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


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


NO
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be disproven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPOrderProof [EQUIVALENT, 52 ms]
        (12) QDP
        (13) QDPOrderProof [EQUIVALENT, 11 ms]
        (14) QDP
        (15) TransformationProof [EQUIVALENT, 4 ms]
        (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 1498 ms]
        (18) QDP
        (19) QDPOrderProof [EQUIVALENT, 2181 ms]
        (20) QDP
        (21) SplitQDPProof [EQUIVALENT, 0 ms]
        (22) AND
            (23) QDP
                (24) SemLabProof [SOUND, 0 ms]
                (25) QDP
                (26) QDPOrderProof [EQUIVALENT, 77 ms]
                (27) QDP
                (28) QDPOrderProof [EQUIVALENT, 82 ms]
                (29) QDP
                (30) QDPOrderProof [EQUIVALENT, 40 ms]
                (31) QDP
                (32) PisEmptyProof [SOUND, 0 ms]
                (33) TRUE
            (34) QDP
                (35) NonTerminationLoopProof [COMPLETE, 12.1 s]
                (36) NO


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

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.

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

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   ACTIVE(f(a, b, X)) -> F(X, X, X)
   ACTIVE(c) -> MARK(a)
   ACTIVE(c) -> MARK(b)
   MARK(f(X1, X2, X3)) -> ACTIVE(f(mark(X1), X2, mark(X3)))
   MARK(f(X1, X2, X3)) -> F(mark(X1), X2, mark(X3))
   MARK(f(X1, X2, X3)) -> MARK(X1)
   MARK(f(X1, X2, X3)) -> MARK(X3)
   MARK(a) -> ACTIVE(a)
   MARK(b) -> ACTIVE(b)
   MARK(c) -> ACTIVE(c)
   F(mark(X1), X2, X3) -> F(X1, X2, X3)
   F(X1, mark(X2), X3) -> F(X1, X2, X3)
   F(X1, X2, mark(X3)) -> F(X1, X2, X3)
   F(active(X1), X2, X3) -> F(X1, X2, X3)
   F(X1, active(X2), X3) -> F(X1, X2, X3)
   F(X1, X2, active(X3)) -> F(X1, X2, X3)

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(X1, mark(X2), X3) -> F(X1, X2, X3)
   F(mark(X1), X2, X3) -> F(X1, X2, X3)
   F(X1, X2, mark(X3)) -> F(X1, X2, X3)
   F(active(X1), X2, X3) -> F(X1, X2, X3)
   F(X1, active(X2), X3) -> F(X1, X2, X3)
   F(X1, X2, active(X3)) -> F(X1, X2, X3)

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(6) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(X1, mark(X2), X3) -> F(X1, X2, X3)
   F(mark(X1), X2, X3) -> F(X1, X2, X3)
   F(X1, X2, mark(X3)) -> F(X1, X2, X3)
   F(active(X1), X2, X3) -> F(X1, X2, X3)
   F(X1, active(X2), X3) -> F(X1, X2, X3)
   F(X1, X2, active(X3)) -> F(X1, X2, X3)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F(X1, mark(X2), X3) -> F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*F(mark(X1), X2, X3) -> F(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*F(X1, X2, mark(X3)) -> F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


*F(active(X1), X2, X3) -> F(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*F(X1, active(X2), X3) -> F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*F(X1, X2, active(X3)) -> F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3


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

(9)
YES

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(f(X1, X2, X3)) -> ACTIVE(f(mark(X1), X2, mark(X3)))
   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(X1, X2, X3)) -> MARK(X1)
   MARK(f(X1, X2, X3)) -> MARK(X3)

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(f(X1, X2, X3)) -> MARK(X3)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(f(x_1, x_2, x_3)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a) =  	[[0A]]
>>>

   <<<
 POL(b) =  	[[0A]]
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(c) =  	[[5A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   active(f(a, b, X)) -> mark(f(X, X, X))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)
   active(c) -> mark(a)
   active(c) -> mark(b)


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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(f(X1, X2, X3)) -> ACTIVE(f(mark(X1), X2, mark(X3)))
   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(X1, X2, X3)) -> MARK(X1)

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(f(X1, X2, X3)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(f(x_1, x_2, x_3)) =  	[[0A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[-I]] 	* 	x_2 	 +  	[[1A]] 	* 	x_3
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(a) =  	[[2A]]
>>>

   <<<
 POL(b) =  	[[0A]]
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(c) =  	[[2A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   active(f(a, b, X)) -> mark(f(X, X, X))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)
   active(c) -> mark(a)
   active(c) -> mark(b)


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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(f(X1, X2, X3)) -> ACTIVE(f(mark(X1), X2, mark(X3)))
   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule MARK(f(X1, X2, X3)) -> ACTIVE(f(mark(X1), X2, mark(X3))) at position [0] we obtained the following new rules [LPAR04]:

