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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 13.3 s]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 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:

   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))

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

   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))

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:

   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))

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

   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))

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:

   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))

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

   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))

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

DOUBLE(s(z0)) ->^+ c8(DOUBLE(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:

   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))

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

   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))

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:

   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))

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

   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))

Rewrite Strategy: INNERMOST