1049.70/291.60 WORST_CASE(Omega(n^1), ?) 1060.99/294.50 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1060.99/294.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1060.99/294.50 1060.99/294.50 1060.99/294.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1060.99/294.50 1060.99/294.50 (0) CpxTRS 1060.99/294.50 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1060.99/294.50 (2) TRS for Loop Detection 1060.99/294.50 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1060.99/294.50 (4) BEST 1060.99/294.50 (5) proven lower bound 1060.99/294.50 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1060.99/294.50 (7) BOUNDS(n^1, INF) 1060.99/294.50 (8) TRS for Loop Detection 1060.99/294.50 1060.99/294.50 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (0) 1060.99/294.50 Obligation: 1060.99/294.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1060.99/294.50 1060.99/294.50 1060.99/294.50 The TRS R consists of the following rules: 1060.99/294.50 1060.99/294.50 a__zeros -> cons(0, zeros) 1060.99/294.50 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 1060.99/294.50 a__U12(tt) -> tt 1060.99/294.50 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 1060.99/294.50 a__U22(tt) -> tt 1060.99/294.50 a__U31(tt, V) -> a__U32(a__isNatList(V)) 1060.99/294.50 a__U32(tt) -> tt 1060.99/294.50 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 1060.99/294.50 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 1060.99/294.50 a__U43(tt) -> tt 1060.99/294.50 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 1060.99/294.50 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 1060.99/294.50 a__U53(tt) -> tt 1060.99/294.50 a__U61(tt, V1, V2) -> a__U62(a__isNat(V1), V2) 1060.99/294.50 a__U62(tt, V2) -> a__U63(a__isNatIList(V2)) 1060.99/294.50 a__U63(tt) -> tt 1060.99/294.50 a__U71(tt, L) -> s(a__length(mark(L))) 1060.99/294.50 a__U81(tt) -> nil 1060.99/294.50 a__U91(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1060.99/294.50 a__and(tt, X) -> mark(X) 1060.99/294.50 a__isNat(0) -> tt 1060.99/294.50 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1060.99/294.50 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1060.99/294.50 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1060.99/294.50 a__isNatIList(zeros) -> tt 1060.99/294.50 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatIListKind(nil) -> tt 1060.99/294.50 a__isNatIListKind(zeros) -> tt 1060.99/294.50 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatKind(0) -> tt 1060.99/294.50 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 1060.99/294.50 a__isNatKind(s(V1)) -> a__isNatKind(V1) 1060.99/294.50 a__isNatList(nil) -> tt 1060.99/294.50 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__length(nil) -> 0 1060.99/294.50 a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 1060.99/294.50 a__take(0, IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) 1060.99/294.50 a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) 1060.99/294.50 mark(zeros) -> a__zeros 1060.99/294.50 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1060.99/294.50 mark(U12(X)) -> a__U12(mark(X)) 1060.99/294.50 mark(isNatList(X)) -> a__isNatList(X) 1060.99/294.50 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1060.99/294.50 mark(U22(X)) -> a__U22(mark(X)) 1060.99/294.50 mark(isNat(X)) -> a__isNat(X) 1060.99/294.50 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1060.99/294.50 mark(U32(X)) -> a__U32(mark(X)) 1060.99/294.50 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1060.99/294.50 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 1060.99/294.50 mark(U43(X)) -> a__U43(mark(X)) 1060.99/294.50 mark(isNatIList(X)) -> a__isNatIList(X) 1060.99/294.50 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1060.99/294.50 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1060.99/294.50 mark(U53(X)) -> a__U53(mark(X)) 1060.99/294.50 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1060.99/294.50 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 1060.99/294.50 mark(U63(X)) -> a__U63(mark(X)) 1060.99/294.50 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1060.99/294.50 mark(length(X)) -> a__length(mark(X)) 1060.99/294.50 mark(U81(X)) -> a__U81(mark(X)) 1060.99/294.50 mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) 1060.99/294.50 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1060.99/294.50 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1060.99/294.50 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1060.99/294.50 mark(isNatKind(X)) -> a__isNatKind(X) 1060.99/294.50 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1060.99/294.50 mark(0) -> 0 1060.99/294.50 mark(tt) -> tt 1060.99/294.50 mark(s(X)) -> s(mark(X)) 1060.99/294.50 mark(nil) -> nil 1060.99/294.50 a__zeros -> zeros 1060.99/294.50 a__U11(X1, X2) -> U11(X1, X2) 1060.99/294.50 a__U12(X) -> U12(X) 1060.99/294.50 a__isNatList(X) -> isNatList(X) 1060.99/294.50 a__U21(X1, X2) -> U21(X1, X2) 1060.99/294.50 a__U22(X) -> U22(X) 1060.99/294.50 a__isNat(X) -> isNat(X) 1060.99/294.50 a__U31(X1, X2) -> U31(X1, X2) 1060.99/294.50 a__U32(X) -> U32(X) 1060.99/294.50 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1060.99/294.50 a__U42(X1, X2) -> U42(X1, X2) 1060.99/294.50 a__U43(X) -> U43(X) 1060.99/294.50 a__isNatIList(X) -> isNatIList(X) 1060.99/294.50 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1060.99/294.50 a__U52(X1, X2) -> U52(X1, X2) 1060.99/294.50 a__U53(X) -> U53(X) 1060.99/294.50 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1060.99/294.50 a__U62(X1, X2) -> U62(X1, X2) 1060.99/294.50 a__U63(X) -> U63(X) 1060.99/294.50 a__U71(X1, X2) -> U71(X1, X2) 1060.99/294.50 a__length(X) -> length(X) 1060.99/294.50 a__U81(X) -> U81(X) 1060.99/294.50 a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) 1060.99/294.50 a__take(X1, X2) -> take(X1, X2) 1060.99/294.50 a__and(X1, X2) -> and(X1, X2) 1060.99/294.50 a__isNatIListKind(X) -> isNatIListKind(X) 1060.99/294.50 a__isNatKind(X) -> isNatKind(X) 1060.99/294.50 1060.99/294.50 S is empty. 1060.99/294.50 Rewrite Strategy: INNERMOST 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1060.99/294.50 Transformed a relative TRS into a decreasing-loop problem. 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (2) 1060.99/294.50 Obligation: 1060.99/294.50 Analyzing the following TRS for decreasing loops: 1060.99/294.50 1060.99/294.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1060.99/294.50 1060.99/294.50 1060.99/294.50 The TRS R consists of the following rules: 1060.99/294.50 1060.99/294.50 a__zeros -> cons(0, zeros) 1060.99/294.50 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 1060.99/294.50 a__U12(tt) -> tt 1060.99/294.50 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 1060.99/294.50 a__U22(tt) -> tt 1060.99/294.50 a__U31(tt, V) -> a__U32(a__isNatList(V)) 1060.99/294.50 a__U32(tt) -> tt 1060.99/294.50 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 1060.99/294.50 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 1060.99/294.50 a__U43(tt) -> tt 1060.99/294.50 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 1060.99/294.50 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 1060.99/294.50 a__U53(tt) -> tt 1060.99/294.50 a__U61(tt, V1, V2) -> a__U62(a__isNat(V1), V2) 1060.99/294.50 a__U62(tt, V2) -> a__U63(a__isNatIList(V2)) 1060.99/294.50 a__U63(tt) -> tt 1060.99/294.50 a__U71(tt, L) -> s(a__length(mark(L))) 1060.99/294.50 a__U81(tt) -> nil 1060.99/294.50 a__U91(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1060.99/294.50 a__and(tt, X) -> mark(X) 1060.99/294.50 a__isNat(0) -> tt 1060.99/294.50 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1060.99/294.50 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1060.99/294.50 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1060.99/294.50 a__isNatIList(zeros) -> tt 1060.99/294.50 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatIListKind(nil) -> tt 1060.99/294.50 a__isNatIListKind(zeros) -> tt 1060.99/294.50 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatKind(0) -> tt 1060.99/294.50 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 1060.99/294.50 a__isNatKind(s(V1)) -> a__isNatKind(V1) 1060.99/294.50 a__isNatList(nil) -> tt 1060.99/294.50 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__length(nil) -> 0 1060.99/294.50 a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 1060.99/294.50 a__take(0, IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) 1060.99/294.50 a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) 1060.99/294.50 mark(zeros) -> a__zeros 1060.99/294.50 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1060.99/294.50 mark(U12(X)) -> a__U12(mark(X)) 1060.99/294.50 mark(isNatList(X)) -> a__isNatList(X) 1060.99/294.50 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1060.99/294.50 mark(U22(X)) -> a__U22(mark(X)) 1060.99/294.50 mark(isNat(X)) -> a__isNat(X) 1060.99/294.50 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1060.99/294.50 mark(U32(X)) -> a__U32(mark(X)) 1060.99/294.50 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1060.99/294.50 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 1060.99/294.50 mark(U43(X)) -> a__U43(mark(X)) 1060.99/294.50 mark(isNatIList(X)) -> a__isNatIList(X) 1060.99/294.50 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1060.99/294.50 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1060.99/294.50 mark(U53(X)) -> a__U53(mark(X)) 1060.99/294.50 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1060.99/294.50 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 1060.99/294.50 mark(U63(X)) -> a__U63(mark(X)) 1060.99/294.50 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1060.99/294.50 mark(length(X)) -> a__length(mark(X)) 1060.99/294.50 mark(U81(X)) -> a__U81(mark(X)) 1060.99/294.50 mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) 1060.99/294.50 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1060.99/294.50 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1060.99/294.50 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1060.99/294.50 mark(isNatKind(X)) -> a__isNatKind(X) 1060.99/294.50 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1060.99/294.50 mark(0) -> 0 1060.99/294.50 mark(tt) -> tt 1060.99/294.50 mark(s(X)) -> s(mark(X)) 1060.99/294.50 mark(nil) -> nil 1060.99/294.50 a__zeros -> zeros 1060.99/294.50 a__U11(X1, X2) -> U11(X1, X2) 1060.99/294.50 a__U12(X) -> U12(X) 1060.99/294.50 a__isNatList(X) -> isNatList(X) 1060.99/294.50 a__U21(X1, X2) -> U21(X1, X2) 1060.99/294.50 a__U22(X) -> U22(X) 1060.99/294.50 a__isNat(X) -> isNat(X) 1060.99/294.50 a__U31(X1, X2) -> U31(X1, X2) 1060.99/294.50 a__U32(X) -> U32(X) 1060.99/294.50 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1060.99/294.50 a__U42(X1, X2) -> U42(X1, X2) 1060.99/294.50 a__U43(X) -> U43(X) 1060.99/294.50 a__isNatIList(X) -> isNatIList(X) 1060.99/294.50 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1060.99/294.50 a__U52(X1, X2) -> U52(X1, X2) 1060.99/294.50 a__U53(X) -> U53(X) 1060.99/294.50 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1060.99/294.50 a__U62(X1, X2) -> U62(X1, X2) 1060.99/294.50 a__U63(X) -> U63(X) 1060.99/294.50 a__U71(X1, X2) -> U71(X1, X2) 1060.99/294.50 a__length(X) -> length(X) 1060.99/294.50 a__U81(X) -> U81(X) 1060.99/294.50 a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) 1060.99/294.50 a__take(X1, X2) -> take(X1, X2) 1060.99/294.50 a__and(X1, X2) -> and(X1, X2) 1060.99/294.50 a__isNatIListKind(X) -> isNatIListKind(X) 1060.99/294.50 a__isNatKind(X) -> isNatKind(X) 1060.99/294.50 1060.99/294.50 S is empty. 1060.99/294.50 Rewrite Strategy: INNERMOST 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1060.99/294.50 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1060.99/294.50 1060.99/294.50 The rewrite sequence 1060.99/294.50 1060.99/294.50 mark(U12(X)) ->^+ a__U12(mark(X)) 1060.99/294.50 1060.99/294.50 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1060.99/294.50 1060.99/294.50 The pumping substitution is [X / U12(X)]. 1060.99/294.50 1060.99/294.50 The result substitution is [ ]. 1060.99/294.50 1060.99/294.50 1060.99/294.50 1060.99/294.50 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (4) 1060.99/294.50 Complex Obligation (BEST) 1060.99/294.50 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (5) 1060.99/294.50 Obligation: 1060.99/294.50 Proved the lower bound n^1 for the following obligation: 1060.99/294.50 1060.99/294.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1060.99/294.50 1060.99/294.50 1060.99/294.50 The TRS R consists of the following rules: 1060.99/294.50 1060.99/294.50 a__zeros -> cons(0, zeros) 1060.99/294.50 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 1060.99/294.50 a__U12(tt) -> tt 1060.99/294.50 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 1060.99/294.50 a__U22(tt) -> tt 1060.99/294.50 a__U31(tt, V) -> a__U32(a__isNatList(V)) 1060.99/294.50 a__U32(tt) -> tt 1060.99/294.50 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 1060.99/294.50 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 1060.99/294.50 a__U43(tt) -> tt 1060.99/294.50 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 1060.99/294.50 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 1060.99/294.50 a__U53(tt) -> tt 1060.99/294.50 a__U61(tt, V1, V2) -> a__U62(a__isNat(V1), V2) 1060.99/294.50 a__U62(tt, V2) -> a__U63(a__isNatIList(V2)) 1060.99/294.50 a__U63(tt) -> tt 1060.99/294.50 a__U71(tt, L) -> s(a__length(mark(L))) 1060.99/294.50 a__U81(tt) -> nil 1060.99/294.50 a__U91(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1060.99/294.50 a__and(tt, X) -> mark(X) 1060.99/294.50 a__isNat(0) -> tt 1060.99/294.50 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1060.99/294.50 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1060.99/294.50 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1060.99/294.50 a__isNatIList(zeros) -> tt 1060.99/294.50 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatIListKind(nil) -> tt 1060.99/294.50 a__isNatIListKind(zeros) -> tt 1060.99/294.50 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatKind(0) -> tt 1060.99/294.50 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 1060.99/294.50 a__isNatKind(s(V1)) -> a__isNatKind(V1) 1060.99/294.50 a__isNatList(nil) -> tt 1060.99/294.50 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__length(nil) -> 0 1060.99/294.50 a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 1060.99/294.50 a__take(0, IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) 1060.99/294.50 a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) 1060.99/294.50 mark(zeros) -> a__zeros 1060.99/294.50 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1060.99/294.50 mark(U12(X)) -> a__U12(mark(X)) 1060.99/294.50 mark(isNatList(X)) -> a__isNatList(X) 1060.99/294.50 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1060.99/294.50 mark(U22(X)) -> a__U22(mark(X)) 1060.99/294.50 mark(isNat(X)) -> a__isNat(X) 1060.99/294.50 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1060.99/294.50 mark(U32(X)) -> a__U32(mark(X)) 1060.99/294.50 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1060.99/294.50 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 1060.99/294.50 mark(U43(X)) -> a__U43(mark(X)) 1060.99/294.50 mark(isNatIList(X)) -> a__isNatIList(X) 1060.99/294.50 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1060.99/294.50 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1060.99/294.50 mark(U53(X)) -> a__U53(mark(X)) 1060.99/294.50 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1060.99/294.50 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 1060.99/294.50 mark(U63(X)) -> a__U63(mark(X)) 1060.99/294.50 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1060.99/294.50 mark(length(X)) -> a__length(mark(X)) 1060.99/294.50 mark(U81(X)) -> a__U81(mark(X)) 1060.99/294.50 mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) 1060.99/294.50 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1060.99/294.50 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1060.99/294.50 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1060.99/294.50 mark(isNatKind(X)) -> a__isNatKind(X) 1060.99/294.50 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1060.99/294.50 mark(0) -> 0 1060.99/294.50 mark(tt) -> tt 1060.99/294.50 mark(s(X)) -> s(mark(X)) 1060.99/294.50 mark(nil) -> nil 1060.99/294.50 a__zeros -> zeros 1060.99/294.50 a__U11(X1, X2) -> U11(X1, X2) 1060.99/294.50 a__U12(X) -> U12(X) 1060.99/294.50 a__isNatList(X) -> isNatList(X) 1060.99/294.50 a__U21(X1, X2) -> U21(X1, X2) 1060.99/294.50 a__U22(X) -> U22(X) 1060.99/294.50 a__isNat(X) -> isNat(X) 1060.99/294.50 a__U31(X1, X2) -> U31(X1, X2) 1060.99/294.50 a__U32(X) -> U32(X) 1060.99/294.50 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1060.99/294.50 a__U42(X1, X2) -> U42(X1, X2) 1060.99/294.50 a__U43(X) -> U43(X) 1060.99/294.50 a__isNatIList(X) -> isNatIList(X) 1060.99/294.50 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1060.99/294.50 a__U52(X1, X2) -> U52(X1, X2) 1060.99/294.50 a__U53(X) -> U53(X) 1060.99/294.50 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1060.99/294.50 a__U62(X1, X2) -> U62(X1, X2) 1060.99/294.50 a__U63(X) -> U63(X) 1060.99/294.50 a__U71(X1, X2) -> U71(X1, X2) 1060.99/294.50 a__length(X) -> length(X) 1060.99/294.50 a__U81(X) -> U81(X) 1060.99/294.50 a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) 1060.99/294.50 a__take(X1, X2) -> take(X1, X2) 1060.99/294.50 a__and(X1, X2) -> and(X1, X2) 1060.99/294.50 a__isNatIListKind(X) -> isNatIListKind(X) 1060.99/294.50 a__isNatKind(X) -> isNatKind(X) 1060.99/294.50 1060.99/294.50 S is empty. 1060.99/294.50 Rewrite Strategy: INNERMOST 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (6) LowerBoundPropagationProof (FINISHED) 1060.99/294.50 Propagated lower bound. 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (7) 1060.99/294.50 BOUNDS(n^1, INF) 1060.99/294.50 1060.99/294.50 ---------------------------------------- 1060.99/294.50 1060.99/294.50 (8) 1060.99/294.50 Obligation: 1060.99/294.50 Analyzing the following TRS for decreasing loops: 1060.99/294.50 1060.99/294.50 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1060.99/294.50 1060.99/294.50 1060.99/294.50 The TRS R consists of the following rules: 1060.99/294.50 1060.99/294.50 a__zeros -> cons(0, zeros) 1060.99/294.50 a__U11(tt, V1) -> a__U12(a__isNatList(V1)) 1060.99/294.50 a__U12(tt) -> tt 1060.99/294.50 a__U21(tt, V1) -> a__U22(a__isNat(V1)) 1060.99/294.50 a__U22(tt) -> tt 1060.99/294.50 a__U31(tt, V) -> a__U32(a__isNatList(V)) 1060.99/294.50 a__U32(tt) -> tt 1060.99/294.50 a__U41(tt, V1, V2) -> a__U42(a__isNat(V1), V2) 1060.99/294.50 a__U42(tt, V2) -> a__U43(a__isNatIList(V2)) 1060.99/294.50 a__U43(tt) -> tt 1060.99/294.50 a__U51(tt, V1, V2) -> a__U52(a__isNat(V1), V2) 1060.99/294.50 a__U52(tt, V2) -> a__U53(a__isNatList(V2)) 1060.99/294.50 a__U53(tt) -> tt 1060.99/294.50 a__U61(tt, V1, V2) -> a__U62(a__isNat(V1), V2) 1060.99/294.50 a__U62(tt, V2) -> a__U63(a__isNatIList(V2)) 1060.99/294.50 a__U63(tt) -> tt 1060.99/294.50 a__U71(tt, L) -> s(a__length(mark(L))) 1060.99/294.50 a__U81(tt) -> nil 1060.99/294.50 a__U91(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1060.99/294.50 a__and(tt, X) -> mark(X) 1060.99/294.50 a__isNat(0) -> tt 1060.99/294.50 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1060.99/294.50 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1060.99/294.50 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1060.99/294.50 a__isNatIList(zeros) -> tt 1060.99/294.50 a__isNatIList(cons(V1, V2)) -> a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatIListKind(nil) -> tt 1060.99/294.50 