/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), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(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


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

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


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

(3) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. 
The certificate found is represented by the following graph.

"[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]
{(31,32,[f_1|0, c_1|0, h_1|0, g_1|0, d_1|0, activate_1|0, n__f_1|1, n__g_1|1, n__d_1|1, g_1|1, d_1|1, n__g_1|2, n__d_1|2]), (31,33,[d_1|1, n__d_1|2]), (31,34,[c_1|1]), (31,35,[f_1|1, n__f_1|2]), (31,36,[d_1|2, n__d_1|3]), (31,37,[c_1|2]), (31,40,[d_1|3, n__d_1|4]), (32,32,[n__f_1|0, n__g_1|0, n__d_1|0]), (33,32,[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]), (33,35,[f_1|1, n__f_1|2]), (33,37,[c_1|2]), (33,40,[d_1|3, n__d_1|4]), (34,32,[n__d_1|1]), (35,32,[activate_1|1, n__f_1|1, g_1|1, n__g_1|1, d_1|1, n__d_1|1, n__g_1|2, n__d_1|2]), (35,35,[f_1|1, n__f_1|2]), (35,37,[c_1|2]), (35,40,[d_1|3, n__d_1|4]), (36,34,[activate_1|2]), (36,32,[d_1|2, n__d_1|2, n__d_1|3]), (37,38,[n__f_1|2]), (38,39,[n__g_1|2]), (39,35,[n__f_1|2]), (40,37,[activate_1|3]), (40,41,[f_1|3, n__f_1|4]), (40,38,[n__f_1|3]), (41,38,[activate_1|3]), (41,39,[g_1|3, n__g_1|3, n__g_1|4])}"
----------------------------------------

(4)
BOUNDS(1, n^1)

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

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

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

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

The rewrite sequence

activate(n__f(X)) ->^+ f(activate(X))

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

The pumping substitution is [X / n__f(X)].

The result substitution is [ ].




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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(f(X)) -> c(n__f(n__g(n__f(X))))
   c(X) -> d(activate(X))
   h(X) -> c(n__d(X))
   f(X) -> n__f(X)
   g(X) -> n__g(X)
   d(X) -> n__d(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__g(X)) -> g(X)
   activate(n__d(X)) -> d(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL