/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 131 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) 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:

   active(f(b, c, x)) -> mark(f(x, x, x))
   active(f(x, y, z)) -> f(x, y, active(z))
   active(d) -> m(b)
   f(x, y, mark(z)) -> mark(f(x, y, z))
   active(d) -> mark(c)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z))
   f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z))
   top(mark(x)) -> top(proper(x))
   top(ok(x)) -> top(active(x))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: active, proper
The following defined symbols can occur below the 0th argument of active: active, proper
The following defined symbols can occur below the 0th argument of proper: active, proper

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(b, c, x)) -> mark(f(x, x, x))
active(f(x, y, z)) -> f(x, y, active(z))
proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z))

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

(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:

   active(d) -> m(b)
   f(x, y, mark(z)) -> mark(f(x, y, z))
   active(d) -> mark(c)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z))
   top(mark(x)) -> top(proper(x))
   top(ok(x)) -> top(active(x))

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

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

(4)
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:

   active(d) -> m(b)
   f(x, y, mark(z)) -> mark(f(x, y, z))
   active(d) -> mark(c)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z))
   top(mark(x)) -> top(proper(x))
   top(ok(x)) -> top(active(x))

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

(5) 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 4. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
d0() -> 0
m0(0) -> 0
b0() -> 0
mark0(0) -> 0
c0() -> 0
ok0(0) -> 0
active0(0) -> 1
f0(0, 0, 0) -> 2
proper0(0) -> 3
top0(0) -> 4
b1() -> 5
m1(5) -> 1
f1(0, 0, 0) -> 6
mark1(6) -> 2
c1() -> 7
mark1(7) -> 1
b1() -> 8
ok1(8) -> 3
c1() -> 9
ok1(9) -> 3
d1() -> 10
ok1(10) -> 3
f1(0, 0, 0) -> 11
ok1(11) -> 2
proper1(0) -> 12
top1(12) -> 4
active1(0) -> 13
top1(13) -> 4
m1(5) -> 13
mark1(6) -> 6
mark1(6) -> 11
mark1(7) -> 13
ok1(8) -> 12
ok1(9) -> 12
ok1(10) -> 12
ok1(11) -> 6
ok1(11) -> 11
proper2(7) -> 14
top2(14) -> 4
active2(8) -> 15
top2(15) -> 4
active2(9) -> 15
active2(10) -> 15
b2() -> 16
m2(16) -> 15
c2() -> 17
mark2(17) -> 15
c2() -> 18
ok2(18) -> 14
proper3(17) -> 19
top3(19) -> 4
active3(18) -> 20
top3(20) -> 4
c3() -> 21
ok3(21) -> 19
active4(21) -> 22
top4(22) -> 4

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

(6)
BOUNDS(1, n^1)

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

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

(8)
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:

   active(f(b, c, x)) -> mark(f(x, x, x))
   active(f(x, y, z)) -> f(x, y, active(z))
   active(d) -> m(b)
   f(x, y, mark(z)) -> mark(f(x, y, z))
   active(d) -> mark(c)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z))
   f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z))
   top(mark(x)) -> top(proper(x))
   top(ok(x)) -> top(active(x))

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

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

The rewrite sequence

f(ok(x), ok(y), ok(z)) ->^+ ok(f(x, y, z))

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

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

The result substitution is [ ].




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

(10)
Complex Obligation (BEST)

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

(11)
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:

   active(f(b, c, x)) -> mark(f(x, x, x))
   active(f(x, y, z)) -> f(x, y, active(z))
   active(d) -> m(b)
   f(x, y, mark(z)) -> mark(f(x, y, z))
   active(d) -> mark(c)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z))
   f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z))
   top(mark(x)) -> top(proper(x))
   top(ok(x)) -> top(active(x))

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
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:

   active(f(b, c, x)) -> mark(f(x, x, x))
   active(f(x, y, z)) -> f(x, y, active(z))
   active(d) -> m(b)
   f(x, y, mark(z)) -> mark(f(x, y, z))
   active(d) -> mark(c)
   proper(b) -> ok(b)
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z))
   f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z))
   top(mark(x)) -> top(proper(x))
   top(ok(x)) -> top(active(x))

S is empty.
Rewrite Strategy: FULL