/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 189 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


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

(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:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(from(x_1)) -> from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(after(x_1, x_2)) -> after(encArg(x_1), encArg(x_2))
   encArg(cons_a__from(x_1)) -> a__from(encArg(x_1))
   encArg(cons_a__after(x_1, x_2)) -> a__after(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__from(x_1) -> a__from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a__after(x_1, x_2) -> a__after(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_after(x_1, x_2) -> after(encArg(x_1), encArg(x_2))


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

(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:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(from(x_1)) -> from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(after(x_1, x_2)) -> after(encArg(x_1), encArg(x_2))
   encArg(cons_a__from(x_1)) -> a__from(encArg(x_1))
   encArg(cons_a__after(x_1, x_2)) -> a__after(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__from(x_1) -> a__from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a__after(x_1, x_2) -> a__after(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_after(x_1, x_2) -> after(encArg(x_1), encArg(x_2))

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(n^1, INF).


The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(from(x_1)) -> from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(after(x_1, x_2)) -> after(encArg(x_1), encArg(x_2))
   encArg(cons_a__from(x_1)) -> a__from(encArg(x_1))
   encArg(cons_a__after(x_1, x_2)) -> a__after(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__from(x_1) -> a__from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a__after(x_1, x_2) -> a__after(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_after(x_1, x_2) -> after(encArg(x_1), encArg(x_2))

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:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(from(x_1)) -> from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(after(x_1, x_2)) -> after(encArg(x_1), encArg(x_2))
   encArg(cons_a__from(x_1)) -> a__from(encArg(x_1))
   encArg(cons_a__after(x_1, x_2)) -> a__after(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__from(x_1) -> a__from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a__after(x_1, x_2) -> a__after(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_after(x_1, x_2) -> after(encArg(x_1), encArg(x_2))

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

mark(from(X)) ->^+ a__from(mark(X))

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

The pumping substitution is [X / from(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:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(from(x_1)) -> from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(after(x_1, x_2)) -> after(encArg(x_1), encArg(x_2))
   encArg(cons_a__from(x_1)) -> a__from(encArg(x_1))
   encArg(cons_a__after(x_1, x_2)) -> a__after(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__from(x_1) -> a__from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a__after(x_1, x_2) -> a__after(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_after(x_1, x_2) -> after(encArg(x_1), encArg(x_2))

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:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__after(0, XS) -> mark(XS)
   a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS))
   mark(from(X)) -> a__from(mark(X))
   mark(after(X1, X2)) -> a__after(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__from(X) -> from(X)
   a__after(X1, X2) -> after(X1, X2)

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(from(x_1)) -> from(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(after(x_1, x_2)) -> after(encArg(x_1), encArg(x_2))
   encArg(cons_a__from(x_1)) -> a__from(encArg(x_1))
   encArg(cons_a__after(x_1, x_2)) -> a__after(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__from(x_1) -> a__from(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_from(x_1) -> from(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a__after(x_1, x_2) -> a__after(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_after(x_1, x_2) -> after(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST