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


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WORST_CASE(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 267 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
        (13) InfiniteLowerBoundProof [FINISHED, 0 ms]
        (14) BOUNDS(INF, INF)


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

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


The TRS R consists of the following rules:

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

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(nil) -> nil
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   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(x_1, x_2) -> 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_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(encArg(x_1))


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

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


The TRS R consists of the following rules:

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

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(nil) -> nil
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   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(x_1, x_2) -> 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_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(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(INF, INF).


The TRS R consists of the following rules:

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

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(nil) -> nil
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   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(x_1, x_2) -> 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_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(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(INF, INF).


The TRS R consists of the following rules:

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

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(nil) -> nil
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   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(x_1, x_2) -> 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_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(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

take(s(X), cons(Y, L)) ->^+ cons(Y, take(X, L))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [X / s(X), L / cons(Y, L)].

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(INF, INF).


The TRS R consists of the following rules:

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

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(nil) -> nil
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   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(x_1, x_2) -> 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_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(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(INF, INF).


The TRS R consists of the following rules:

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

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(nil) -> nil
   encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2))
   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(x_1, x_2) -> 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_inf(x_1) -> inf(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_length(x_1) -> length(encArg(x_1))

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

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

The rewrite sequence

inf(X) ->^+ cons(X, inf(s(X)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [ ].

The result substitution is [X / s(X)].




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(14)
BOUNDS(INF, INF)