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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) QDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) QDP
(9) MNOCProof [EQUIVALENT, 4 ms]
(10) QDP
(11) QDPOrderProof [EQUIVALENT, 108 ms]
(12) QDP
(13) QDPOrderProof [EQUIVALENT, 59 ms]
(14) QDP
(15) DependencyGraphProof [EQUIVALENT, 0 ms]
(16) TRUE


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

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

   a(a(x1)) -> b(b(x1))
   c(c(b(x1))) -> d(c(a(x1)))
   a(x1) -> d(c(c(x1)))
   c(d(x1)) -> b(c(x1))

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(x1)) -> b(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))
   d(c(x1)) -> c(b(x1))

Q is empty.

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

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

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(a(x1)) -> B(b(x1))
   A(a(x1)) -> B(x1)
   B(c(c(x1))) -> A(c(d(x1)))
   B(c(c(x1))) -> D(x1)
   A(x1) -> D(x1)
   D(c(x1)) -> B(x1)

The TRS R consists of the following rules:

   a(a(x1)) -> b(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))
   d(c(x1)) -> c(b(x1))

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(x1) -> D(x1)
   D(c(x1)) -> B(x1)
   B(c(c(x1))) -> A(c(d(x1)))
   B(c(c(x1))) -> D(x1)

The TRS R consists of the following rules:

   a(a(x1)) -> b(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))
   d(c(x1)) -> c(b(x1))

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

(7) 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.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A(x1) -> D(x1)
   D(c(x1)) -> B(x1)
   B(c(c(x1))) -> A(c(d(x1)))
   B(c(c(x1))) -> D(x1)

The TRS R consists of the following rules:

   d(c(x1)) -> c(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))

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

(9) MNOCProof (EQUIVALENT)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
----------------------------------------

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

   A(x1) -> D(x1)
   D(c(x1)) -> B(x1)
   B(c(c(x1))) -> A(c(d(x1)))
   B(c(c(x1))) -> D(x1)

The TRS R consists of the following rules:

   d(c(x1)) -> c(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))

The set Q consists of the following terms:

   d(c(x0))
   b(c(c(x0)))
   a(x0)

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.

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

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

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

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

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

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

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

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


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

   d(c(x1)) -> c(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))


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

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

   A(x1) -> D(x1)
   D(c(x1)) -> B(x1)
   B(c(c(x1))) -> A(c(d(x1)))

The TRS R consists of the following rules:

   d(c(x1)) -> c(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))

The set Q consists of the following terms:

   d(c(x0))
   b(c(c(x0)))
   a(x0)

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.

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

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

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

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

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

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

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

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


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

   d(c(x1)) -> c(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))


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

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

   D(c(x1)) -> B(x1)
   B(c(c(x1))) -> A(c(d(x1)))

The TRS R consists of the following rules:

   d(c(x1)) -> c(b(x1))
   b(c(c(x1))) -> a(c(d(x1)))
   a(x1) -> c(c(d(x1)))

The set Q consists of the following terms:

   d(c(x0))
   b(c(c(x0)))
   a(x0)

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

(15) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(16)
TRUE