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proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt3-2.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 94 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 101 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 2 ms]
(6) YES


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

   p(0, y) -> y
   p(s(x), y) -> s(p(x, y))

The relative TRS consists of the following S rules:

   p(x, y) -> p(x, s(y))


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(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(p(x_1, x_2)) = 1 + x_1 + x_2
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   p(0, y) -> y
Rules from S:
none




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

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

   p(s(x), y) -> s(p(x, y))

The relative TRS consists of the following S rules:

   p(x, y) -> p(x, s(y))


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

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

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

   <<<
 POL(s(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:

   p(s(x), y) -> s(p(x, y))
Rules from S:
none




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

   p(x, y) -> p(x, s(y))


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(5) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES