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


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WORST_CASE(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 556 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 28 ms]
(8) BOUNDS(INF, INF)


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

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


The TRS R consists of the following rules:

   pairNs -> cons(0, n__incr(oddNs))
   oddNs -> incr(pairNs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   zip(nil, XS) -> nil
   zip(X, nil) -> nil
   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
   tail(cons(X, XS)) -> activate(XS)
   repItems(nil) -> nil
   repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS))))
   incr(X) -> n__incr(X)
   take(X1, X2) -> n__take(X1, X2)
   zip(X1, X2) -> n__zip(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   repItems(X) -> n__repItems(X)
   activate(n__incr(X)) -> incr(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__zip(X1, X2)) -> zip(X1, X2)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__repItems(X)) -> repItems(X)
   activate(X) -> 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(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(n__zip(x_1, x_2)) -> n__zip(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__repItems(x_1)) -> n__repItems(encArg(x_1))
   encArg(cons_pairNs) -> pairNs
   encArg(cons_oddNs) -> oddNs
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_repItems(x_1)) -> repItems(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_pairNs -> pairNs
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_oddNs -> oddNs
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2))
   encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_n__zip(x_1, x_2) -> n__zip(encArg(x_1), encArg(x_2))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_repItems(x_1) -> repItems(encArg(x_1))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__repItems(x_1) -> n__repItems(encArg(x_1))


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

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


The TRS R consists of the following rules:

   pairNs -> cons(0, n__incr(oddNs))
   oddNs -> incr(pairNs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   zip(nil, XS) -> nil
   zip(X, nil) -> nil
   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
   tail(cons(X, XS)) -> activate(XS)
   repItems(nil) -> nil
   repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS))))
   incr(X) -> n__incr(X)
   take(X1, X2) -> n__take(X1, X2)
   zip(X1, X2) -> n__zip(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   repItems(X) -> n__repItems(X)
   activate(n__incr(X)) -> incr(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__zip(X1, X2)) -> zip(X1, X2)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__repItems(X)) -> repItems(X)
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(n__zip(x_1, x_2)) -> n__zip(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__repItems(x_1)) -> n__repItems(encArg(x_1))
   encArg(cons_pairNs) -> pairNs
   encArg(cons_oddNs) -> oddNs
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_repItems(x_1)) -> repItems(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_pairNs -> pairNs
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_oddNs -> oddNs
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2))
   encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_n__zip(x_1, x_2) -> n__zip(encArg(x_1), encArg(x_2))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_repItems(x_1) -> repItems(encArg(x_1))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__repItems(x_1) -> n__repItems(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(INF, INF).


The TRS R consists of the following rules:

   pairNs -> cons(0, n__incr(oddNs))
   oddNs -> incr(pairNs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   zip(nil, XS) -> nil
   zip(X, nil) -> nil
   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
   tail(cons(X, XS)) -> activate(XS)
   repItems(nil) -> nil
   repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS))))
   incr(X) -> n__incr(X)
   take(X1, X2) -> n__take(X1, X2)
   zip(X1, X2) -> n__zip(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   repItems(X) -> n__repItems(X)
   activate(n__incr(X)) -> incr(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__zip(X1, X2)) -> zip(X1, X2)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__repItems(X)) -> repItems(X)
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(n__zip(x_1, x_2)) -> n__zip(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__repItems(x_1)) -> n__repItems(encArg(x_1))
   encArg(cons_pairNs) -> pairNs
   encArg(cons_oddNs) -> oddNs
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_repItems(x_1)) -> repItems(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_pairNs -> pairNs
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_oddNs -> oddNs
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2))
   encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_n__zip(x_1, x_2) -> n__zip(encArg(x_1), encArg(x_2))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_repItems(x_1) -> repItems(encArg(x_1))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__repItems(x_1) -> n__repItems(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(INF, INF).


The TRS R consists of the following rules:

   pairNs -> cons(0, n__incr(oddNs))
   oddNs -> incr(pairNs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   zip(nil, XS) -> nil
   zip(X, nil) -> nil
   zip(cons(X, XS), cons(Y, YS)) -> cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
   tail(cons(X, XS)) -> activate(XS)
   repItems(nil) -> nil
   repItems(cons(X, XS)) -> cons(X, n__cons(X, n__repItems(activate(XS))))
   incr(X) -> n__incr(X)
   take(X1, X2) -> n__take(X1, X2)
   zip(X1, X2) -> n__zip(X1, X2)
   cons(X1, X2) -> n__cons(X1, X2)
   repItems(X) -> n__repItems(X)
   activate(n__incr(X)) -> incr(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(n__zip(X1, X2)) -> zip(X1, X2)
   activate(n__cons(X1, X2)) -> cons(X1, X2)
   activate(n__repItems(X)) -> repItems(X)
   activate(X) -> X

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

   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(n__zip(x_1, x_2)) -> n__zip(encArg(x_1), encArg(x_2))
   encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2))
   encArg(n__repItems(x_1)) -> n__repItems(encArg(x_1))
   encArg(cons_pairNs) -> pairNs
   encArg(cons_oddNs) -> oddNs
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_zip(x_1, x_2)) -> zip(encArg(x_1), encArg(x_2))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_repItems(x_1)) -> repItems(encArg(x_1))
   encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_pairNs -> pairNs
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_n__incr(x_1) -> n__incr(encArg(x_1))
   encode_oddNs -> oddNs
   encode_incr(x_1) -> incr(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_activate(x_1) -> activate(encArg(x_1))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2))
   encode_zip(x_1, x_2) -> zip(encArg(x_1), encArg(x_2))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_n__zip(x_1, x_2) -> n__zip(encArg(x_1), encArg(x_2))
   encode_tail(x_1) -> tail(encArg(x_1))
   encode_repItems(x_1) -> repItems(encArg(x_1))
   encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2))
   encode_n__repItems(x_1) -> n__repItems(encArg(x_1))

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

(7) InfiniteLowerBoundProof (FINISHED)
The following loop proves infinite runtime complexity:

The rewrite sequence

oddNs ->^+ incr(cons(0, n__incr(oddNs)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].

The pumping substitution is [ ].

The result substitution is [ ].




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

(8)
BOUNDS(INF, INF)