/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 167 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:

   f(tt, x) -> f(isList(x), x)
   isList(Cons(x, xs)) -> isList(xs)
   isList(nil) -> tt

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(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(xs) -> xs
   encArg(nil) -> nil
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_xs -> xs
   encode_nil -> nil


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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:

   f(tt, x) -> f(isList(x), x)
   isList(Cons(x, xs)) -> isList(xs)
   isList(nil) -> tt

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

   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(xs) -> xs
   encArg(nil) -> nil
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_xs -> xs
   encode_nil -> nil

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:

   f(tt, x) -> f(isList(x), x)
   isList(Cons(x, xs)) -> isList(xs)
   isList(nil) -> tt

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

   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(xs) -> xs
   encArg(nil) -> nil
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_xs -> xs
   encode_nil -> nil

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:

   f(tt, x) -> f(isList(x), x)
   isList(Cons(x, xs)) -> isList(xs)
   isList(nil) -> tt

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

   encArg(tt) -> tt
   encArg(Cons(x_1, x_2)) -> Cons(encArg(x_1), encArg(x_2))
   encArg(xs) -> xs
   encArg(nil) -> nil
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_isList(x_1)) -> isList(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_tt -> tt
   encode_isList(x_1) -> isList(encArg(x_1))
   encode_Cons(x_1, x_2) -> Cons(encArg(x_1), encArg(x_2))
   encode_xs -> xs
   encode_nil -> nil

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

The rewrite sequence

f(tt, nil) ->^+ f(tt, nil)

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

The pumping substitution is [ ].

The result substitution is [ ].




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(8)
BOUNDS(INF, INF)