Platform claims not to support CPU time, falling back to wall time.
Cannot find out cpu time on external processes, falling back to wall time!
proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt3-5.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 2530 ms]
(2) RelTRS
(3) RIsEmptyProof [EQUIVALENT, 46 ms]
(4) YES


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

   f(g(f(x))) -> f(g(g(g(f(x)))))

The relative TRS consists of the following S rules:

   g(x) -> g(g(x))
   g(x) -> g(g(f(g(g(x)))))
   f(x) -> g(g(f(g(g(x)))))


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

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

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

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

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




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

   g(x) -> g(g(x))
   g(x) -> g(g(f(g(g(x)))))
   f(x) -> g(g(f(g(g(x)))))


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(3) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES