5.03/2.13 WORST_CASE(NON_POLY, ?) 5.03/2.15 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.03/2.15 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.03/2.15 5.03/2.15 5.03/2.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.03/2.15 5.03/2.15 (0) CpxTRS 5.03/2.15 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 5.03/2.15 (2) TRS for Loop Detection 5.03/2.15 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 5.03/2.15 (4) BEST 5.03/2.15 (5) proven lower bound 5.03/2.15 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 5.03/2.15 (7) BOUNDS(n^1, INF) 5.03/2.15 (8) TRS for Loop Detection 5.03/2.15 (9) DecreasingLoopProof [FINISHED, 323 ms] 5.03/2.15 (10) BOUNDS(EXP, INF) 5.03/2.15 5.03/2.15 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (0) 5.03/2.15 Obligation: 5.03/2.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.03/2.15 5.03/2.15 5.03/2.15 The TRS R consists of the following rules: 5.03/2.15 5.03/2.15 a__U11(tt, N, XS) -> a__U12(tt, N, XS) 5.03/2.15 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__U21(tt, X) -> a__U22(tt, X) 5.03/2.15 a__U22(tt, X) -> mark(X) 5.03/2.15 a__U31(tt, N) -> a__U32(tt, N) 5.03/2.15 a__U32(tt, N) -> mark(N) 5.03/2.15 a__U41(tt, N, XS) -> a__U42(tt, N, XS) 5.03/2.15 a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 5.03/2.15 a__U51(tt, Y) -> a__U52(tt, Y) 5.03/2.15 a__U52(tt, Y) -> mark(Y) 5.03/2.15 a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) 5.03/2.15 a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) 5.03/2.15 a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) 5.03/2.15 a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 5.03/2.15 a__U71(tt, XS) -> a__U72(tt, XS) 5.03/2.15 a__U72(tt, XS) -> mark(XS) 5.03/2.15 a__U81(tt, N, XS) -> a__U82(tt, N, XS) 5.03/2.15 a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__afterNth(N, XS) -> a__U11(tt, N, XS) 5.03/2.15 a__fst(pair(X, Y)) -> a__U21(tt, X) 5.03/2.15 a__head(cons(N, XS)) -> a__U31(tt, N) 5.03/2.15 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 5.03/2.15 a__sel(N, XS) -> a__U41(tt, N, XS) 5.03/2.15 a__snd(pair(X, Y)) -> a__U51(tt, Y) 5.03/2.15 a__splitAt(0, XS) -> pair(nil, mark(XS)) 5.03/2.15 a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) 5.03/2.15 a__tail(cons(N, XS)) -> a__U71(tt, XS) 5.03/2.15 a__take(N, XS) -> a__U81(tt, N, XS) 5.03/2.15 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 5.03/2.15 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 5.03/2.15 mark(snd(X)) -> a__snd(mark(X)) 5.03/2.15 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 5.03/2.15 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 5.03/2.15 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 5.03/2.15 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.03/2.15 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 5.03/2.15 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.03/2.15 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.03/2.15 mark(head(X)) -> a__head(mark(X)) 5.03/2.15 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 5.03/2.15 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 5.03/2.15 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 5.03/2.15 mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) 5.03/2.15 mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) 5.03/2.15 mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) 5.03/2.15 mark(U64(X1, X2)) -> a__U64(mark(X1), X2) 5.03/2.15 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 5.03/2.15 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 5.03/2.15 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 5.03/2.15 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 5.03/2.15 mark(fst(X)) -> a__fst(mark(X)) 5.03/2.15 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 5.03/2.15 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 5.03/2.15 mark(tail(X)) -> a__tail(mark(X)) 5.03/2.15 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 5.03/2.15 mark(tt) -> tt 5.03/2.15 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 5.03/2.15 mark(cons(X1, X2)) -> cons(mark(X1), X2) 5.03/2.15 mark(s(X)) -> s(mark(X)) 5.03/2.15 mark(0) -> 0 5.03/2.15 mark(nil) -> nil 5.03/2.15 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 5.03/2.15 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 5.03/2.15 a__snd(X) -> snd(X) 5.03/2.15 a__splitAt(X1, X2) -> splitAt(X1, X2) 5.03/2.15 a__U21(X1, X2) -> U21(X1, X2) 5.03/2.15 a__U22(X1, X2) -> U22(X1, X2) 5.03/2.15 a__U31(X1, X2) -> U31(X1, X2) 5.03/2.15 a__U32(X1, X2) -> U32(X1, X2) 5.03/2.15 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.03/2.15 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.03/2.15 a__head(X) -> head(X) 5.03/2.15 a__afterNth(X1, X2) -> afterNth(X1, X2) 5.03/2.15 a__U51(X1, X2) -> U51(X1, X2) 5.03/2.15 a__U52(X1, X2) -> U52(X1, X2) 5.03/2.15 a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) 5.03/2.15 a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) 5.03/2.15 a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) 5.03/2.15 a__U64(X1, X2) -> U64(X1, X2) 5.03/2.15 a__U71(X1, X2) -> U71(X1, X2) 5.03/2.15 a__U72(X1, X2) -> U72(X1, X2) 5.03/2.15 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 5.03/2.15 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 5.03/2.15 a__fst(X) -> fst(X) 5.03/2.15 a__natsFrom(X) -> natsFrom(X) 5.03/2.15 a__sel(X1, X2) -> sel(X1, X2) 5.03/2.15 a__tail(X) -> tail(X) 5.03/2.15 a__take(X1, X2) -> take(X1, X2) 5.03/2.15 5.03/2.15 S is empty. 5.03/2.15 Rewrite Strategy: FULL 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 5.03/2.15 Transformed a relative TRS into a decreasing-loop problem. 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (2) 5.03/2.15 Obligation: 5.03/2.15 Analyzing the following TRS for decreasing loops: 5.03/2.15 5.03/2.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.03/2.15 5.03/2.15 5.03/2.15 The TRS R consists of the following rules: 5.03/2.15 5.03/2.15 a__U11(tt, N, XS) -> a__U12(tt, N, XS) 5.03/2.15 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__U21(tt, X) -> a__U22(tt, X) 5.03/2.15 a__U22(tt, X) -> mark(X) 5.03/2.15 a__U31(tt, N) -> a__U32(tt, N) 5.03/2.15 a__U32(tt, N) -> mark(N) 5.03/2.15 a__U41(tt, N, XS) -> a__U42(tt, N, XS) 5.03/2.15 a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 5.03/2.15 a__U51(tt, Y) -> a__U52(tt, Y) 5.03/2.15 a__U52(tt, Y) -> mark(Y) 5.03/2.15 a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) 5.03/2.15 a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) 5.03/2.15 a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) 5.03/2.15 a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 5.03/2.15 a__U71(tt, XS) -> a__U72(tt, XS) 5.03/2.15 a__U72(tt, XS) -> mark(XS) 5.03/2.15 a__U81(tt, N, XS) -> a__U82(tt, N, XS) 5.03/2.15 a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__afterNth(N, XS) -> a__U11(tt, N, XS) 5.03/2.15 a__fst(pair(X, Y)) -> a__U21(tt, X) 5.03/2.15 a__head(cons(N, XS)) -> a__U31(tt, N) 5.03/2.15 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 5.03/2.15 a__sel(N, XS) -> a__U41(tt, N, XS) 5.03/2.15 a__snd(pair(X, Y)) -> a__U51(tt, Y) 5.03/2.15 a__splitAt(0, XS) -> pair(nil, mark(XS)) 5.03/2.15 a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) 5.03/2.15 a__tail(cons(N, XS)) -> a__U71(tt, XS) 5.03/2.15 a__take(N, XS) -> a__U81(tt, N, XS) 5.03/2.15 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 5.03/2.15 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 5.03/2.15 mark(snd(X)) -> a__snd(mark(X)) 5.03/2.15 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 5.03/2.15 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 5.03/2.15 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 5.03/2.15 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.03/2.15 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 5.03/2.15 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.03/2.15 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.03/2.15 mark(head(X)) -> a__head(mark(X)) 5.03/2.15 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 5.03/2.15 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 5.03/2.15 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 5.03/2.15 mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) 5.03/2.15 mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) 5.03/2.15 mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) 5.03/2.15 mark(U64(X1, X2)) -> a__U64(mark(X1), X2) 5.03/2.15 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 5.03/2.15 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 5.03/2.15 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 5.03/2.15 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 5.03/2.15 mark(fst(X)) -> a__fst(mark(X)) 5.03/2.15 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 5.03/2.15 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 5.03/2.15 mark(tail(X)) -> a__tail(mark(X)) 5.03/2.15 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 5.03/2.15 mark(tt) -> tt 5.03/2.15 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 5.03/2.15 mark(cons(X1, X2)) -> cons(mark(X1), X2) 5.03/2.15 mark(s(X)) -> s(mark(X)) 5.03/2.15 mark(0) -> 0 5.03/2.15 mark(nil) -> nil 5.03/2.15 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 5.03/2.15 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 5.03/2.15 a__snd(X) -> snd(X) 5.03/2.15 a__splitAt(X1, X2) -> splitAt(X1, X2) 5.03/2.15 a__U21(X1, X2) -> U21(X1, X2) 5.03/2.15 a__U22(X1, X2) -> U22(X1, X2) 5.03/2.15 a__U31(X1, X2) -> U31(X1, X2) 5.03/2.15 a__U32(X1, X2) -> U32(X1, X2) 5.03/2.15 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.03/2.15 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.03/2.15 a__head(X) -> head(X) 5.03/2.15 a__afterNth(X1, X2) -> afterNth(X1, X2) 5.03/2.15 a__U51(X1, X2) -> U51(X1, X2) 5.03/2.15 a__U52(X1, X2) -> U52(X1, X2) 5.03/2.15 a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) 5.03/2.15 a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) 5.03/2.15 a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) 5.03/2.15 a__U64(X1, X2) -> U64(X1, X2) 5.03/2.15 a__U71(X1, X2) -> U71(X1, X2) 5.03/2.15 a__U72(X1, X2) -> U72(X1, X2) 5.03/2.15 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 5.03/2.15 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 5.03/2.15 a__fst(X) -> fst(X) 5.03/2.15 a__natsFrom(X) -> natsFrom(X) 5.03/2.15 a__sel(X1, X2) -> sel(X1, X2) 5.03/2.15 a__tail(X) -> tail(X) 5.03/2.15 a__take(X1, X2) -> take(X1, X2) 5.03/2.15 5.03/2.15 S is empty. 5.03/2.15 Rewrite Strategy: FULL 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (3) DecreasingLoopProof (LOWER BOUND(ID)) 5.03/2.15 The following loop(s) give(s) rise to the lower bound Omega(n^1): 5.03/2.15 5.03/2.15 The rewrite sequence 5.03/2.15 5.03/2.15 mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) 5.03/2.15 5.03/2.15 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 5.03/2.15 5.03/2.15 The pumping substitution is [X1 / afterNth(X1, X2)]. 5.03/2.15 5.03/2.15 The result substitution is [ ]. 