WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 850 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 1 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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(0)
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:

   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))

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

   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

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

   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))

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

   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(4)
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:

   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))

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

   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

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

The rewrite sequence

G(d, s(z0)) ->^+ c1(G(d, z0))

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

The pumping substitution is [z0 / s(z0)].

The result substitution is [ ].




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(6)
Complex Obligation (BEST)

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(7)
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:

   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))

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

   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
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:

   G(z0, 0) -> c
   G(d, s(z0)) -> c1(G(d, z0))
   G(h, s(0)) -> c2
   G(h, s(s(z0))) -> c3(G(h, z0))
   DOUBLE(z0) -> c4(G(d, z0))
   HALF(z0) -> c5(G(h, z0))
   F(s(z0), z1) -> c6(F(half(s(z0)), double(z1)), HALF(s(z0)))
   F(s(z0), z1) -> c7(F(half(s(z0)), double(z1)), DOUBLE(z1))
   F(s(0), z0) -> c8
   ID(z0) -> c9(F(z0, s(0)))

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

   g(z0, 0) -> 0
   g(d, s(z0)) -> s(s(g(d, z0)))
   g(h, s(0)) -> 0
   g(h, s(s(z0))) -> s(g(h, z0))
   double(z0) -> g(d, z0)
   half(z0) -> g(h, z0)
   f(s(z0), z1) -> f(half(s(z0)), double(z1))
   f(s(0), z0) -> z0
   id(z0) -> f(z0, s(0))

Rewrite Strategy: INNERMOST