/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CpxTrsMatchBoundsTAProof [FINISHED, 77 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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
L0(0) -> 0
N0(0, 0) -> 0
00() -> 0
s0(0) -> 0
max0(0) -> 1
L1(0) -> 4
L1(0) -> 5
N1(4, 5) -> 3
max1(3) -> 2
s1(2) -> 1
N1(0, 0) -> 8
max1(8) -> 7
L1(7) -> 6
N1(4, 6) -> 3
max1(3) -> 1
s1(2) -> 7
max1(3) -> 7
L1(2) -> 5
0 -> 1
0 -> 7
0 -> 2
7 -> 1
7 -> 2
2 -> 1
2 -> 7

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

(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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

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

max(N(L(s(x)), L(s(y)))) ->^+ s(max(N(L(x), L(y))))

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

The pumping substitution is [x / s(x), y / s(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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

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:

   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

S is empty.
Rewrite Strategy: FULL