5.03/2.15 5.03/2.15 5.03/2.15 5.03/2.15 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (4) 5.03/2.15 Complex Obligation (BEST) 5.03/2.15 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (5) 5.03/2.15 Obligation: 5.03/2.15 Proved the lower bound n^1 for the following obligation: 5.03/2.15 5.03/2.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.03/2.15 5.03/2.15 5.03/2.15 The TRS R consists of the following rules: 5.03/2.15 5.03/2.15 a__U11(tt, N, XS) -> a__U12(tt, N, XS) 5.03/2.15 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__U21(tt, X) -> a__U22(tt, X) 5.03/2.15 a__U22(tt, X) -> mark(X) 5.03/2.15 a__U31(tt, N) -> a__U32(tt, N) 5.03/2.15 a__U32(tt, N) -> mark(N) 5.03/2.15 a__U41(tt, N, XS) -> a__U42(tt, N, XS) 5.03/2.15 a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 5.03/2.15 a__U51(tt, Y) -> a__U52(tt, Y) 5.03/2.15 a__U52(tt, Y) -> mark(Y) 5.03/2.15 a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) 5.03/2.15 a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) 5.03/2.15 a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) 5.03/2.15 a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 5.03/2.15 a__U71(tt, XS) -> a__U72(tt, XS) 5.03/2.15 a__U72(tt, XS) -> mark(XS) 5.03/2.15 a__U81(tt, N, XS) -> a__U82(tt, N, XS) 5.03/2.15 a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__afterNth(N, XS) -> a__U11(tt, N, XS) 5.03/2.15 a__fst(pair(X, Y)) -> a__U21(tt, X) 5.03/2.15 a__head(cons(N, XS)) -> a__U31(tt, N) 5.03/2.15 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 5.03/2.15 a__sel(N, XS) -> a__U41(tt, N, XS) 5.03/2.15 a__snd(pair(X, Y)) -> a__U51(tt, Y) 5.03/2.15 a__splitAt(0, XS) -> pair(nil, mark(XS)) 5.03/2.15 a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) 5.03/2.15 a__tail(cons(N, XS)) -> a__U71(tt, XS) 5.03/2.15 a__take(N, XS) -> a__U81(tt, N, XS) 5.03/2.15 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 5.03/2.15 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 5.03/2.15 mark(snd(X)) -> a__snd(mark(X)) 5.03/2.15 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 5.03/2.15 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 5.03/2.15 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 5.03/2.15 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.03/2.15 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 5.03/2.15 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.03/2.15 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.03/2.15 mark(head(X)) -> a__head(mark(X)) 5.03/2.15 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 5.03/2.15 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 5.03/2.15 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 5.03/2.15 mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) 5.03/2.15 mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) 5.03/2.15 mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) 5.03/2.15 mark(U64(X1, X2)) -> a__U64(mark(X1), X2) 5.03/2.15 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 5.03/2.15 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 5.03/2.15 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 5.03/2.15 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 5.03/2.15 mark(fst(X)) -> a__fst(mark(X)) 5.03/2.15 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 5.03/2.15 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 5.03/2.15 mark(tail(X)) -> a__tail(mark(X)) 5.03/2.15 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 5.03/2.15 mark(tt) -> tt 5.03/2.15 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 5.03/2.15 mark(cons(X1, X2)) -> cons(mark(X1), X2) 5.03/2.15 mark(s(X)) -> s(mark(X)) 5.03/2.15 mark(0) -> 0 5.03/2.15 