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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 87 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 28 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 1 ms]
(8) YES


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

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

The relative TRS consists of the following S rules:

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


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(b(x_1)) =  	[[1], [1]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c(x_1)) =  	[[1], [1]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[1], [1]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b(b(x1)) -> c(d(x1))
Rules from S:
none




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

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

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

The relative TRS consists of the following S rules:

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


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

(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(c(x_1)) =  	[[2], [0]] 	 +  	[[1, 0], [2, 1]] 	* 	x_1
>>>

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

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

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

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   c(c(x1)) -> d(d(d(x1)))
Rules from S:
none




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

(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

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

The relative TRS consists of the following S rules:

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


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

(5) 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)) = 1 + x_1
   POL(c(x_1)) = 1 + x_1
   POL(d(x_1)) = 1 + x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   d(d(d(x1))) -> a(c(x1))
Rules from S:
none




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

(6)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

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


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

(7) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(8)
YES