/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CpxTrsMatchBoundsTAProof [FINISHED, 45 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(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
00() -> 0
s0(0) -> 0
*0(0, 0) -> 0
+0(0, 0) -> 0
f0(0) -> 1
01() -> 2
s1(2) -> 1
s1(2) -> 3
s1(3) -> 1
s1(3) -> 4
01() -> 6
f1(6) -> 5
*1(4, 5) -> 1
s1(3) -> 7
f1(0) -> 8
+1(7, 8) -> 1
f1(0) -> 9
f1(0) -> 10
*1(9, 10) -> 1
s1(2) -> 8
s1(2) -> 9
s1(2) -> 10
02() -> 11
s2(11) -> 5
s1(3) -> 8
s1(3) -> 9
s1(3) -> 10
*1(4, 5) -> 8
*1(4, 5) -> 9
*1(4, 5) -> 10
+1(7, 8) -> 8
+1(7, 8) -> 9
+1(7, 8) -> 10
*1(9, 10) -> 8
*1(9, 10) -> 9
*1(9, 10) -> 10

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

(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(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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

f(+(x, y)) ->^+ *(f(x), f(y))

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

The pumping substitution is [x / +(x, y)].

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(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

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(0) -> s(0)
   f(s(0)) -> s(s(0))
   f(s(0)) -> *(s(s(0)), f(0))
   f(+(x, s(0))) -> +(s(s(0)), f(x))
   f(+(x, y)) -> *(f(x), f(y))

S is empty.
Rewrite Strategy: FULL