YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) QDPOrderProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPOrderProof [EQUIVALENT, 33 ms]
        (12) QDP
        (13) QDPOrderProof [EQUIVALENT, 376 ms]
        (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES


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

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

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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:

   A(b(b(x1))) -> A(b(x1))
   A(P(x1)) -> A(x(x1))
   A(x(x1)) -> A(x1)
   B(P(x1)) -> B(Q(x1))
   B(P(x1)) -> Q^1(x1)
   Q^1(x(x1)) -> A(Q(x1))
   Q^1(x(x1)) -> Q^1(x1)
   Q^1(a(x1)) -> B(b(a(x1)))
   Q^1(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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 1 less node.
----------------------------------------

(4)
Complex Obligation (AND)

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

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

   A(P(x1)) -> A(x(x1))
   A(x(x1)) -> A(x1)
   A(b(b(x1))) -> A(b(x1))

The TRS R consists of the following rules:

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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

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


The following pairs can be oriented strictly and are deleted.

   A(P(x1)) -> A(x(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A(x_1)) = x_1
   POL(P(x_1)) = 1 + x_1
   POL(Q(x_1)) = 0
   POL(a(x_1)) = 0
   POL(b(x_1)) = 0
   POL(x(x_1)) = x_1

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

   b(P(x1)) -> b(Q(x1))


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

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

   A(x(x1)) -> A(x1)
   A(b(b(x1))) -> A(b(x1))

The TRS R consists of the following rules:

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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:
*A(x(x1)) -> A(x1)
The graph contains the following edges 1 > 1


*A(b(b(x1))) -> A(b(x1))
The graph contains the following edges 1 > 1


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

(9)
YES

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

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

   Q^1(x(x1)) -> Q^1(x1)
   Q^1(a(x1)) -> B(b(a(x1)))
   B(P(x1)) -> B(Q(x1))
   B(P(x1)) -> Q^1(x1)
   Q^1(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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.

   Q^1(a(x1)) -> B(b(a(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B(x_1)) = x_1
   POL(P(x_1)) = 1
   POL(Q(x_1)) = 1
   POL(Q^1(x_1)) = 1
   POL(a(x_1)) = 1
   POL(b(x_1)) = 0
   POL(x(x_1)) = x_1

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

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))


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

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

   Q^1(x(x1)) -> Q^1(x1)
   B(P(x1)) -> B(Q(x1))
   B(P(x1)) -> Q^1(x1)
   Q^1(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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.

   B(P(x1)) -> B(Q(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(Q^1(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

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

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

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

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

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

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


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

   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))
   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))


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

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

   Q^1(x(x1)) -> Q^1(x1)
   B(P(x1)) -> Q^1(x1)
   Q^1(a(x1)) -> B(a(x1))

The TRS R consists of the following rules:

   a(b(b(x1))) -> P(a(b(x1)))
   a(P(x1)) -> P(a(x(x1)))
   a(x(x1)) -> x(a(x1))
   b(P(x1)) -> b(Q(x1))
   Q(x(x1)) -> a(Q(x1))
   Q(a(x1)) -> b(b(a(x1)))

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

(15) 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:
*Q^1(x(x1)) -> Q^1(x1)
The graph contains the following edges 1 > 1


*Q^1(a(x1)) -> B(a(x1))
The graph contains the following edges 1 >= 1


*B(P(x1)) -> Q^1(x1)
The graph contains the following edges 1 > 1


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

(16)
YES