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


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 342 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 30 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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> cons(x, l)
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

S is empty.
Rewrite Strategy: INNERMOST
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(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(nil) -> nil
   encArg(true) -> true
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(cons_is_empty(x_1)) -> is_empty(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_tl(x_1)) -> tl(encArg(x_1))
   encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2))
   encArg(cons_ifappend(x_1, x_2, x_3)) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_is_empty(x_1) -> is_empty(encArg(x_1))
   encode_nil -> nil
   encode_true -> true
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_tl(x_1) -> tl(encArg(x_1))
   encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2))
   encode_ifappend(x_1, x_2, x_3) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))


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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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> cons(x, l)
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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

   encArg(nil) -> nil
   encArg(true) -> true
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(cons_is_empty(x_1)) -> is_empty(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_tl(x_1)) -> tl(encArg(x_1))
   encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2))
   encArg(cons_ifappend(x_1, x_2, x_3)) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_is_empty(x_1) -> is_empty(encArg(x_1))
   encode_nil -> nil
   encode_true -> true
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_tl(x_1) -> tl(encArg(x_1))
   encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2))
   encode_ifappend(x_1, x_2, x_3) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> cons(x, l)
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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

   encArg(nil) -> nil
   encArg(true) -> true
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(cons_is_empty(x_1)) -> is_empty(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_tl(x_1)) -> tl(encArg(x_1))
   encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2))
   encArg(cons_ifappend(x_1, x_2, x_3)) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_is_empty(x_1) -> is_empty(encArg(x_1))
   encode_nil -> nil
   encode_true -> true
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_tl(x_1) -> tl(encArg(x_1))
   encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2))
   encode_ifappend(x_1, x_2, x_3) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> cons(x, l)
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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

   encArg(nil) -> nil
   encArg(true) -> true
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(false) -> false
   encArg(cons_is_empty(x_1)) -> is_empty(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_tl(x_1)) -> tl(encArg(x_1))
   encArg(cons_append(x_1, x_2)) -> append(encArg(x_1), encArg(x_2))
   encArg(cons_ifappend(x_1, x_2, x_3)) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_is_empty(x_1) -> is_empty(encArg(x_1))
   encode_nil -> nil
   encode_true -> true
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_false -> false
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_tl(x_1) -> tl(encArg(x_1))
   encode_append(x_1, x_2) -> append(encArg(x_1), encArg(x_2))
   encode_ifappend(x_1, x_2, x_3) -> ifappend(encArg(x_1), encArg(x_2), encArg(x_3))

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

The rewrite sequence

ifappend(cons(x1_0, l2_0), l2, false) ->^+ cons(hd(cons(x1_0, l2_0)), ifappend(cons(x1_0, l2_0), l2, false))

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

The pumping substitution is [ ].

The result substitution is [ ].




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(8)
BOUNDS(INF, INF)