/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 313 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__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(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(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(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__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

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:

   a__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(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__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(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(length(X)) ->^+ a__length(mark(X))

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

The pumping substitution is [X / length(X)].

The result substitution is [ ].




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

(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__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(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__eq(0, 0) -> true
   a__eq(s(X), s(Y)) -> a__eq(X, Y)
   a__eq(X, Y) -> false
   a__inf(X) -> cons(X, inf(s(X)))
   a__take(0, X) -> nil
   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
   a__length(nil) -> 0
   a__length(cons(X, L)) -> s(length(L))
   mark(eq(X1, X2)) -> a__eq(X1, X2)
   mark(inf(X)) -> a__inf(mark(X))
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(length(X)) -> a__length(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(s(X)) -> s(X)
   mark(false) -> false
   mark(cons(X1, X2)) -> cons(X1, X2)
   mark(nil) -> nil
   a__eq(X1, X2) -> eq(X1, X2)
   a__inf(X) -> inf(X)
   a__take(X1, X2) -> take(X1, X2)
   a__length(X) -> length(X)

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(inf(x_1)) -> inf(encArg(x_1))
   encArg(nil) -> nil
   encArg(take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2))
   encArg(length(x_1)) -> length(encArg(x_1))
   encArg(eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__eq(x_1, x_2)) -> a__eq(encArg(x_1), encArg(x_2))
   encArg(cons_a__inf(x_1)) -> a__inf(encArg(x_1))
   encArg(cons_a__take(x_1, x_2)) -> a__take(encArg(x_1), encArg(x_2))
   encArg(cons_a__length(x_1)) -> a__length(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__eq(x_1, x_2) -> a__eq(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_a__inf(x_1) -> a__inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_inf(x_1) -> inf(encArg(x_1))
   encode_a__take(x_1, x_2) -> a__take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_a__length(x_1) -> a__length(encArg(x_1))
   encode_length(x_1) -> length(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST