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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 38 ms]
(2) RelTRS
(3) SIsEmptyProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 1 ms]
(8) AND
    (9) QDP
        (10) UsableRulesProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPOrderProof [EQUIVALENT, 16 ms]
        (13) QDP
        (14) QDPOrderProof [EQUIVALENT, 215 ms]
        (15) QDP
        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
        (17) TRUE
    (18) QDP
        (19) UsableRulesProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPOrderProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) PisEmptyProof [EQUIVALENT, 0 ms]
        (24) YES


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

(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

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

The relative TRS consists of the following S rules:

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


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a(x_1)) = 1 + x_1
   POL(b(x_1)) = x_1
   POL(c(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   c(b(a(x1))) -> c(b(b(x1)))
   b(a(c(x1))) -> b(b(c(x1)))
   a(c(a(x1))) -> b(c(b(x1)))
Rules from S:

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




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

(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

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

S is empty.

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

(3) SIsEmptyProof (EQUIVALENT)
The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

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

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

   C(c(c(x1))) -> B(c(c(x1)))
   B(c(b(x1))) -> C(b(c(x1)))
   B(c(b(x1))) -> B(c(x1))
   B(c(b(x1))) -> C(x1)
   A(c(b(x1))) -> B(a(c(x1)))
   A(c(b(x1))) -> A(c(x1))
   A(c(b(x1))) -> C(x1)

The TRS R consists of the following rules:

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

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

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(b(x1))) -> C(b(c(x1)))
   C(c(c(x1))) -> B(c(c(x1)))
   B(c(b(x1))) -> B(c(x1))
   B(c(b(x1))) -> C(x1)

The TRS R consists of the following rules:

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

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

(10) 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.
----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(b(x1))) -> C(b(c(x1)))
   C(c(c(x1))) -> B(c(c(x1)))
   B(c(b(x1))) -> B(c(x1))
   B(c(b(x1))) -> C(x1)

The TRS R consists of the following rules:

   b(c(b(x1))) -> c(b(c(x1)))
   c(c(c(x1))) -> b(c(c(x1)))

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

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


The following pairs can be oriented strictly and are deleted.

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

   POL(B(x_1)) = 4 + 2*x_1
   POL(C(x_1)) = 4 + 2*x_1
   POL(b(x_1)) = 2 + 4*x_1
   POL(c(x_1)) = 2 + 4*x_1

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

   c(c(c(x1))) -> b(c(c(x1)))
   b(c(b(x1))) -> c(b(c(x1)))


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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B(c(b(x1))) -> C(b(c(x1)))
   C(c(c(x1))) -> B(c(c(x1)))

The TRS R consists of the following rules:

   b(c(b(x1))) -> c(b(c(x1)))
   c(c(c(x1))) -> b(c(c(x1)))

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

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


The following pairs can be oriented strictly and are deleted.

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

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

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

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

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


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

   c(c(c(x1))) -> b(c(c(x1)))
   b(c(b(x1))) -> c(b(c(x1)))


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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   C(c(c(x1))) -> B(c(c(x1)))

The TRS R consists of the following rules:

   b(c(b(x1))) -> c(b(c(x1)))
   c(c(c(x1))) -> b(c(c(x1)))

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

(16) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(17)
TRUE

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

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

   A(c(b(x1))) -> A(c(x1))

The TRS R consists of the following rules:

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

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

(19) 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.
----------------------------------------

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

   A(c(b(x1))) -> A(c(x1))

The TRS R consists of the following rules:

   b(c(b(x1))) -> c(b(c(x1)))
   c(c(c(x1))) -> b(c(c(x1)))

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

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


The following pairs can be oriented strictly and are deleted.

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

   POL(A(x_1)) = x_1
   POL(b(x_1)) = 1 + x_1
   POL(c(x_1)) = 1 + x_1

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

   c(c(c(x1))) -> b(c(c(x1)))
   b(c(b(x1))) -> c(b(c(x1)))


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

(22)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   b(c(b(x1))) -> c(b(c(x1)))
   c(c(c(x1))) -> b(c(c(x1)))

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

(23) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(24)
YES