4.28/1.84 WORST_CASE(NON_POLY, ?) 4.38/1.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.38/1.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.38/1.85 4.38/1.85 4.38/1.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.38/1.85 4.38/1.85 (0) CpxTRS 4.38/1.85 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.38/1.85 (2) TRS for Loop Detection 4.38/1.85 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.38/1.85 (4) BEST 4.38/1.85 (5) proven lower bound 4.38/1.85 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 4.38/1.85 (7) BOUNDS(n^1, INF) 4.38/1.85 (8) TRS for Loop Detection 4.38/1.85 (9) DecreasingLoopProof [FINISHED, 125 ms] 4.38/1.85 (10) BOUNDS(EXP, INF) 4.38/1.85 4.38/1.85 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (0) 4.38/1.85 Obligation: 4.38/1.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.38/1.85 4.38/1.85 4.38/1.85 The TRS R consists of the following rules: 4.38/1.85 4.38/1.85 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 4.38/1.85 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 4.38/1.85 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 4.38/1.85 a__and(tt, X) -> mark(X) 4.38/1.85 a__fst(pair(X, Y)) -> mark(X) 4.38/1.85 a__head(cons(N, XS)) -> mark(N) 4.38/1.85 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 4.38/1.85 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 4.38/1.85 a__snd(pair(X, Y)) -> mark(Y) 4.38/1.85 a__splitAt(0, XS) -> pair(nil, mark(XS)) 4.38/1.85 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 4.38/1.85 a__tail(cons(N, XS)) -> mark(XS) 4.38/1.85 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 4.38/1.85 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 4.38/1.85 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 4.38/1.85 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 4.38/1.85 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 4.38/1.85 mark(snd(X)) -> a__snd(mark(X)) 4.38/1.85 mark(and(X1, X2)) -> a__and(mark(X1), X2) 4.38/1.85 mark(fst(X)) -> a__fst(mark(X)) 4.38/1.85 mark(head(X)) -> a__head(mark(X)) 4.38/1.85 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 4.38/1.85 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 4.38/1.85 mark(tail(X)) -> a__tail(mark(X)) 4.38/1.85 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 4.38/1.85 mark(tt) -> tt 4.38/1.85 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 4.38/1.85 mark(cons(X1, X2)) -> cons(mark(X1), X2) 4.38/1.85 mark(s(X)) -> s(mark(X)) 4.38/1.85 mark(0) -> 0 4.38/1.85 mark(nil) -> nil 4.38/1.85 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 4.38/1.85 a__U12(X1, X2) -> U12(X1, X2) 4.38/1.85 a__splitAt(X1, X2) -> splitAt(X1, X2) 4.38/1.85 a__afterNth(X1, X2) -> afterNth(X1, X2) 4.38/1.85 a__snd(X) -> snd(X) 4.38/1.85 a__and(X1, X2) -> and(X1, X2) 4.38/1.85 a__fst(X) -> fst(X) 4.38/1.85 a__head(X) -> head(X) 4.38/1.85 a__natsFrom(X) -> natsFrom(X) 4.38/1.85 a__sel(X1, X2) -> sel(X1, X2) 4.38/1.85 a__tail(X) -> tail(X) 4.38/1.85 a__take(X1, X2) -> take(X1, X2) 4.38/1.85 4.38/1.85 S is empty. 4.38/1.85 Rewrite Strategy: FULL 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.38/1.85 Transformed a relative TRS into a decreasing-loop problem. 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (2) 4.38/1.85 Obligation: 4.38/1.85 Analyzing the following TRS for decreasing loops: 4.38/1.85 4.38/1.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.38/1.85 4.38/1.85 4.38/1.85 The TRS R consists of the following rules: 4.38/1.85 4.38/1.85 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 4.38/1.85 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 4.38/1.85 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 4.38/1.85 a__and(tt, X) -> mark(X) 4.38/1.85 a__fst(pair(X, Y)) -> mark(X) 4.38/1.85 a__head(cons(N, XS)) -> mark(N) 4.38/1.85 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 4.38/1.85 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 4.38/1.85 a__snd(pair(X, Y)) -> mark(Y) 4.38/1.85 a__splitAt(0, XS) -> pair(nil, mark(XS)) 4.38/1.85 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 4.38/1.85 a__tail(cons(N, XS)) -> mark(XS) 4.38/1.85 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 4.38/1.85 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 4.38/1.85 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 4.38/1.85 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 4.38/1.85 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 4.38/1.85 mark(snd(X)) -> a__snd(mark(X)) 4.38/1.85 mark(and(X1, X2)) -> a__and(mark(X1), X2) 4.38/1.85 mark(fst(X)) -> a__fst(mark(X)) 4.38/1.85 mark(head(X)) -> a__head(mark(X)) 4.38/1.85 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 4.38/1.85 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 4.38/1.85 mark(tail(X)) -> a__tail(mark(X)) 4.38/1.85 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 4.38/1.85 mark(tt) -> tt 4.38/1.85 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 4.38/1.85 mark(cons(X1, X2)) -> cons(mark(X1), X2) 4.38/1.85 mark(s(X)) -> s(mark(X)) 4.38/1.85 mark(0) -> 0 4.38/1.85 mark(nil) -> nil 4.38/1.85 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 4.38/1.85 a__U12(X1, X2) -> U12(X1, X2) 4.38/1.85 a__splitAt(X1, X2) -> splitAt(X1, X2) 4.38/1.85 a__afterNth(X1, X2) -> afterNth(X1, X2) 4.38/1.85 a__snd(X) -> snd(X) 4.38/1.85 a__and(X1, X2) -> and(X1, X2) 4.38/1.85 a__fst(X) -> fst(X) 4.38/1.85 a__head(X) -> head(X) 4.38/1.85 a__natsFrom(X) -> natsFrom(X) 4.38/1.85 a__sel(X1, X2) -> sel(X1, X2) 4.38/1.85 a__tail(X) -> tail(X) 4.38/1.85 a__take(X1, X2) -> take(X1, X2) 4.38/1.85 4.38/1.85 S is empty. 4.38/1.85 Rewrite Strategy: FULL 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (3) DecreasingLoopProof (LOWER BOUND(ID)) 4.38/1.85 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.38/1.85 4.38/1.85 The rewrite sequence 4.38/1.85 4.38/1.85 mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) 4.38/1.85 4.38/1.85 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 4.38/1.85 4.38/1.85 The pumping substitution is [X1 / afterNth(X1, X2)]. 4.38/1.85 4.38/1.85 The result substitution is [ ]. 4.38/1.85 4.38/1.85 4.38/1.85 4.38/1.85 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (4) 4.38/1.85 Complex Obligation (BEST) 4.38/1.85 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (5) 4.38/1.85 Obligation: 4.38/1.85 Proved the lower bound n^1 for the following obligation: 4.38/1.85 4.38/1.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.38/1.85 4.38/1.85 4.38/1.85 The TRS R consists of the following rules: 4.38/1.85 4.38/1.85 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 4.38/1.85 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 4.38/1.85 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 4.38/1.85 a__and(tt, X) -> mark(X) 4.38/1.85 a__fst(pair(X, Y)) -> mark(X) 4.38/1.85 a__head(cons(N, XS)) -> mark(N) 4.38/1.85 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 4.38/1.85 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 4.38/1.85 a__snd(pair(X, Y)) -> mark(Y) 4.38/1.85 a__splitAt(0, XS) -> pair(nil, mark(XS)) 4.38/1.85 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 4.38/1.85 a__tail(cons(N, XS)) -> mark(XS) 4.38/1.85 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 4.38/1.85 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 4.38/1.85 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 4.38/1.85 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 4.38/1.85 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 4.38/1.85 mark(snd(X)) -> a__snd(mark(X)) 4.38/1.85 mark(and(X1, X2)) -> a__and(mark(X1), X2) 4.38/1.85 mark(fst(X)) -> a__fst(mark(X)) 4.38/1.85 mark(head(X)) -> a__head(mark(X)) 4.38/1.85 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 4.38/1.85 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 4.38/1.85 mark(tail(X)) -> a__tail(mark(X)) 4.38/1.85 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 4.38/1.85 mark(tt) -> tt 4.38/1.85 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 4.38/1.85 mark(cons(X1, X2)) -> cons(mark(X1), X2) 4.38/1.85 mark(s(X)) -> s(mark(X)) 4.38/1.85 mark(0) -> 0 4.38/1.85 mark(nil) -> nil 4.38/1.85 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 4.38/1.85 a__U12(X1, X2) -> U12(X1, X2) 4.38/1.85 a__splitAt(X1, X2) -> splitAt(X1, X2) 4.38/1.85 a__afterNth(X1, X2) -> afterNth(X1, X2) 4.38/1.85 a__snd(X) -> snd(X) 4.38/1.85 a__and(X1, X2) -> and(X1, X2) 4.38/1.85 a__fst(X) -> fst(X) 4.38/1.85 a__head(X) -> head(X) 4.38/1.85 a__natsFrom(X) -> natsFrom(X) 4.38/1.85 a__sel(X1, X2) -> sel(X1, X2) 4.38/1.85 a__tail(X) -> tail(X) 4.38/1.85 a__take(X1, X2) -> take(X1, X2) 4.38/1.85 4.38/1.85 S is empty. 4.38/1.85 Rewrite Strategy: FULL 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (6) LowerBoundPropagationProof (FINISHED) 4.38/1.85 Propagated lower bound. 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (7) 4.38/1.85 BOUNDS(n^1, INF) 4.38/1.85 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (8) 4.38/1.85 Obligation: 4.38/1.85 Analyzing the following TRS for decreasing loops: 4.38/1.85 4.38/1.85 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 4.38/1.85 4.38/1.85 4.38/1.85 The TRS R consists of the following rules: 4.38/1.85 4.38/1.85 a__U11(tt, N, X, XS) -> a__U12(a__splitAt(mark(N), mark(XS)), X) 4.38/1.85 a__U12(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) 4.38/1.85 a__afterNth(N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) 4.38/1.85 a__and(tt, X) -> mark(X) 4.38/1.85 a__fst(pair(X, Y)) -> mark(X) 4.38/1.85 a__head(cons(N, XS)) -> mark(N) 4.38/1.85 a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) 4.38/1.85 a__sel(N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) 4.38/1.85 a__snd(pair(X, Y)) -> mark(Y) 4.38/1.85 a__splitAt(0, XS) -> pair(nil, mark(XS)) 4.38/1.85 a__splitAt(s(N), cons(X, XS)) -> a__U11(tt, N, X, XS) 4.38/1.85 a__tail(cons(N, XS)) -> mark(XS) 4.38/1.85 a__take(N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) 4.38/1.85 mark(U11(X1, X2, X3, X4)) -> a__U11(mark(X1), X2, X3, X4) 4.38/1.85 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 4.38/1.85 mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) 4.38/1.85 mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) 4.38/1.85 mark(snd(X)) -> a__snd(mark(X)) 4.38/1.85 mark(and(X1, X2)) -> a__and(mark(X1), X2) 4.38/1.85 mark(fst(X)) -> a__fst(mark(X)) 4.38/1.85 mark(head(X)) -> a__head(mark(X)) 4.38/1.85 mark(natsFrom(X)) -> a__natsFrom(mark(X)) 4.38/1.85 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 4.38/1.85 mark(tail(X)) -> a__tail(mark(X)) 4.38/1.85 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 4.38/1.85 mark(tt) -> tt 4.38/1.85 mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) 4.38/1.85 mark(cons(X1, X2)) -> cons(mark(X1), X2) 4.38/1.85 mark(s(X)) -> s(mark(X)) 4.38/1.85 mark(0) -> 0 4.38/1.85 mark(nil) -> nil 4.38/1.85 a__U11(X1, X2, X3, X4) -> U11(X1, X2, X3, X4) 4.38/1.85 a__U12(X1, X2) -> U12(X1, X2) 4.38/1.85 a__splitAt(X1, X2) -> splitAt(X1, X2) 4.38/1.85 a__afterNth(X1, X2) -> afterNth(X1, X2) 4.38/1.85 a__snd(X) -> snd(X) 4.38/1.85 a__and(X1, X2) -> and(X1, X2) 4.38/1.85 a__fst(X) -> fst(X) 4.38/1.85 a__head(X) -> head(X) 4.38/1.85 a__natsFrom(X) -> natsFrom(X) 4.38/1.85 a__sel(X1, X2) -> sel(X1, X2) 4.38/1.85 a__tail(X) -> tail(X) 4.38/1.85 a__take(X1, X2) -> take(X1, X2) 4.38/1.85 4.38/1.85 S is empty. 4.38/1.85 Rewrite Strategy: FULL 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (9) DecreasingLoopProof (FINISHED) 4.38/1.85 The following loop(s) give(s) rise to the lower bound EXP: 4.38/1.85 4.38/1.85 The rewrite sequence 4.38/1.85 4.38/1.85 mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) 4.38/1.85 4.38/1.85 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 4.38/1.85 4.38/1.85 The pumping substitution is [X / natsFrom(X)]. 4.38/1.85 4.38/1.85 The result substitution is [ ]. 4.38/1.85 4.38/1.85 4.38/1.85 4.38/1.85 The rewrite sequence 4.38/1.85 4.38/1.85 mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) 4.38/1.85 4.38/1.85 gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. 4.38/1.85 4.38/1.85 The pumping substitution is [X / natsFrom(X)]. 4.38/1.85 4.38/1.85 The result substitution is [ ]. 4.38/1.85 4.38/1.85 4.38/1.85 4.38/1.85 4.38/1.85 ---------------------------------------- 4.38/1.85 4.38/1.85 (10) 4.38/1.85 BOUNDS(EXP, INF) 4.45/2.07 EOF