/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 351 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(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), 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(h) -> h
   encArg(1) -> 1
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_a(x_1, x_2, x_3, x_4)) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_a(x_1, x_2, x_3, x_4) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_h -> h
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(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:

   a(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))

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

   encArg(h) -> h
   encArg(1) -> 1
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_a(x_1, x_2, x_3, x_4)) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_a(x_1, x_2, x_3, x_4) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_h -> h
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(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:

   a(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))

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

   encArg(h) -> h
   encArg(1) -> 1
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_a(x_1, x_2, x_3, x_4)) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_a(x_1, x_2, x_3, x_4) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_h -> h
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST
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(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(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))

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

   encArg(h) -> h
   encArg(1) -> 1
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_a(x_1, x_2, x_3, x_4)) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_a(x_1, x_2, x_3, x_4) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_h -> h
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(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

app(cons(x, l), k) ->^+ cons(x, app(l, k))

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

The pumping substitution is [l / cons(x, 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(n^1, INF).


The TRS R consists of the following rules:

   a(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))

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

   encArg(h) -> h
   encArg(1) -> 1
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_a(x_1, x_2, x_3, x_4)) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_a(x_1, x_2, x_3, x_4) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_h -> h
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST
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(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(h, h, h, x) -> s(x)
   a(l, x, s(y), h) -> a(l, x, y, s(h))
   a(l, x, s(y), s(z)) -> a(l, x, y, a(l, x, s(y), z))
   a(l, s(x), h, z) -> a(l, x, z, z)
   a(s(l), h, h, z) -> a(l, z, h, z)
   +(x, h) -> x
   +(h, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   s(h) -> 1
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h, h), l))

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

   encArg(h) -> h
   encArg(1) -> 1
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_a(x_1, x_2, x_3, x_4)) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encode_a(x_1, x_2, x_3, x_4) -> a(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_h -> h
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))

Rewrite Strategy: INNERMOST