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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 134 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 12 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 3 ms]
(6) YES


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

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

   top(left(car(x, y), car(old, z))) -> top(right(y, car(old, z)))
   top(left(car(x, car(old, y)), z)) -> top(right(car(old, y), z))
   top(right(x, car(y, car(old, z)))) -> top(left(x, car(old, z)))
   top(right(car(old, x), car(y, z))) -> top(left(car(old, x), z))
   top(left(bot, car(old, x))) -> top(right(bot, car(old, x)))
   top(right(car(old, x), bot)) -> top(left(car(old, x), bot))

The relative TRS consists of the following S rules:

   top(left(car(x, y), z)) -> top(left(y, z))
   top(right(x, car(y, z))) -> top(right(x, z))
   bot -> car(new, bot)


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

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

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

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

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

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

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

   <<<
 POL(bot) =  	[[0], [0]]
>>>

   <<<
 POL(new) =  	[[0], [0]]
>>>

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

   top(left(car(x, y), car(old, z))) -> top(right(y, car(old, z)))
   top(left(car(x, car(old, y)), z)) -> top(right(car(old, y), z))
   top(left(bot, car(old, x))) -> top(right(bot, car(old, x)))
Rules from S:
none




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

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

   top(right(x, car(y, car(old, z)))) -> top(left(x, car(old, z)))
   top(right(car(old, x), car(y, z))) -> top(left(car(old, x), z))
   top(right(car(old, x), bot)) -> top(left(car(old, x), bot))

The relative TRS consists of the following S rules:

   top(left(car(x, y), z)) -> top(left(y, z))
   top(right(x, car(y, z))) -> top(right(x, z))
   bot -> car(new, bot)


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

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

   POL(bot) = 0
   POL(car(x_1, x_2)) = x_1 + x_2
   POL(left(x_1, x_2)) = x_1 + x_2
   POL(new) = 0
   POL(old) = 0
   POL(right(x_1, x_2)) = 1 + x_1 + x_2
   POL(top(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   top(right(x, car(y, car(old, z)))) -> top(left(x, car(old, z)))
   top(right(car(old, x), car(y, z))) -> top(left(car(old, x), z))
   top(right(car(old, x), bot)) -> top(left(car(old, x), bot))
Rules from S:
none




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

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

   top(left(car(x, y), z)) -> top(left(y, z))
   top(right(x, car(y, z))) -> top(right(x, z))
   bot -> car(new, bot)


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

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

(6)
YES