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proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt2-7.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RootLabelingProof [EQUIVALENT, 2 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 679 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 432 ms]
(8) RelTRS
(9) RIsEmptyProof [EQUIVALENT, 0 ms]
(10) 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))


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(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   f(g(f(x))) -> f(g(g(g(f(x)))))

The set of rules S is 
   g(x) -> g(g(x))

We have obtained the following relative TRS:
The set of rules R is 
   f(g(f(x))) -> f(g(g(g(f(x)))))

The set of rules S is 
   g(x) -> g(g(x))


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(2)
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))


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(3) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled


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

   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))

The relative TRS consists of the following S rules:

   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))


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

   <<<
 POL(f_{g_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [2, 2]] 	* 	x_1
>>>

   <<<
 POL(g_{f_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [2, 2]] 	* 	x_1
>>>

   <<<
 POL(f_{f_1}(x_1)) =  	[[2], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(g_{g_1}(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:

   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
Rules from S:
none




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

   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))

The relative TRS consists of the following S rules:

   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))


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

   <<<
 POL(f_{g_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(g_{f_1}(x_1)) =  	[[2], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(g_{g_1}(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:

   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
Rules from S:
none




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(8)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))


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(9) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES