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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 194 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


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(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(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(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0


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(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
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(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

g(e(x), e(y)) ->^+ e(g(x, y))

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

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

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

   encArg(e(x_1)) -> e(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2)) -> d(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e(x_1) -> e(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1, x_2) -> d(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0

Rewrite Strategy: INNERMOST