   (MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2))),MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2))))
   (MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2))),MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2))))
   (MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2)),MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2)))
   (MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2))),MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2))))
   (MARK(f(f(x0, x1, x2), y1, y2)) -> ACTIVE(f(active(f(mark(x0), x1, mark(x2))), y1, mark(y2))),MARK(f(f(x0, x1, x2), y1, y2)) -> ACTIVE(f(active(f(mark(x0), x1, mark(x2))), y1, mark(y2))))
   (MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2))),MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2))))
   (MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2))),MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2))))
   (MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2))),MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2))))
   (MARK(f(y0, y1, f(x0, x1, x2))) -> ACTIVE(f(mark(y0), y1, active(f(mark(x0), x1, mark(x2))))),MARK(f(y0, y1, f(x0, x1, x2))) -> ACTIVE(f(mark(y0), y1, active(f(mark(x0), x1, mark(x2))))))
   (MARK(f(y0, y1, a)) -> ACTIVE(f(mark(y0), y1, active(a))),MARK(f(y0, y1, a)) -> ACTIVE(f(mark(y0), y1, active(a))))
   (MARK(f(y0, y1, b)) -> ACTIVE(f(mark(y0), y1, active(b))),MARK(f(y0, y1, b)) -> ACTIVE(f(mark(y0), y1, active(b))))
   (MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c))),MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c))))


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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2)))
   MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2))
   MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(f(x0, x1, x2), y1, y2)) -> ACTIVE(f(active(f(mark(x0), x1, mark(x2))), y1, mark(y2)))
   MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2)))
   MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2)))
   MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2)))
   MARK(f(y0, y1, f(x0, x1, x2))) -> ACTIVE(f(mark(y0), y1, active(f(mark(x0), x1, mark(x2)))))
   MARK(f(y0, y1, a)) -> ACTIVE(f(mark(y0), y1, active(a)))
   MARK(f(y0, y1, b)) -> ACTIVE(f(mark(y0), y1, active(b)))
   MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c)))

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(f(y0, y1, a)) -> ACTIVE(f(mark(y0), y1, active(a)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(f(x_1, x_2, x_3)) =  	[[0A], [2A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [-I, -I, -I]] 	* 	x_1 	 +  	[[-I, -I, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_2 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [1A, -I, -I]] 	* 	x_3
>>>

   <<<
 POL(a) =  	[[0A], [0A], [0A]]
>>>

   <<<
 POL(b) =  	[[2A], [0A], [0A]]
>>>

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(c) =  	[[2A], [0A], [-I]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   active(f(a, b, X)) -> mark(f(X, X, X))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   active(c) -> mark(a)
   active(c) -> mark(b)


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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2)))
   MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2))
   MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(f(x0, x1, x2), y1, y2)) -> ACTIVE(f(active(f(mark(x0), x1, mark(x2))), y1, mark(y2)))
   MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2)))
   MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2)))
   MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2)))
   MARK(f(y0, y1, f(x0, x1, x2))) -> ACTIVE(f(mark(y0), y1, active(f(mark(x0), x1, mark(x2)))))
   MARK(f(y0, y1, b)) -> ACTIVE(f(mark(y0), y1, active(b)))
   MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c)))

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(f(y0, y1, b)) -> ACTIVE(f(mark(y0), y1, active(b)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(f(x_1, x_2, x_3)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1 	 +  	[[1A, -I, -I], [-I, -I, -I], [0A, -I, -I]] 	* 	x_2 	 +  	[[1A, -I, -I], [0A, -I, 0A], [1A, -I, -I]] 	* 	x_3
>>>

   <<<
 POL(a) =  	[[2A], [0A], [-I]]
>>>

   <<<
 POL(b) =  	[[0A], [0A], [0A]]
>>>

   <<<
 POL(MARK(x_1)) =  	[[2A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(c) =  	[[2A], [0A], [0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   active(f(a, b, X)) -> mark(f(X, X, X))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   active(c) -> mark(a)
   active(c) -> mark(b)


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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2)))
   MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2))
   MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(f(x0, x1, x2), y1, y2)) -> ACTIVE(f(active(f(mark(x0), x1, mark(x2))), y1, mark(y2)))
   MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2)))
   MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2)))
   MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2)))
   MARK(f(y0, y1, f(x0, x1, x2))) -> ACTIVE(f(mark(y0), y1, active(f(mark(x0), x1, mark(x2)))))
   MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c)))

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(21) SplitQDPProof (EQUIVALENT)
We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem

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

(22)
Complex Obligation (AND)

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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2)))
   MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2))
   MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(f(x0, x1, x2), y1, y2)) -> ACTIVE(f(active(f(mark(x0), x1, mark(x2))), y1, mark(y2)))
   MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2)))
   MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2)))
   MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2)))
   MARK(f(y0, y1, f(x0, x1, x2))) -> ACTIVE(f(mark(y0), y1, active(f(mark(x0), x1, mark(x2)))))
   MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c)))

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(24) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
active: x<sub>0</sub>
a: 1
b: 1
c: 1
MARK: 0
f: 0
ACTIVE: 0
mark: x<sub>0</sub>
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ACTIVE.0(f.1-1-0(a., b., X)) -> MARK.0(f.0-0-0(X, X, X))
   MARK.0(f.0-0-0(x0, x1, y2)) -> ACTIVE.0(f.0-0-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-0-1(x0, x1, y2)) -> ACTIVE.0(f.0-0-1(x0, x1, mark.1(y2)))
   MARK.0(f.0-1-0(x0, x1, y2)) -> ACTIVE.0(f.0-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-1-1(x0, x1, y2)) -> ACTIVE.0(f.0-1-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-0-0(x0, x1, y2)) -> ACTIVE.0(f.1-0-0(x0, x1, mark.0(y2)))
   MARK.0(f.1-0-1(x0, x1, y2)) -> ACTIVE.0(f.1-0-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-1-0(x0, x1, y2)) -> ACTIVE.0(f.1-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.1-1-1(x0, x1, y2)) -> ACTIVE.0(f.1-1-1(x0, x1, mark.1(y2)))
   ACTIVE.0(f.1-1-1(a., b., X)) -> MARK.0(f.1-1-1(X, X, X))
   MARK.0(f.0-0-0(y0, mark.0(x1), y2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-0-1(y0, mark.0(x1), y2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-0-0(y0, mark.0(x1), y2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-1(y0, mark.0(x1), y2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, x1, x2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, x2))
   MARK.0(f.0-0-1(y0, x1, x2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, x2))
   MARK.0(f.0-1-0(y0, x1, x2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, x2))
   MARK.0(f.0-1-1(y0, x1, x2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, x2))
   MARK.0(f.1-0-0(y0, x1, x2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, x2))
   MARK.0(f.1-0-1(y0, x1, x2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, x2))
   MARK.0(f.1-1-0(y0, x1, x2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, x2))
   MARK.0(f.1-1-1(y0, x1, x2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, x2))
   MARK.0(f.0-0-0(y0, active.0(x1), y2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-0-1(y0, active.0(x1), y2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-0-0(y0, active.0(x1), y2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-1(y0, active.0(x1), y2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.1-0-0(a., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(a.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(a., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(a., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(a.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(a., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-0-0(b., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(b.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(b., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(b., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(b.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(b., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-0-0(c., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(c., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(c., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(c., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-1(y0, y1, c.)) -> ACTIVE.0(f.0-0-1(mark.0(y0), y1, active.1(c.)))
   MARK.0(f.0-1-1(y0, y1, c.)) -> ACTIVE.0(f.0-1-1(mark.0(y0), y1, active.1(c.)))
   MARK.0(f.1-0-1(y0, y1, c.)) -> ACTIVE.0(f.1-0-1(mark.1(y0), y1, active.1(c.)))
   MARK.0(f.1-1-1(y0, y1, c.)) -> ACTIVE.0(f.1-1-1(mark.1(y0), y1, active.1(c.)))

The TRS R consists of the following rules:

   active.0(f.1-1-0(a., b., X)) -> mark.0(f.0-0-0(X, X, X))
   active.0(f.1-1-1(a., b., X)) -> mark.0(f.1-1-1(X, X, X))
   active.1(c.) -> mark.1(a.)
   active.1(c.) -> mark.1(b.)
   mark.0(f.0-0-0(X1, X2, X3)) -> active.0(f.0-0-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-0-1(X1, X2, X3)) -> active.0(f.0-0-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.0-1-0(X1, X2, X3)) -> active.0(f.0-1-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-1-1(X1, X2, X3)) -> active.0(f.0-1-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.1-0-0(X1, X2, X3)) -> active.0(f.1-0-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-0-1(X1, X2, X3)) -> active.0(f.1-0-1(mark.1(X1), X2, mark.1(X3)))
   mark.0(f.1-1-0(X1, X2, X3)) -> active.0(f.1-1-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-1-1(X1, X2, X3)) -> active.0(f.1-1-1(mark.1(X1), X2, mark.1(X3)))
   mark.1(a.) -> active.1(a.)
   mark.1(b.) -> active.1(b.)
   mark.1(c.) -> active.1(c.)
   f.0-0-0(mark.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(mark.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(mark.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(mark.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(mark.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(mark.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(mark.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, mark.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, mark.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, mark.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, mark.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, mark.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, mark.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, mark.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, mark.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, mark.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, mark.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, mark.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, mark.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, mark.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, mark.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(active.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(active.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(active.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(active.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(active.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(active.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(active.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, active.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, active.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, active.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, active.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, active.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, active.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, active.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, active.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, active.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, active.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, active.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, active.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, active.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, active.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(26) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE.0(f.1-1-0(a., b., X)) -> MARK.0(f.0-0-0(X, X, X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(ACTIVE.0(x_1)) = x_1
   POL(MARK.0(x_1)) = x_1
   POL(a.) = 0
   POL(active.0(x_1)) = 0
   POL(active.1(x_1)) = 0
   POL(b.) = 0
   POL(c.) = 0
   POL(f.0-0-0(x_1, x_2, x_3)) = 0
   POL(f.0-0-1(x_1, x_2, x_3)) = 0
   POL(f.0-1-0(x_1, x_2, x_3)) = 0
   POL(f.0-1-1(x_1, x_2, x_3)) = 0
   POL(f.1-0-0(x_1, x_2, x_3)) = 0
   POL(f.1-0-1(x_1, x_2, x_3)) = 0
   POL(f.1-1-0(x_1, x_2, x_3)) = 1
   POL(f.1-1-1(x_1, x_2, x_3)) = 0
   POL(mark.0(x_1)) = 0
   POL(mark.1(x_1)) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f.0-0-0(X1, mark.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-0(mark.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-0(X1, X2, mark.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-0(active.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-0(X1, active.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-0(X1, X2, active.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, mark.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-0-1(mark.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-0-1(X1, X2, mark.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-0-1(active.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-0-1(X1, active.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-0-1(X1, X2, active.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, mark.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(mark.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(X1, X2, mark.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(active.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(X1, active.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(X1, X2, active.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, mark.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(mark.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(X1, X2, mark.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(active.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(X1, active.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(X1, X2, active.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, mark.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-0(mark.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-0(X1, X2, mark.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-0(active.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-0(X1, active.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-0(X1, X2, active.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, mark.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-0-1(mark.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-0-1(X1, X2, mark.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-0-1(active.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-0-1(X1, active.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-0-1(X1, X2, active.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, mark.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(mark.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(X1, X2, mark.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(active.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(X1, active.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(X1, X2, active.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)


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

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK.0(f.0-0-0(x0, x1, y2)) -> ACTIVE.0(f.0-0-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-0-1(x0, x1, y2)) -> ACTIVE.0(f.0-0-1(x0, x1, mark.1(y2)))
   MARK.0(f.0-1-0(x0, x1, y2)) -> ACTIVE.0(f.0-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-1-1(x0, x1, y2)) -> ACTIVE.0(f.0-1-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-0-0(x0, x1, y2)) -> ACTIVE.0(f.1-0-0(x0, x1, mark.0(y2)))
   MARK.0(f.1-0-1(x0, x1, y2)) -> ACTIVE.0(f.1-0-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-1-0(x0, x1, y2)) -> ACTIVE.0(f.1-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.1-1-1(x0, x1, y2)) -> ACTIVE.0(f.1-1-1(x0, x1, mark.1(y2)))
   ACTIVE.0(f.1-1-1(a., b., X)) -> MARK.0(f.1-1-1(X, X, X))
   MARK.0(f.0-0-0(y0, mark.0(x1), y2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-0-1(y0, mark.0(x1), y2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-0-0(y0, mark.0(x1), y2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-1(y0, mark.0(x1), y2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, x1, x2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, x2))
   MARK.0(f.0-0-1(y0, x1, x2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, x2))
   MARK.0(f.0-1-0(y0, x1, x2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, x2))
   MARK.0(f.0-1-1(y0, x1, x2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, x2))
   MARK.0(f.1-0-0(y0, x1, x2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, x2))
   MARK.0(f.1-0-1(y0, x1, x2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, x2))
   MARK.0(f.1-1-0(y0, x1, x2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, x2))
   MARK.0(f.1-1-1(y0, x1, x2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, x2))
   MARK.0(f.0-0-0(y0, active.0(x1), y2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-0-1(y0, active.0(x1), y2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-0-0(y0, active.0(x1), y2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-1(y0, active.0(x1), y2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.1-0-0(a., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(a.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(a., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(a., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(a.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(a., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-0-0(b., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(b.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(b., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(b., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(b.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(b., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-0-0(c., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(c., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(c., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(c., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-1(y0, y1, c.)) -> ACTIVE.0(f.0-0-1(mark.0(y0), y1, active.1(c.)))
   MARK.0(f.0-1-1(y0, y1, c.)) -> ACTIVE.0(f.0-1-1(mark.0(y0), y1, active.1(c.)))
   MARK.0(f.1-0-1(y0, y1, c.)) -> ACTIVE.0(f.1-0-1(mark.1(y0), y1, active.1(c.)))
   MARK.0(f.1-1-1(y0, y1, c.)) -> ACTIVE.0(f.1-1-1(mark.1(y0), y1, active.1(c.)))

The TRS R consists of the following rules:

   active.0(f.1-1-0(a., b., X)) -> mark.0(f.0-0-0(X, X, X))
   active.0(f.1-1-1(a., b., X)) -> mark.0(f.1-1-1(X, X, X))
   active.1(c.) -> mark.1(a.)
   active.1(c.) -> mark.1(b.)
   mark.0(f.0-0-0(X1, X2, X3)) -> active.0(f.0-0-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-0-1(X1, X2, X3)) -> active.0(f.0-0-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.0-1-0(X1, X2, X3)) -> active.0(f.0-1-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-1-1(X1, X2, X3)) -> active.0(f.0-1-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.1-0-0(X1, X2, X3)) -> active.0(f.1-0-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-0-1(X1, X2, X3)) -> active.0(f.1-0-1(mark.1(X1), X2, mark.1(X3)))
   mark.0(f.1-1-0(X1, X2, X3)) -> active.0(f.1-1-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-1-1(X1, X2, X3)) -> active.0(f.1-1-1(mark.1(X1), X2, mark.1(X3)))
   mark.1(a.) -> active.1(a.)
   mark.1(b.) -> active.1(b.)
   mark.1(c.) -> active.1(c.)
   f.0-0-0(mark.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(mark.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(mark.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(mark.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(mark.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(mark.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(mark.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, mark.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, mark.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, mark.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, mark.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, mark.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, mark.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, mark.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, mark.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, mark.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, mark.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, mark.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, mark.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, mark.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, mark.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(active.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(active.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(active.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(active.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(active.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(active.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(active.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, active.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, active.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, active.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, active.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, active.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, active.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, active.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, active.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, active.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, active.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, active.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, active.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, active.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, active.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(28) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK.0(f.0-0-0(x0, x1, y2)) -> ACTIVE.0(f.0-0-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-0-1(x0, x1, y2)) -> ACTIVE.0(f.0-0-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-0-0(x0, x1, y2)) -> ACTIVE.0(f.1-0-0(x0, x1, mark.0(y2)))
   MARK.0(f.1-0-1(x0, x1, y2)) -> ACTIVE.0(f.1-0-1(x0, x1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, mark.0(x1), y2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-0-1(y0, mark.0(x1), y2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-0(y0, mark.0(x1), y2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-1(y0, mark.0(x1), y2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, x1, x2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, x2))
   MARK.0(f.0-0-1(y0, x1, x2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, x2))
   MARK.0(f.1-0-0(y0, x1, x2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, x2))
   MARK.0(f.1-0-1(y0, x1, x2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, x2))
   MARK.0(f.0-0-0(y0, active.0(x1), y2)) -> ACTIVE.0(f.0-0-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-0-1(y0, active.0(x1), y2)) -> ACTIVE.0(f.0-0-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-0-0(y0, active.0(x1), y2)) -> ACTIVE.0(f.1-0-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-0-1(y0, active.0(x1), y2)) -> ACTIVE.0(f.1-0-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-1(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-1(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-1(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-1(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-0-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-0-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-0-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.1-0-0(a., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(a.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(a., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(a., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(a.), y1, mark.0(y2)))
   MARK.0(f.1-0-0(b., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(b.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(b., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(b., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(b.), y1, mark.0(y2)))
   MARK.0(f.1-0-0(c., y1, y2)) -> ACTIVE.0(f.1-0-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.1-0-1(c., y1, y2)) -> ACTIVE.0(f.1-0-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.0-0-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-0-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-0-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-0-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-0-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-0-1(y0, y1, c.)) -> ACTIVE.0(f.0-0-1(mark.0(y0), y1, active.1(c.)))
   MARK.0(f.1-0-1(y0, y1, c.)) -> ACTIVE.0(f.1-0-1(mark.1(y0), y1, active.1(c.)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(ACTIVE.0(x_1)) = 0
   POL(MARK.0(x_1)) = x_1
   POL(a.) = 1
   POL(active.0(x_1)) = 0
   POL(active.1(x_1)) = 0
   POL(b.) = 1
   POL(c.) = 0
   POL(f.0-0-0(x_1, x_2, x_3)) = 1 + x_2
   POL(f.0-0-1(x_1, x_2, x_3)) = 1 + x_2
   POL(f.0-1-0(x_1, x_2, x_3)) = x_2 + x_3
   POL(f.0-1-1(x_1, x_2, x_3)) = x_1
   POL(f.1-0-0(x_1, x_2, x_3)) = 1 + x_1
   POL(f.1-0-1(x_1, x_2, x_3)) = 1 + x_1 + x_2
   POL(f.1-1-0(x_1, x_2, x_3)) = x_1
   POL(f.1-1-1(x_1, x_2, x_3)) = 0
   POL(mark.0(x_1)) = 1
   POL(mark.1(x_1)) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)


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

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK.0(f.0-1-0(x0, x1, y2)) -> ACTIVE.0(f.0-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-1-1(x0, x1, y2)) -> ACTIVE.0(f.0-1-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-1-0(x0, x1, y2)) -> ACTIVE.0(f.1-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.1-1-1(x0, x1, y2)) -> ACTIVE.0(f.1-1-1(x0, x1, mark.1(y2)))
   ACTIVE.0(f.1-1-1(a., b., X)) -> MARK.0(f.1-1-1(X, X, X))
   MARK.0(f.0-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, x1, x2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, x2))
   MARK.0(f.0-1-1(y0, x1, x2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, x2))
   MARK.0(f.1-1-0(y0, x1, x2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, x2))
   MARK.0(f.1-1-1(y0, x1, x2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, x2))
   MARK.0(f.0-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.1-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.1-1-1(a., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-1-1(b., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-1-0(c., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.1-1-1(c., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.0-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-1(y0, y1, c.)) -> ACTIVE.0(f.0-1-1(mark.0(y0), y1, active.1(c.)))
   MARK.0(f.1-1-1(y0, y1, c.)) -> ACTIVE.0(f.1-1-1(mark.1(y0), y1, active.1(c.)))

The TRS R consists of the following rules:

   active.0(f.1-1-0(a., b., X)) -> mark.0(f.0-0-0(X, X, X))
   active.0(f.1-1-1(a., b., X)) -> mark.0(f.1-1-1(X, X, X))
   active.1(c.) -> mark.1(a.)
   active.1(c.) -> mark.1(b.)
   mark.0(f.0-0-0(X1, X2, X3)) -> active.0(f.0-0-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-0-1(X1, X2, X3)) -> active.0(f.0-0-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.0-1-0(X1, X2, X3)) -> active.0(f.0-1-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-1-1(X1, X2, X3)) -> active.0(f.0-1-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.1-0-0(X1, X2, X3)) -> active.0(f.1-0-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-0-1(X1, X2, X3)) -> active.0(f.1-0-1(mark.1(X1), X2, mark.1(X3)))
   mark.0(f.1-1-0(X1, X2, X3)) -> active.0(f.1-1-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-1-1(X1, X2, X3)) -> active.0(f.1-1-1(mark.1(X1), X2, mark.1(X3)))
   mark.1(a.) -> active.1(a.)
   mark.1(b.) -> active.1(b.)
   mark.1(c.) -> active.1(c.)
   f.0-0-0(mark.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(mark.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(mark.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(mark.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(mark.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(mark.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(mark.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, mark.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, mark.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, mark.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, mark.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, mark.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, mark.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, mark.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, mark.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, mark.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, mark.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, mark.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, mark.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, mark.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, mark.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(active.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(active.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(active.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(active.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(active.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(active.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(active.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, active.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, active.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, active.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, active.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, active.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, active.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, active.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, active.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, active.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, active.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, active.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, active.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, active.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, active.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(30) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK.0(f.0-1-0(x0, x1, y2)) -> ACTIVE.0(f.0-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-1-1(x0, x1, y2)) -> ACTIVE.0(f.0-1-1(x0, x1, mark.1(y2)))
   MARK.0(f.1-1-0(x0, x1, y2)) -> ACTIVE.0(f.1-1-0(x0, x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-0(y0, x1, x2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, x2))
   MARK.0(f.0-1-1(y0, x1, x2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, x2))
   MARK.0(f.1-1-0(y0, x1, x2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, x2))
   MARK.0(f.0-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-0(mark.0(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.0-1-1(mark.0(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-0(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-0(mark.1(y0), x1, mark.0(y2)))
   MARK.0(f.0-1-0(f.0-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.0-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.0-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-0-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.1-0-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-0(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-0(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2))), y1, mark.1(y2)))
   MARK.0(f.0-1-0(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-0(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.0(y2)))
   MARK.0(f.0-1-1(f.1-1-1(x0, x1, x2), y1, y2)) -> ACTIVE.0(f.0-1-1(active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2))), y1, mark.1(y2)))
   MARK.0(f.1-1-0(c., y1, y2)) -> ACTIVE.0(f.1-1-0(active.1(c.), y1, mark.0(y2)))
   MARK.0(f.0-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.0-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.0-1-0(mark.0(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-0-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-0(mark.0(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.0-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.0-1-1(mark.0(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-0-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-0-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-0(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-0(mark.1(x0), x1, mark.0(x2)))))
   MARK.0(f.1-1-0(y0, y1, f.1-1-1(x0, x1, x2))) -> ACTIVE.0(f.1-1-0(mark.1(y0), y1, active.0(f.1-1-1(mark.1(x0), x1, mark.1(x2)))))
   MARK.0(f.0-1-1(y0, y1, c.)) -> ACTIVE.0(f.0-1-1(mark.0(y0), y1, active.1(c.)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(ACTIVE.0(x_1)) = x_1
   POL(MARK.0(x_1)) = 1
   POL(a.) = 0
   POL(active.0(x_1)) = 0
   POL(active.1(x_1)) = 0
   POL(b.) = 0
   POL(c.) = 0
   POL(f.0-0-0(x_1, x_2, x_3)) = x_2
   POL(f.0-0-1(x_1, x_2, x_3)) = x_2
   POL(f.0-1-0(x_1, x_2, x_3)) = 0
   POL(f.0-1-1(x_1, x_2, x_3)) = 0
   POL(f.1-0-0(x_1, x_2, x_3)) = x_2
   POL(f.1-0-1(x_1, x_2, x_3)) = x_2
   POL(f.1-1-0(x_1, x_2, x_3)) = 0
   POL(f.1-1-1(x_1, x_2, x_3)) = 1
   POL(mark.0(x_1)) = 0
   POL(mark.1(x_1)) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f.0-1-0(X1, mark.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(mark.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(X1, X2, mark.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(active.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(X1, active.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-0(X1, X2, active.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, mark.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(mark.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(X1, X2, mark.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(active.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(X1, active.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.0-1-1(X1, X2, active.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-1-0(X1, mark.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(mark.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(X1, X2, mark.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(active.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(X1, active.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-0(X1, X2, active.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)


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

(31)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK.0(f.1-1-1(x0, x1, y2)) -> ACTIVE.0(f.1-1-1(x0, x1, mark.1(y2)))
   ACTIVE.0(f.1-1-1(a., b., X)) -> MARK.0(f.1-1-1(X, X, X))
   MARK.0(f.1-1-1(y0, mark.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-1(y0, x1, x2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, x2))
   MARK.0(f.1-1-1(y0, active.1(x1), y2)) -> ACTIVE.0(f.1-1-1(mark.1(y0), x1, mark.1(y2)))
   MARK.0(f.1-1-1(a., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(a.), y1, mark.1(y2)))
   MARK.0(f.1-1-1(b., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(b.), y1, mark.1(y2)))
   MARK.0(f.1-1-1(c., y1, y2)) -> ACTIVE.0(f.1-1-1(active.1(c.), y1, mark.1(y2)))
   MARK.0(f.1-1-1(y0, y1, c.)) -> ACTIVE.0(f.1-1-1(mark.1(y0), y1, active.1(c.)))

The TRS R consists of the following rules:

   active.0(f.1-1-0(a., b., X)) -> mark.0(f.0-0-0(X, X, X))
   active.0(f.1-1-1(a., b., X)) -> mark.0(f.1-1-1(X, X, X))
   active.1(c.) -> mark.1(a.)
   active.1(c.) -> mark.1(b.)
   mark.0(f.0-0-0(X1, X2, X3)) -> active.0(f.0-0-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-0-1(X1, X2, X3)) -> active.0(f.0-0-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.0-1-0(X1, X2, X3)) -> active.0(f.0-1-0(mark.0(X1), X2, mark.0(X3)))
   mark.0(f.0-1-1(X1, X2, X3)) -> active.0(f.0-1-1(mark.0(X1), X2, mark.1(X3)))
   mark.0(f.1-0-0(X1, X2, X3)) -> active.0(f.1-0-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-0-1(X1, X2, X3)) -> active.0(f.1-0-1(mark.1(X1), X2, mark.1(X3)))
   mark.0(f.1-1-0(X1, X2, X3)) -> active.0(f.1-1-0(mark.1(X1), X2, mark.0(X3)))
   mark.0(f.1-1-1(X1, X2, X3)) -> active.0(f.1-1-1(mark.1(X1), X2, mark.1(X3)))
   mark.1(a.) -> active.1(a.)
   mark.1(b.) -> active.1(b.)
   mark.1(c.) -> active.1(c.)
   f.0-0-0(mark.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(mark.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(mark.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(mark.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(mark.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(mark.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(mark.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(mark.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, mark.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, mark.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, mark.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, mark.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, mark.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, mark.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, mark.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, mark.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, mark.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, mark.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, mark.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, mark.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, mark.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, mark.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, mark.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, mark.1(X3)) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(active.0(X1), X2, X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(active.0(X1), X2, X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(active.0(X1), X2, X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(active.0(X1), X2, X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(active.1(X1), X2, X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(active.1(X1), X2, X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(active.1(X1), X2, X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(active.1(X1), X2, X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, active.0(X2), X3) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, active.0(X2), X3) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, active.1(X2), X3) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, active.1(X2), X3) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, active.0(X2), X3) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, active.0(X2), X3) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, active.1(X2), X3) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, active.1(X2), X3) -> f.1-1-1(X1, X2, X3)
   f.0-0-0(X1, X2, active.0(X3)) -> f.0-0-0(X1, X2, X3)
   f.0-0-1(X1, X2, active.1(X3)) -> f.0-0-1(X1, X2, X3)
   f.0-1-0(X1, X2, active.0(X3)) -> f.0-1-0(X1, X2, X3)
   f.0-1-1(X1, X2, active.1(X3)) -> f.0-1-1(X1, X2, X3)
   f.1-0-0(X1, X2, active.0(X3)) -> f.1-0-0(X1, X2, X3)
   f.1-0-1(X1, X2, active.1(X3)) -> f.1-0-1(X1, X2, X3)
   f.1-1-0(X1, X2, active.0(X3)) -> f.1-1-0(X1, X2, X3)
   f.1-1-1(X1, X2, active.1(X3)) -> f.1-1-1(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(32) PisEmptyProof (SOUND)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(33)
TRUE

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(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2)))
   ACTIVE(f(a, b, X)) -> MARK(f(X, X, X))
   MARK(f(y0, mark(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(y0, x1, x2)) -> ACTIVE(f(mark(y0), x1, x2))
   MARK(f(y0, active(x1), y2)) -> ACTIVE(f(mark(y0), x1, mark(y2)))
   MARK(f(a, y1, y2)) -> ACTIVE(f(active(a), y1, mark(y2)))
   MARK(f(b, y1, y2)) -> ACTIVE(f(active(b), y1, mark(y2)))
   MARK(f(c, y1, y2)) -> ACTIVE(f(active(c), y1, mark(y2)))
   MARK(f(y0, y1, c)) -> ACTIVE(f(mark(y0), y1, active(c)))

The TRS R consists of the following rules:

   active(f(a, b, X)) -> mark(f(X, X, X))
   active(c) -> mark(a)
   active(c) -> mark(b)
   mark(f(X1, X2, X3)) -> active(f(mark(X1), X2, mark(X3)))
   mark(a) -> active(a)
   mark(b) -> active(b)
   mark(c) -> active(c)
   f(mark(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, mark(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, mark(X3)) -> f(X1, X2, X3)
   f(active(X1), X2, X3) -> f(X1, X2, X3)
   f(X1, active(X2), X3) -> f(X1, X2, X3)
   f(X1, X2, active(X3)) -> f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(35) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = ACTIVE(f(active(c), active(c), mark(X3))) evaluates to  t =ACTIVE(f(X3, X3, mark(X3)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [X3 / active(c)]

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Rewriting sequence

ACTIVE(f(active(c), active(c), mark(active(c)))) -> ACTIVE(f(active(c), mark(b), mark(active(c))))
with rule active(c) -> mark(b) at position [0,1] and matcher [ ]

ACTIVE(f(active(c), mark(b), mark(active(c)))) -> ACTIVE(f(mark(a), mark(b), mark(active(c))))
with rule active(c) -> mark(a) at position [0,0] and matcher [ ]

ACTIVE(f(mark(a), mark(b), mark(active(c)))) -> ACTIVE(f(mark(a), mark(b), active(c)))
with rule f(X1, X2, mark(X3)) -> f(X1, X2, X3) at position [0] and matcher [X1 / mark(a), X2 / mark(b), X3 / active(c)]

ACTIVE(f(mark(a), mark(b), active(c))) -> ACTIVE(f(mark(a), b, active(c)))
with rule f(X1, mark(X2), X3') -> f(X1, X2, X3') at position [0] and matcher [X1 / mark(a), X2 / b, X3' / active(c)]

ACTIVE(f(mark(a), b, active(c))) -> ACTIVE(f(a, b, active(c)))
with rule f(mark(X1), X2, X3) -> f(X1, X2, X3) at position [0] and matcher [X1 / a, X2 / b, X3 / active(c)]

ACTIVE(f(a, b, active(c))) -> MARK(f(active(c), active(c), active(c)))
with rule ACTIVE(f(a, b, X)) -> MARK(f(X, X, X)) at position [] and matcher [X / active(c)]

MARK(f(active(c), active(c), active(c))) -> ACTIVE(f(active(c), active(c), mark(active(c))))
with rule MARK(f(x0, x1, y2)) -> ACTIVE(f(x0, x1, mark(y2)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




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(36)
NO