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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 43 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 8 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 0 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))


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
p/2(YES,YES)
0/0)
s/1)YES(

Quasi precedence:
0 > p_2


Status:
p_2: [2,1]
0: multiset status

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




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

(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.
----------------------------------------

(6)
YES