/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 95 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 1 ms]
(8) BOUNDS(INF, INF)


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


The TRS R consists of the following rules:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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

(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)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_length) -> length
   encArg(cons_length1) -> length1
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_length -> length
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_length1 -> length1


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

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


The TRS R consists of the following rules:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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

   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_length) -> length
   encArg(cons_length1) -> length1
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_length -> length
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_length1 -> length1

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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

   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_length) -> length
   encArg(cons_length1) -> length1
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_length -> length
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_length1 -> length1

Rewrite Strategy: FULL
----------------------------------------

(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 (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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

   encArg(cons(x_1)) -> cons(encArg(x_1))
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_from(x_1)) -> from(encArg(x_1))
   encArg(cons_length) -> length
   encArg(cons_length1) -> length1
   encode_from(x_1) -> from(encArg(x_1))
   encode_cons(x_1) -> cons(encArg(x_1))
   encode_length -> length
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_length1 -> length1

Rewrite Strategy: FULL
----------------------------------------

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

The rewrite sequence

length ->^+ s(length)

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

The pumping substitution is [ ].

The result substitution is [ ].




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(8)
BOUNDS(INF, INF)