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Cannot find out cpu time on external processes, falling back to wall time!
proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt3-1.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 457 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 195 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 27 ms]
(6) YES


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

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

   f(g(x), y, z) -> f(x, y, g(z))

The relative TRS consists of the following S rules:

   f(x, y, g(z)) -> f(x, g(y), z)
   f(x, a, z) -> f(x, g(a), z)
   f(x, y, z) -> f(x, y, g(z))


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

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

   <<<
 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
>>>

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

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

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   f(x, a, z) -> f(x, g(a), z)




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

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

   f(g(x), y, z) -> f(x, y, g(z))

The relative TRS consists of the following S rules:

   f(x, y, g(z)) -> f(x, g(y), z)
   f(x, y, z) -> f(x, y, g(z))


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

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

   <<<
 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
>>>

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

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

   f(g(x), y, z) -> f(x, y, g(z))
Rules from S:
none




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

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

   f(x, y, g(z)) -> f(x, g(y), z)
   f(x, y, z) -> f(x, y, g(z))


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

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

(6)
YES