/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 411 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 0 ms]
(8) BOUNDS(INF, INF)


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(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:

   nats -> cons(0, n__incr(nats))
   pairs -> cons(0, n__incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   activate(n__incr(X)) -> incr(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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_pairs) -> pairs
   encArg(cons_odds) -> odds
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_nats -> nats
   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_pairs -> pairs
   encode_odds -> odds
   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_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))


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(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:

   nats -> cons(0, n__incr(nats))
   pairs -> cons(0, n__incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   activate(n__incr(X)) -> incr(X)
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_pairs) -> pairs
   encArg(cons_odds) -> odds
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_nats -> nats
   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_pairs -> pairs
   encode_odds -> odds
   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_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

Rewrite Strategy: INNERMOST
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(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:

   nats -> cons(0, n__incr(nats))
   pairs -> cons(0, n__incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   activate(n__incr(X)) -> incr(X)
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_pairs) -> pairs
   encArg(cons_odds) -> odds
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_nats -> nats
   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_pairs -> pairs
   encode_odds -> odds
   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_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

Rewrite Strategy: INNERMOST
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(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:

   nats -> cons(0, n__incr(nats))
   pairs -> cons(0, n__incr(odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   activate(n__incr(X)) -> incr(X)
   activate(X) -> X

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(n__incr(x_1)) -> n__incr(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_nats) -> nats
   encArg(cons_pairs) -> pairs
   encArg(cons_odds) -> odds
   encArg(cons_incr(x_1)) -> incr(encArg(x_1))
   encArg(cons_head(x_1)) -> head(encArg(x_1))
   encArg(cons_tail(x_1)) -> tail(encArg(x_1))
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_nats -> nats
   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_pairs -> pairs
   encode_odds -> odds
   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_head(x_1) -> head(encArg(x_1))
   encode_tail(x_1) -> tail(encArg(x_1))

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

The rewrite sequence

nats ->^+ cons(0, n__incr(nats))

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

The pumping substitution is [ ].

The result substitution is [ ].




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(8)
BOUNDS(INF, INF)