62 a__U11(X1, X2) -> U11(X1, X2) 625.50/291.62 a__U12(X1, X2) -> U12(X1, X2) 625.50/291.62 a__isNatIListKind(X) -> isNatIListKind(X) 625.50/291.62 a__U13(X) -> U13(X) 625.50/291.62 a__isNatList(X) -> isNatList(X) 625.50/291.62 a__U21(X1, X2) -> U21(X1, X2) 625.50/291.62 a__U22(X1, X2) -> U22(X1, X2) 625.50/291.62 a__isNatKind(X) -> isNatKind(X) 625.50/291.62 a__U23(X) -> U23(X) 625.50/291.62 a__isNat(X) -> isNat(X) 625.50/291.62 a__U31(X1, X2) -> U31(X1, X2) 625.50/291.62 a__U32(X1, X2) -> U32(X1, X2) 625.50/291.62 a__U33(X) -> U33(X) 625.50/291.62 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 625.50/291.62 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 625.50/291.62 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 625.50/291.62 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 625.50/291.62 a__U45(X1, X2) -> U45(X1, X2) 625.50/291.62 a__U46(X) -> U46(X) 625.50/291.62 a__isNatIList(X) -> isNatIList(X) 625.50/291.62 a__U51(X1, X2) -> U51(X1, X2) 625.50/291.62 a__U52(X) -> U52(X) 625.50/291.62 a__U61(X) -> U61(X) 625.50/291.62 a__U71(X) -> U71(X) 625.50/291.62 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 625.50/291.62 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 625.50/291.62 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 625.50/291.62 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 625.50/291.62 a__U85(X1, X2) -> U85(X1, X2) 625.50/291.62 a__U86(X) -> U86(X) 625.50/291.62 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 625.50/291.62 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 625.50/291.62 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 625.50/291.62 a__U94(X1, X2) -> U94(X1, X2) 625.50/291.62 a__length(X) -> length(X) 625.50/291.62 625.50/291.62 S is empty. 625.50/291.62 Rewrite Strategy: FULL 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 625.50/291.62 Transformed a relative TRS into a decreasing-loop problem. 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (2) 625.50/291.62 Obligation: 625.50/291.62 Analyzing the following TRS for decreasing loops: 625.50/291.62 625.50/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 625.50/291.62 625.50/291.62 625.50/291.62 The TRS R consists of the following rules: 625.50/291.62 625.50/291.62 a__zeros -> cons(0, zeros) 625.50/291.62 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 625.50/291.62 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 625.50/291.62 a__U13(tt) -> tt 625.50/291.62 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 625.50/291.62 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 625.50/291.62 a__U23(tt) -> tt 625.50/291.62 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 625.50/291.62 a__U32(tt, V) -> a__U33(a__isNatList(V)) 625.50/291.62 a__U33(tt) -> tt 625.50/291.62 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 625.50/291.62 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 625.50/291.62 a__U46(tt) -> tt 625.50/291.62 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 625.50/291.62 a__U52(tt) -> tt 625.50/291.62 a__U61(tt) -> tt 625.50/291.62 a__U71(tt) -> tt 625.50/291.62 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 625.50/291.62 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 625.50/291.62 a__U86(tt) -> tt 625.50/291.62 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 625.50/291.62 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 625.50/291.62 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 625.50/291.62 a__U94(tt, L) -> s(a__length(mark(L))) 625.50/291.62 a__isNat(0) -> tt 625.50/291.62 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 625.50/291.62 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 625.50/291.62 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 625.50/291.62 a__isNatIList(zeros) -> tt 625.50/291.62 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 625.50/291.62 a__isNatIListKind(nil) -> tt 625.50/291.62 a__isNatIListKind(zeros) -> tt 625.50/291.62 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 625.50/291.62 a__isNatKind(0) -> tt 625.50/291.62 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 625.50/291.62 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 625.50/291.62 a__isNatList(nil) -> tt 625.50/291.62 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 625.50/291.62 a__length(nil) -> 0 625.50/291.62 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 625.50/291.62 mark(zeros) -> a__zeros 625.50/291.62 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 625.50/291.62 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 625.50/291.62 