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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 48 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

   <<<
 POL(c(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:

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




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

   <<<
 POL(c(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:

   a(b(x1)) -> a(x1)
Rules from S:
none




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

   <<<
 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[2, 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:

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


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

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

(8)
YES