/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(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 78 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(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(bot) -> bot
   encArg(up(x_1)) -> up(encArg(x_1))
   encArg(cons_rec(x_1)) -> rec(encArg(x_1))
   encArg(cons_sent(x_1)) -> sent(encArg(x_1))
   encArg(cons_no(x_1)) -> no(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encode_rec(x_1) -> rec(encArg(x_1))
   encode_sent(x_1) -> sent(encArg(x_1))
   encode_no(x_1) -> no(encArg(x_1))
   encode_bot -> bot
   encode_up(x_1) -> up(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_check(x_1) -> check(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

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

   encArg(bot) -> bot
   encArg(up(x_1)) -> up(encArg(x_1))
   encArg(cons_rec(x_1)) -> rec(encArg(x_1))
   encArg(cons_sent(x_1)) -> sent(encArg(x_1))
   encArg(cons_no(x_1)) -> no(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encode_rec(x_1) -> rec(encArg(x_1))
   encode_sent(x_1) -> sent(encArg(x_1))
   encode_no(x_1) -> no(encArg(x_1))
   encode_bot -> bot
   encode_up(x_1) -> up(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_check(x_1) -> check(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(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(n^1, INF).


The TRS R consists of the following rules:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

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

   encArg(bot) -> bot
   encArg(up(x_1)) -> up(encArg(x_1))
   encArg(cons_rec(x_1)) -> rec(encArg(x_1))
   encArg(cons_sent(x_1)) -> sent(encArg(x_1))
   encArg(cons_no(x_1)) -> no(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encode_rec(x_1) -> rec(encArg(x_1))
   encode_sent(x_1) -> sent(encArg(x_1))
   encode_no(x_1) -> no(encArg(x_1))
   encode_bot -> bot
   encode_up(x_1) -> up(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_check(x_1) -> check(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(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(n^1, INF).


The TRS R consists of the following rules:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

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

   encArg(bot) -> bot
   encArg(up(x_1)) -> up(encArg(x_1))
   encArg(cons_rec(x_1)) -> rec(encArg(x_1))
   encArg(cons_sent(x_1)) -> sent(encArg(x_1))
   encArg(cons_no(x_1)) -> no(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encode_rec(x_1) -> rec(encArg(x_1))
   encode_sent(x_1) -> sent(encArg(x_1))
   encode_no(x_1) -> no(encArg(x_1))
   encode_bot -> bot
   encode_up(x_1) -> up(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_check(x_1) -> check(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

rec(up(x)) ->^+ up(rec(x))

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

The pumping substitution is [x / up(x)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

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

   encArg(bot) -> bot
   encArg(up(x_1)) -> up(encArg(x_1))
   encArg(cons_rec(x_1)) -> rec(encArg(x_1))
   encArg(cons_sent(x_1)) -> sent(encArg(x_1))
   encArg(cons_no(x_1)) -> no(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encode_rec(x_1) -> rec(encArg(x_1))
   encode_sent(x_1) -> sent(encArg(x_1))
   encode_no(x_1) -> no(encArg(x_1))
   encode_bot -> bot
   encode_up(x_1) -> up(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_check(x_1) -> check(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

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

   encArg(bot) -> bot
   encArg(up(x_1)) -> up(encArg(x_1))
   encArg(cons_rec(x_1)) -> rec(encArg(x_1))
   encArg(cons_sent(x_1)) -> sent(encArg(x_1))
   encArg(cons_no(x_1)) -> no(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encArg(cons_check(x_1)) -> check(encArg(x_1))
   encode_rec(x_1) -> rec(encArg(x_1))
   encode_sent(x_1) -> sent(encArg(x_1))
   encode_no(x_1) -> no(encArg(x_1))
   encode_bot -> bot
   encode_up(x_1) -> up(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))
   encode_check(x_1) -> check(encArg(x_1))

Rewrite Strategy: INNERMOST