a__isNatIListKind(zeros) -> tt 1060.99/294.50 a__isNatIListKind(cons(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatIListKind(take(V1, V2)) -> a__and(a__isNatKind(V1), isNatIListKind(V2)) 1060.99/294.50 a__isNatKind(0) -> tt 1060.99/294.50 a__isNatKind(length(V1)) -> a__isNatIListKind(V1) 1060.99/294.50 a__isNatKind(s(V1)) -> a__isNatKind(V1) 1060.99/294.50 a__isNatList(nil) -> tt 1060.99/294.50 a__isNatList(cons(V1, V2)) -> a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__isNatList(take(V1, V2)) -> a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2) 1060.99/294.50 a__length(nil) -> 0 1060.99/294.50 a__length(cons(N, L)) -> a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) 1060.99/294.50 a__take(0, IL) -> a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL))) 1060.99/294.50 a__take(s(M), cons(N, IL)) -> a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) 1060.99/294.50 mark(zeros) -> a__zeros 1060.99/294.50 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1060.99/294.50 mark(U12(X)) -> a__U12(mark(X)) 1060.99/294.50 mark(isNatList(X)) -> a__isNatList(X) 1060.99/294.50 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1060.99/294.50 mark(U22(X)) -> a__U22(mark(X)) 1060.99/294.50 mark(isNat(X)) -> a__isNat(X) 1060.99/294.50 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1060.99/294.50 mark(U32(X)) -> a__U32(mark(X)) 1060.99/294.50 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1060.99/294.50 mark(U42(X1, X2)) -> a__U42(mark(X1), X2) 1060.99/294.50 mark(U43(X)) -> a__U43(mark(X)) 1060.99/294.50 mark(isNatIList(X)) -> a__isNatIList(X) 1060.99/294.50 mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3) 1060.99/294.50 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 1060.99/294.50 mark(U53(X)) -> a__U53(mark(X)) 1060.99/294.50 mark(U61(X1, X2, X3)) -> a__U61(mark(X1), X2, X3) 1060.99/294.50 mark(U62(X1, X2)) -> a__U62(mark(X1), X2) 1060.99/294.50 mark(U63(X)) -> a__U63(mark(X)) 1060.99/294.50 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 1060.99/294.50 mark(length(X)) -> a__length(mark(X)) 1060.99/294.50 mark(U81(X)) -> a__U81(mark(X)) 1060.99/294.50 mark(U91(X1, X2, X3, X4)) -> a__U91(mark(X1), X2, X3, X4) 1060.99/294.50 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1060.99/294.50 mark(and(X1, X2)) -> a__and(mark(X1), X2) 1060.99/294.50 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1060.99/294.50 mark(isNatKind(X)) -> a__isNatKind(X) 1060.99/294.50 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1060.99/294.50 mark(0) -> 0 1060.99/294.50 mark(tt) -> tt 1060.99/294.50 mark(s(X)) -> s(mark(X)) 1060.99/294.50 mark(nil) -> nil 1060.99/294.50 a__zeros -> zeros 1060.99/294.50 a__U11(X1, X2) -> U11(X1, X2) 1060.99/294.50 a__U12(X) -> U12(X) 1060.99/294.50 a__isNatList(X) -> isNatList(X) 1060.99/294.50 a__U21(X1, X2) -> U21(X1, X2) 1060.99/294.50 a__U22(X) -> U22(X) 1060.99/294.50 a__isNat(X) -> isNat(X) 1060.99/294.50 a__U31(X1, X2) -> U31(X1, X2) 1060.99/294.50 a__U32(X) -> U32(X) 1060.99/294.50 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1060.99/294.50 a__U42(X1, X2) -> U42(X1, X2) 1060.99/294.50 a__U43(X) -> U43(X) 1060.99/294.50 a__isNatIList(X) -> isNatIList(X) 1060.99/294.50 a__U51(X1, X2, X3) -> U51(X1, X2, X3) 1060.99/294.50 a__U52(X1, X2) -> U52(X1, X2) 1060.99/294.50 a__U53(X) -> U53(X) 1060.99/294.50 a__U61(X1, X2, X3) -> U61(X1, X2, X3) 1060.99/294.50 a__U62(X1, X2) -> U62(X1, X2) 1060.99/294.50 a__U63(X) -> U63(X) 1060.99/294.50 a__U71(X1, X2) -> U71(X1, X2) 1060.99/294.50 a__length(X) -> length(X) 1060.99/294.50 a__U81(X) -> U81(X) 1060.99/294.50 a__U91(X1, X2, X3, X4) -> U91(X1, X2, X3, X4) 1060.99/294.50 a__take(X1, X2) -> take(X1, X2) 1060.99/294.50 a__and(X1, X2) -> and(X1, X2) 1060.99/294.50 a__isNatIListKind(X) -> isNatIListKind(X) 1060.99/294.50 a__isNatKind(X) -> isNatKind(X) 1060.99/294.50 1060.99/294.50 S is empty. 1060.99/294.50 Rewrite Strategy: INNERMOST 1061.27/294.57 EOF