/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 800 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
   active(fst(pair(XS, YS))) -> mark(XS)
   active(snd(pair(XS, YS))) -> mark(YS)
   active(splitAt(0, XS)) -> mark(pair(nil, XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
   active(head(cons(N, XS))) -> mark(N)
   active(tail(cons(N, XS))) -> mark(XS)
   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(fst(X)) -> fst(active(X))
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   fst(mark(X)) -> mark(fst(X))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(fst(X)) -> fst(proper(X))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   fst(ok(X)) -> ok(fst(X))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_fst(x_1)) -> fst(encArg(x_1))
   encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_snd(x_1)) -> snd(encArg(x_1))
   encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2))
   encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_natsFrom(x_1) -> natsFrom(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fst(x_1) -> fst(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_snd(x_1) -> snd(encArg(x_1))
   encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
   active(fst(pair(XS, YS))) -> mark(XS)
   active(snd(pair(XS, YS))) -> mark(YS)
   active(splitAt(0, XS)) -> mark(pair(nil, XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
   active(head(cons(N, XS))) -> mark(N)
   active(tail(cons(N, XS))) -> mark(XS)
   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(fst(X)) -> fst(active(X))
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   fst(mark(X)) -> mark(fst(X))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(fst(X)) -> fst(proper(X))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   fst(ok(X)) -> ok(fst(X))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_fst(x_1)) -> fst(encArg(x_1))
   encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_snd(x_1)) -> snd(encArg(x_1))
   encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2))
   encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_natsFrom(x_1) -> natsFrom(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fst(x_1) -> fst(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_snd(x_1) -> snd(encArg(x_1))
   encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
   active(fst(pair(XS, YS))) -> mark(XS)
   active(snd(pair(XS, YS))) -> mark(YS)
   active(splitAt(0, XS)) -> mark(pair(nil, XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
   active(head(cons(N, XS))) -> mark(N)
   active(tail(cons(N, XS))) -> mark(XS)
   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(fst(X)) -> fst(active(X))
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   fst(mark(X)) -> mark(fst(X))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(fst(X)) -> fst(proper(X))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   fst(ok(X)) -> ok(fst(X))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_fst(x_1)) -> fst(encArg(x_1))
   encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_snd(x_1)) -> snd(encArg(x_1))
   encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2))
   encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_natsFrom(x_1) -> natsFrom(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fst(x_1) -> fst(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_snd(x_1) -> snd(encArg(x_1))
   encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
   active(fst(pair(XS, YS))) -> mark(XS)
   active(snd(pair(XS, YS))) -> mark(YS)
   active(splitAt(0, XS)) -> mark(pair(nil, XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
   active(head(cons(N, XS))) -> mark(N)
   active(tail(cons(N, XS))) -> mark(XS)
   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(fst(X)) -> fst(active(X))
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   fst(mark(X)) -> mark(fst(X))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(fst(X)) -> fst(proper(X))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   fst(ok(X)) -> ok(fst(X))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_fst(x_1)) -> fst(encArg(x_1))
   encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_snd(x_1)) -> snd(encArg(x_1))
   encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2))
   encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_natsFrom(x_1) -> natsFrom(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fst(x_1) -> fst(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_snd(x_1) -> snd(encArg(x_1))
   encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

take(ok(X1), ok(X2)) ->^+ ok(take(X1, X2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X1 / ok(X1), X2 / ok(X2)].

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
   active(fst(pair(XS, YS))) -> mark(XS)
   active(snd(pair(XS, YS))) -> mark(YS)
   active(splitAt(0, XS)) -> mark(pair(nil, XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
   active(head(cons(N, XS))) -> mark(N)
   active(tail(cons(N, XS))) -> mark(XS)
   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(fst(X)) -> fst(active(X))
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   fst(mark(X)) -> mark(fst(X))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(fst(X)) -> fst(proper(X))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   fst(ok(X)) -> ok(fst(X))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_fst(x_1)) -> fst(encArg(x_1))
   encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_snd(x_1)) -> snd(encArg(x_1))
   encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2))
   encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_natsFrom(x_1) -> natsFrom(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fst(x_1) -> fst(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_snd(x_1) -> snd(encArg(x_1))
   encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
   active(fst(pair(XS, YS))) -> mark(XS)
   active(snd(pair(XS, YS))) -> mark(YS)
   active(splitAt(0, XS)) -> mark(pair(nil, XS))
   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
   active(head(cons(N, XS))) -> mark(N)
   active(tail(cons(N, XS))) -> mark(XS)
   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
   active(natsFrom(X)) -> natsFrom(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(fst(X)) -> fst(active(X))
   active(pair(X1, X2)) -> pair(active(X1), X2)
   active(pair(X1, X2)) -> pair(X1, active(X2))
   active(snd(X)) -> snd(active(X))
   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   natsFrom(mark(X)) -> mark(natsFrom(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   fst(mark(X)) -> mark(fst(X))
   pair(mark(X1), X2) -> mark(pair(X1, X2))
   pair(X1, mark(X2)) -> mark(pair(X1, X2))
   snd(mark(X)) -> mark(snd(X))
   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(natsFrom(X)) -> natsFrom(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(fst(X)) -> fst(proper(X))
   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
   proper(snd(X)) -> snd(proper(X))
   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   natsFrom(ok(X)) -> ok(natsFrom(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   fst(ok(X)) -> ok(fst(X))
   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
   snd(ok(X)) -> ok(snd(X))
   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The (relative) TRS S consists of the following rules:

   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_fst(x_1)) -> fst(encArg(x_1))
   encArg(cons_pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_snd(x_1)) -> snd(encArg(x_1))
   encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2))
   encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2))
   encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_natsFrom(x_1) -> natsFrom(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_fst(x_1) -> fst(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_snd(x_1) -> snd(encArg(x_1))
   encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2))
   encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST