/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 12 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(x))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(2)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(x))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

exp(s(x)) ->^+ double(exp(x))

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

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

The result substitution is [ ].




----------------------------------------

(4)
Complex Obligation (BEST)

----------------------------------------

(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(x))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(7)
BOUNDS(n^1, INF)

----------------------------------------

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(x))

S is empty.
Rewrite Strategy: INNERMOST