/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 218 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 0 ms]
(8) BOUNDS(INF, INF)


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

(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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0, X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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(true) -> true
   encArg(false) -> false
   encArg(cons) -> cons
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s) -> s
   encArg(cons_eq) -> eq
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encode_eq -> eq
   encode_true -> true
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons -> cons
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s -> s
   encode_length(x_1) -> length(encArg(x_1))


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

(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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0, X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons) -> cons
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s) -> s
   encArg(cons_eq) -> eq
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encode_eq -> eq
   encode_true -> true
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons -> cons
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s -> s
   encode_length(x_1) -> length(encArg(x_1))

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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0, X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons) -> cons
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s) -> s
   encArg(cons_eq) -> eq
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encode_eq -> eq
   encode_true -> true
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons -> cons
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s -> s
   encode_length(x_1) -> length(encArg(x_1))

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:

   eq -> true
   eq -> eq
   eq -> false
   inf(X) -> cons
   take(0, X) -> nil
   take(s, cons) -> cons
   length(nil) -> 0
   length(cons) -> s

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

   encArg(true) -> true
   encArg(false) -> false
   encArg(cons) -> cons
   encArg(0) -> 0
   encArg(nil) -> nil
   encArg(s) -> s
   encArg(cons_eq) -> eq
   encArg(cons_inf(x_1)) -> inf(encArg(x_1))
   encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encode_eq -> eq
   encode_true -> true
   encode_false -> false
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_cons -> cons
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_nil -> nil
   encode_s -> s
   encode_length(x_1) -> length(encArg(x_1))

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

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

The rewrite sequence

eq ->^+ eq

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)