mark(nil) -> nil 5.03/2.15 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 5.03/2.15 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 5.03/2.15 a__snd(X) -> snd(X) 5.03/2.15 a__splitAt(X1, X2) -> splitAt(X1, X2) 5.03/2.15 a__U21(X1, X2) -> U21(X1, X2) 5.03/2.15 a__U22(X1, X2) -> U22(X1, X2) 5.03/2.15 a__U31(X1, X2) -> U31(X1, X2) 5.03/2.15 a__U32(X1, X2) -> U32(X1, X2) 5.03/2.15 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.03/2.15 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.03/2.15 a__head(X) -> head(X) 5.03/2.15 a__afterNth(X1, X2) -> afterNth(X1, X2) 5.03/2.15 a__U51(X1, X2) -> U51(X1, X2) 5.03/2.15 a__U52(X1, X2) -> U52(X1, X2) 5.03/2.15 a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) 5.03/2.15 a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) 5.03/2.15 a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) 5.03/2.15 a__U64(X1, X2) -> U64(X1, X2) 5.03/2.15 a__U71(X1, X2) -> U71(X1, X2) 5.03/2.15 a__U72(X1, X2) -> U72(X1, X2) 5.03/2.15 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 5.03/2.15 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 5.03/2.15 a__fst(X) -> fst(X) 5.03/2.15 a__natsFrom(X) -> natsFrom(X) 5.03/2.15 a__sel(X1, X2) -> sel(X1, X2) 5.03/2.15 a__tail(X) -> tail(X) 5.03/2.15 a__take(X1, X2) -> take(X1, X2) 5.03/2.15 5.03/2.15 S is empty. 5.03/2.15 Rewrite Strategy: FULL 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (6) LowerBoundPropagationProof (FINISHED) 5.03/2.15 Propagated lower bound. 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (7) 5.03/2.15 BOUNDS(n^1, INF) 5.03/2.15 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (8) 5.03/2.15 Obligation: 5.03/2.15 Analyzing the following TRS for decreasing loops: 5.03/2.15 5.03/2.15 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 5.03/2.15 5.03/2.15 5.03/2.15 The TRS R consists of the following rules: 5.03/2.15 5.03/2.15 a__U11(tt, N, XS) -> a__U12(tt, N, XS) 5.03/2.15 a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__U21(tt, X) -> a__U22(tt, X) 5.03/2.15 a__U22(tt, X) -> mark(X) 5.03/2.15 a__U31(tt, N) -> a__U32(tt, N) 5.03/2.15 a__U32(tt, N) -> mark(N) 5.03/2.15 a__U41(tt, N, XS) -> a__U42(tt, N, XS) 5.03/2.15 a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 5.03/2.15 a__U51(tt, Y) -> a__U52(tt, Y) 5.03/2.15 a__U52(tt, Y) -> mark(Y) 5.03/2.15 a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) 5.03/2.15 a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) 5.03/2.15 a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) 5.03/2.15 a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 5.03/2.15 a__U71(tt, XS) -> a__U72(tt, XS) 5.03/2.15 a__U72(tt, XS) -> mark(XS) 5.03/2.15 a__U81(tt, N, XS) -> a__U82(tt, N, XS) 5.03/2.15 a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 5.03/2.15 a__afterNth(N, XS) -> a__U11(tt, N, XS) 5.03/2.15 a__fst(pair(X, Y)) -> a__U21(tt, X) 5.03/2.15 a__head(cons(N, XS)) -> a__U31(tt, N) 5.03/2.15 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 5.03/2.15 a__sel(N, XS) -> a__U41(tt, N, XS) 5.03/2.15 a__snd(pair(X, Y)) -> a__U51(tt, Y) 5.03/2.15 a__splitAt(0, XS) -> pair(nil, mark(XS)) 5.03/2.15 a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) 5.03/2.15 a__tail(cons(N, XS)) -> a__U71(tt, XS) 5.03/2.15 a__take(N, XS) -> a__U81(tt, N, XS) 5.03/2.15 mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 5.03/2.15 mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) 5.03/2.15 mark(snd(X)) -> a__snd(mark(X)) 5.03/2.15 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 5.03/2.15 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 5.03/2.15 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 5.03/2.15 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 5.03/2.15 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 5.03/2.15 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 5.03/2.15 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 5.03/2.15 mark(head(X)) -> a__head(mark(X)) 5.03/2.15 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 5.03/2.15 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 5.03/2.15 mark(U52(X1, X2)) -> a__U52(mark(X1), X2) 5.03/2.15 mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) 5.03/2.15 mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) 5.03/2.15 mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) 5.03/2.15 mark(U64(X1, X2)) -> a__U64(mark(X1), X2) 5.03/2.15 mark(U71(X1, X2)) -> a__U71(mark(X1), X2) 5.03/2.15 mark(U72(X1, X2)) -> a__U72(mark(X1), X2) 5.03/2.15 mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) 5.03/2.15 mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) 5.03/2.15 mark(fst(X)) -> a__fst(mark(X)) 5.03/2.15 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 5.03/2.15 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 5.03/2.15 mark(tail(X)) -> a__tail(mark(X)) 5.03/2.15 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 5.03/2.15 mark(tt) -> tt 5.03/2.15 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 5.03/2.15 mark(cons(X1, X2)) -> cons(mark(X1), X2) 5.03/2.15 mark(s(X)) -> s(mark(X)) 5.03/2.15 mark(0) -> 0 5.03/2.15 mark(nil) -> nil 5.03/2.15 a__U11(X1, X2, X3) -> U11(X1, X2, X3) 5.03/2.15 a__U12(X1, X2, X3) -> U12(X1, X2, X3) 5.03/2.15 a__snd(X) -> snd(X) 5.03/2.15 a__splitAt(X1, X2) -> splitAt(X1, X2) 5.03/2.15 a__U21(X1, X2) -> U21(X1, X2) 5.03/2.15 a__U22(X1, X2) -> U22(X1, X2) 5.03/2.15 a__U31(X1, X2) -> U31(X1, X2) 5.03/2.15 a__U32(X1, X2) -> U32(X1, X2) 5.03/2.15 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 5.03/2.15 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 5.03/2.15 a__head(X) -> head(X) 5.03/2.15 a__afterNth(X1, X2) -> afterNth(X1, X2) 5.03/2.15 a__U51(X1, X2) -> U51(X1, X2) 5.03/2.15 a__U52(X1, X2) -> U52(X1, X2) 5.03/2.15 a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) 5.03/2.15 a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) 5.03/2.15 a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) 5.03/2.15 a__U64(X1, X2) -> U64(X1, X2) 5.03/2.15 a__U71(X1, X2) -> U71(X1, X2) 5.03/2.15 a__U72(X1, X2) -> U72(X1, X2) 5.03/2.15 a__U81(X1, X2, X3) -> U81(X1, X2, X3) 5.03/2.15 a__U82(X1, X2, X3) -> U82(X1, X2, X3) 5.03/2.15 a__fst(X) -> fst(X) 5.03/2.15 a__natsFrom(X) -> natsFrom(X) 5.03/2.15 a__sel(X1, X2) -> sel(X1, X2) 5.03/2.15 a__tail(X) -> tail(X) 5.03/2.15 a__take(X1, X2) -> take(X1, X2) 5.03/2.15 5.03/2.15 S is empty. 5.03/2.15 Rewrite Strategy: FULL 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (9) DecreasingLoopProof (FINISHED) 5.03/2.15 The following loop(s) give(s) rise to the lower bound EXP: 5.03/2.15 5.03/2.15 The rewrite sequence 5.03/2.15 5.03/2.15 mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) 5.03/2.15 5.03/2.15 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 5.03/2.15 5.03/2.15 The pumping substitution is [X / natsFrom(X)]. 5.03/2.15 5.03/2.15 The result substitution is [ ]. 5.03/2.15 5.03/2.15 5.03/2.15 5.03/2.15 The rewrite sequence 5.03/2.15 5.03/2.15 mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) 5.03/2.15 5.03/2.15 gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. 5.03/2.15 5.03/2.15 The pumping substitution is [X / natsFrom(X)]. 5.03/2.15 5.03/2.15 The result substitution is [ ]. 5.03/2.15 5.03/2.15 5.03/2.15 5.03/2.15 5.03/2.15 ---------------------------------------- 5.03/2.15 5.03/2.15 (10) 5.03/2.15 BOUNDS(EXP, INF) 5.36/2.20 EOF