/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(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 330 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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_avg(x_1)) -> avg(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_length(x_1) -> length(encArg(x_1))
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_avg(x_1) -> avg(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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_avg(x_1)) -> avg(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_length(x_1) -> length(encArg(x_1))
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_avg(x_1) -> avg(encArg(x_1))

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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_avg(x_1)) -> avg(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_length(x_1) -> length(encArg(x_1))
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_avg(x_1) -> avg(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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_avg(x_1)) -> avg(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_length(x_1) -> length(encArg(x_1))
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_avg(x_1) -> avg(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

-(s(x), s(y)) ->^+ -(x, y)

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

The pumping substitution is [x / s(x), y / s(y)].

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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_avg(x_1)) -> avg(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_length(x_1) -> length(encArg(x_1))
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_avg(x_1) -> avg(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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(:(x_1, x_2)) -> :(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_length(x_1)) -> length(encArg(x_1))
   encArg(cons_hd(x_1)) -> hd(encArg(x_1))
   encArg(cons_avg(x_1)) -> avg(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_:(x_1, x_2) -> :(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_length(x_1) -> length(encArg(x_1))
   encode_hd(x_1) -> hd(encArg(x_1))
   encode_avg(x_1) -> avg(encArg(x_1))

Rewrite Strategy: INNERMOST