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 625.50/291.62 mark(U13(X)) -> a__U13(mark(X)) 625.50/291.62 mark(isNatList(X)) -> a__isNatList(X) 625.50/291.62 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 625.50/291.62 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 625.50/291.62 mark(isNatKind(X)) -> a__isNatKind(X) 625.50/291.62 mark(U23(X)) -> a__U23(mark(X)) 625.50/291.62 mark(isNat(X)) -> a__isNat(X) 625.50/291.62 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 625.50/291.62 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 625.50/291.62 mark(U33(X)) -> a__U33(mark(X)) 625.50/291.62 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 625.50/291.62 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 625.50/291.62 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 625.50/291.62 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 625.50/291.62 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 625.50/291.62 mark(U46(X)) -> a__U46(mark(X)) 625.50/291.62 mark(isNatIList(X)) -> a__isNatIList(X) 625.50/291.62 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 625.50/291.62 mark(U52(X)) -> a__U52(mark(X)) 625.50/291.62 mark(U61(X)) -> a__U61(mark(X)) 625.50/291.62 mark(U71(X)) -> a__U71(mark(X)) 625.50/291.62 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 625.50/291.62 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 625.50/291.62 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 625.50/291.62 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 625.50/291.62 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 625.50/291.62 mark(U86(X)) -> a__U86(mark(X)) 625.50/291.62 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 625.50/291.62 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 625.50/291.62 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 625.50/291.62 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 625.50/291.62 mark(length(X)) -> a__length(mark(X)) 625.50/291.62 mark(cons(X1, X2)) -> cons(mark(X1), X2) 625.50/291.62 mark(0) -> 0 625.50/291.62 mark(tt) -> tt 625.50/291.62 mark(s(X)) -> s(mark(X)) 625.50/291.62 mark(nil) -> nil 625.50/291.62 a__zeros -> zeros 625.50/291.62 a__U11(X1, X2) -> U11(X1, X2) 625.50/291.62 a__U12(X1, X2) -> U12(X1, X2) 625.50/291.62 a__isNatIListKind(X) -> isNatIListKind(X) 625.50/291.62 a__U13(X) -> U13(X) 625.50/291.62 a__isNatList(X) -> isNatList(X) 625.50/291.62 a__U21(X1, X2) -> U21(X1, X2) 625.50/291.62 a__U22(X1, X2) -> U22(X1, X2) 625.50/291.62 a__isNatKind(X) -> isNatKind(X) 625.50/291.62 a__U23(X) -> U23(X) 625.50/291.62 a__isNat(X) -> isNat(X) 625.50/291.62 a__U31(X1, X2) -> U31(X1, X2) 625.50/291.62 a__U32(X1, X2) -> U32(X1, X2) 625.50/291.62 a__U33(X) -> U33(X) 625.50/291.62 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 625.50/291.62 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 625.50/291.62 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 625.50/291.62 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 625.50/291.62 a__U45(X1, X2) -> U45(X1, X2) 625.50/291.62 a__U46(X) -> U46(X) 625.50/291.62 a__isNatIList(X) -> isNatIList(X) 625.50/291.62 a__U51(X1, X2) -> U51(X1, X2) 625.50/291.62 a__U52(X) -> U52(X) 625.50/291.62 a__U61(X) -> U61(X) 625.50/291.62 a__U71(X) -> U71(X) 625.50/291.62 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 625.50/291.62 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 625.50/291.62 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 625.50/291.62 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 625.50/291.62 a__U85(X1, X2) -> U85(X1, X2) 625.50/291.62 a__U86(X) -> U86(X) 625.50/291.62 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 625.50/291.62 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 625.50/291.62 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 625.50/291.62 a__U94(X1, X2) -> U94(X1, X2) 625.50/291.62 a__length(X) -> length(X) 625.50/291.62 625.50/291.62 S is empty. 625.50/291.62 Rewrite Strategy: FULL 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (3) DecreasingLoopProof (LOWER BOUND(ID)) 625.50/291.62 The following loop(s) give(s) rise to the lower bound Omega(n^1): 625.50/291.62 625.50/291.62 The rewrite sequence 625.50/291.62 625.50/291.62 mark(U71(X)) ->^+ a__U71(mark(X)) 625.50/291.62 625.50/291.62 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 625.50/291.62 625.50/291.62 The pumping substitution is [X / U71(X)]. 625.50/291.62 625.50/291.62 The result substitution is [ ]. 625.50/291.62 625.50/291.62 625.50/291.62 625.50/291.62 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (4) 625.50/291.62 Complex Obligation (BEST) 625.50/291.62 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (5) 625.50/291.62 Obligation: 625.50/291.62 Proved the lower bound n^1 for the following obligation: 625.50/291.62 625.50/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 625.50/291.62 625.50/291.62 625.50/291.62 The TRS R consists of the following rules: 625.50/291.62 625.50/291.62 a__zeros -> cons(0, zeros) 625.50/291.62 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 625.50/291.62 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 625.50/291.62 a__U13(tt) -> tt 625.50/291.62 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 625.50/291.62 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 625.50/291.62 a__U23(tt) -> tt 625.50/291.62 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 625.50/291.62 a__U32(tt, V) -> a__U33(a__isNatList(V)) 625.50/291.62 a__U33(tt) -> tt 625.50/291.62 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 625.50/291.62 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 625.50/291.62 a__U46(tt) -> tt 625.50/291.62 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 625.50/291.62 a__U52(tt) -> tt 625.50/291.62 a__U61(tt) -> tt 625.50/291.62 a__U71(tt) -> tt 625.50/291.62 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 625.50/291.62 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 625.50/291.62 a__U86(tt) -> tt 625.50/291.62 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 625.50/291.62 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 625.50/291.62 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 625.50/291.62 a__U94(tt, L) -> s(a__length(mark(L))) 625.50/291.62 a__isNat(0) -> tt 625.50/291.62 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 625.50/291.62 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 625.50/291.62 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 625.50/291.62 a__isNatIList(zeros) -> tt 625.50/291.62 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 625.50/291.62 a__isNatIListKind(nil) -> tt 625.50/291.62 a__isNatIListKind(zeros) -> tt 625.50/291.62 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 625.50/291.62 a__isNatKind(0) -> tt 625.50/291.62 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 625.50/291.62 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 625.50/291.62 a__isNatList(nil) -> tt 625.50/291.62 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 625.50/291.62 a__length(nil) -> 0 625.50/291.62 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 625.50/291.62 mark(zeros) -> a__zeros 625.50/291.62 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 625.50/291.62 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 625.50/291.62 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 625.50/291.62 mark(U13(X)) -> a__U13(mark(X)) 625.50/291.62 mark(isNatList(X)) -> a__isNatList(X) 625.50/291.62 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 625.50/291.62 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 625.50/291.62 mark(isNatKind(X)) -> a__isNatKind(X) 625.50/291.62 mark(U23(X)) -> a__U23(mark(X)) 625.50/291.62 mark(isNat(X)) -> a__isNat(X) 625.50/291.62 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 625.50/291.62 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 625.50/291.62 mark(U33(X)) -> a__U33(mark(X)) 625.50/291.62 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 625.50/291.62 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 625.50/291.62 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 625.50/291.62 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 625.50/291.62 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 625.50/291.62 mark(U46(X)) -> a__U46(mark(X)) 625.50/291.62 mark(isNatIList(X)) -> a__isNatIList(X) 625.50/291.62 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 625.50/291.62 mark(U52(X)) -> a__U52(mark(X)) 625.50/291.62 mark(U61(X)) -> a__U61(mark(X)) 625.50/291.62 mark(U71(X)) -> a__U71(mark(X)) 625.50/291.62 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 625.50/291.62 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 625.50/291.62 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 625.50/291.62 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 625.50/291.62 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 625.50/291.62 mark(U86(X)) -> a__U86(mark(X)) 625.50/291.62 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 625.50/291.62 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 625.50/291.62 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 625.50/291.62 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 625.50/291.62 mark(length(X)) -> a__length(mark(X)) 625.50/291.62 mark(cons(X1, X2)) -> cons(mark(X1), X2) 625.50/291.62 mark(0) -> 0 625.50/291.62 mark(tt) -> tt 625.50/291.62 mark(s(X)) -> s(mark(X)) 625.50/291.62 mark(nil) -> nil 625.50/291.62 a__zeros -> zeros 625.50/291.62 a__U11(X1, X2) -> U11(X1, X2) 625.50/291.62 a__U12(X1, X2) -> U12(X1, X2) 625.50/291.62 a__isNatIListKind(X) -> isNatIListKind(X) 625.50/291.62 a__U13(X) -> U13(X) 625.50/291.62 a__isNatList(X) -> isNatList(X) 625.50/291.62 a__U21(X1, X2) -> U21(X1, X2) 625.50/291.62 a__U22(X1, X2) -> U22(X1, X2) 625.50/291.62 a__isNatKind(X) -> isNatKind(X) 625.50/291.62 a__U23(X) -> U23(X) 625.50/291.62 a__isNat(X) -> isNat(X) 625.50/291.62 a__U31(X1, X2) -> U31(X1, X2) 625.50/291.62 a__U32(X1, X2) -> U32(X1, X2) 625.50/291.62 a__U33(X) -> U33(X) 625.50/291.62 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 625.50/291.62 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 625.50/291.62 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 625.50/291.62 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 625.50/291.62 a__U45(X1, X2) -> U45(X1, X2) 625.50/291.62 a__U46(X) -> U46(X) 625.50/291.62 a__isNatIList(X) -> isNatIList(X) 625.50/291.62 a__U51(X1, X2) -> U51(X1, X2) 625.50/291.62 a__U52(X) -> U52(X) 625.50/291.62 a__U61(X) -> U61(X) 625.50/291.62 a__U71(X) -> U71(X) 625.50/291.62 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 625.50/291.62 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 625.50/291.62 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 625.50/291.62 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 625.50/291.62 a__U85(X1, X2) -> U85(X1, X2) 625.50/291.62 a__U86(X) -> U86(X) 625.50/291.62 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 625.50/291.62 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 625.50/291.62 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 625.50/291.62 a__U94(X1, X2) -> U94(X1, X2) 625.50/291.62 a__length(X) -> length(X) 625.50/291.62 625.50/291.62 S is empty. 625.50/291.62 Rewrite Strategy: FULL 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (6) LowerBoundPropagationProof (FINISHED) 625.50/291.62 Propagated lower bound. 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (7) 625.50/291.62 BOUNDS(n^1, INF) 625.50/291.62 625.50/291.62 ---------------------------------------- 625.50/291.62 625.50/291.62 (8) 625.50/291.62 Obligation: 625.50/291.62 Analyzing the following TRS for decreasing loops: 625.50/291.62 625.50/291.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 625.50/291.62 625.50/291.62 625.50/291.62 The TRS R consists of the following rules: 625.50/291.62 625.50/291.62 a__zeros -> cons(0, zeros) 625.50/291.62 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 625.50/291.62 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 625.50/291.62 a__U13(tt) -> tt 625.50/291.62 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 625.50/291.62 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 625.50/291.62 a__U23(tt) -> tt 625.50/291.62 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 625.50/291.62 a__U32(tt, V) -> a__U33(a__isNatList(V)) 625.50/291.62 a__U33(tt) -> tt 625.50/291.62 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 625.50/291.62 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 625.50/291.62 a__U46(tt) -> tt 625.50/291.62 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 625.50/291.62 a__U52(tt) -> tt 625.50/291.62 a__U61(tt) -> tt 625.50/291.62 a__U71(tt) -> tt 625.50/291.62 a__U81(tt, V1, V2) -> a__U82(a__isNatKind(V1), V1, V2) 625.50/291.62 a__U82(tt, V1, V2) -> a__U83(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U83(tt, V1, V2) -> a__U84(a__isNatIListKind(V2), V1, V2) 625.50/291.62 a__U84(tt, V1, V2) -> a__U85(a__isNat(V1), V2) 625.50/291.62 a__U85(tt, V2) -> a__U86(a__isNatList(V2)) 625.50/291.62 a__U86(tt) -> tt 625.50/291.62 a__U91(tt, L, N) -> a__U92(a__isNatIListKind(L), L, N) 625.50/291.62 a__U92(tt, L, N) -> a__U93(a__isNat(N), L, N) 625.50/291.62 a__U93(tt, L, N) -> a__U94(a__isNatKind(N), L) 625.50/291.62 a__U94(tt, L) -> s(a__length(mark(L))) 625.50/291.62 a__isNat(0) -> tt 625.50/291.62 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 625.50/291.62 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 625.50/291.62 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 625.50/291.62 a__isNatIList(zeros) -> tt 625.50/291.62 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 625.50/291.62 a__isNatIListKind(nil) -> tt 625.50/291.62 a__isNatIListKind(zeros) -> tt 625.50/291.62 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 625.50/291.62 a__isNatKind(0) -> tt 625.50/291.62 a__isNatKind(length(V1)) -> a__U61(a__isNatIListKind(V1)) 625.50/291.62 a__isNatKind(s(V1)) -> a__U71(a__isNatKind(V1)) 625.50/291.62 a__isNatList(nil) -> tt 625.50/291.62 a__isNatList(cons(V1, V2)) -> a__U81(a__isNatKind(V1), V1, V2) 625.50/291.62 a__length(nil) -> 0 625.50/291.62 a__length(cons(N, L)) -> a__U91(a__isNatList(L), L, N) 625.50/291.62 mark(zeros) -> a__zeros 625.50/291.62 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 625.50/291.62 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 625.50/291.62 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 625.50/291.62 mark(U13(X)) -> a__U13(mark(X)) 625.50/291.62 mark(isNatList(X)) -> a__isNatList(X) 625.50/291.62 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 625.50/291.62 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 625.50/291.62 mark(isNatKind(X)) -> a__isNatKind(X) 625.50/291.62 mark(U23(X)) -> a__U23(mark(X)) 625.50/291.62 mark(isNat(X)) -> a__isNat(X) 625.50/291.62 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 625.50/291.62 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 625.50/291.62 mark(U33(X)) -> a__U33(mark(X)) 625.50/291.62 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 625.50/291.62 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 625.50/291.62 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 625.50/291.62 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 625.50/291.62 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 625.50/291.62 mark(U46(X)) -> a__U46(mark(X)) 625.50/291.62 mark(isNatIList(X)) -> a__isNatIList(X) 625.50/291.62 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 625.50/291.62 mark(U52(X)) -> a__U52(mark(X)) 625.50/291.62 mark(U61(X)) -> a__U61(mark(X)) 625.50/291.62 mark(U71(X)) -> a__U71(mark(X)) 625.50/291.62 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 625.50/291.62 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 625.50/291.62 mark(U83(X1, X2, X3)) -> a__U83(mark(X1), X2, X3) 625.50/291.62 mark(U84(X1, X2, X3)) -> a__U84(mark(X1), X2, X3) 625.50/291.62 mark(U85(X1, X2)) -> a__U85(mark(X1), X2) 625.50/291.62 mark(U86(X)) -> a__U86(mark(X)) 625.50/291.62 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 625.50/291.62 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 625.50/291.62 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 625.50/291.62 mark(U94(X1, X2)) -> a__U94(mark(X1), X2) 625.50/291.62 mark(length(X)) -> a__length(mark(X)) 625.50/291.62 mark(cons(X1, X2)) -> cons(mark(X1), X2) 625.50/291.62 mark(0) -> 0 625.50/291.62 mark(tt) -> tt 625.50/291.62 mark(s(X)) -> s(mark(X)) 625.50/291.62 mark(nil) -> nil 625.50/291.62 a__zeros -> zeros 625.50/291.62 a__U11(X1, X2) -> U11(X1, X2) 625.50/291.62 a__U12(X1, X2) -> U12(X1, X2) 625.50/291.62 a__isNatIListKind(X) -> isNatIListKind(X) 625.50/291.62 a__U13(X) -> U13(X) 625.50/291.62 a__isNatList(X) -> isNatList(X) 625.50/291.62 a__U21(X1, X2) -> U21(X1, X2) 625.50/291.62 a__U22(X1, X2) -> U22(X1, X2) 625.50/291.62 a__isNatKind(X) -> isNatKind(X) 625.50/291.62 a__U23(X) -> U23(X) 625.50/291.62 a__isNat(X) -> isNat(X) 625.50/291.62 a__U31(X1, X2) -> U31(X1, X2) 625.50/291.62 a__U32(X1, X2) -> U32(X1, X2) 625.50/291.62 a__U33(X) -> U33(X) 625.50/291.62 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 625.50/291.62 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 625.50/291.62 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 625.50/291.62 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 625.50/291.62 a__U45(X1, X2) -> U45(X1, X2) 625.50/291.63 a__U46(X) -> U46(X) 625.50/291.63 a__isNatIList(X) -> isNatIList(X) 625.50/291.63 a__U51(X1, X2) -> U51(X1, X2) 625.50/291.63 a__U52(X) -> U52(X) 625.50/291.63 a__U61(X) -> U61(X) 625.50/291.63 a__U71(X) -> U71(X) 625.50/291.63 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 625.50/291.63 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 625.50/291.63 a__U83(X1, X2, X3) -> U83(X1, X2, X3) 625.50/291.63 a__U84(X1, X2, X3) -> U84(X1, X2, X3) 625.50/291.63 a__U85(X1, X2) -> U85(X1, X2) 625.50/291.63 a__U86(X) -> U86(X) 625.50/291.63 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 625.50/291.63 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 625.50/291.63 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 625.50/291.63 a__U94(X1, X2) -> U94(X1, X2) 625.50/291.63 a__length(X) -> length(X) 625.50/291.63 625.50/291.63 S is empty. 625.50/291.63 Rewrite Strategy: FULL 625.67/291.67 EOF