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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 169 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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2, x_3)) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_cond(x_1, x_2, x_3, x_4)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_d(x_1, x_2, x_3) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_cond(x_1, x_2, x_3, x_4) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_false -> false


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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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2, x_3)) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_cond(x_1, x_2, x_3, x_4)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_d(x_1, x_2, x_3) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_cond(x_1, x_2, x_3, x_4) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_false -> false

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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2, x_3)) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_cond(x_1, x_2, x_3, x_4)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_d(x_1, x_2, x_3) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_cond(x_1, x_2, x_3, x_4) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_false -> false

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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2, x_3)) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_cond(x_1, x_2, x_3, x_4)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_d(x_1, x_2, x_3) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_cond(x_1, x_2, x_3, x_4) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_false -> false

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

plus(n, s(m)) ->^+ s(plus(n, m))

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

The pumping substitution is [m / s(m)].

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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2, x_3)) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_cond(x_1, x_2, x_3, x_4)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_d(x_1, x_2, x_3) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_cond(x_1, x_2, x_3, x_4) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_false -> false

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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_d(x_1, x_2, x_3)) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_cond(x_1, x_2, x_3, x_4)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_d(x_1, x_2, x_3) -> d(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_cond(x_1, x_2, x_3, x_4) -> cond(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_false -> false

Rewrite Strategy: INNERMOST