4.49/1.94	YES
4.49/1.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.49/1.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.49/1.95	
4.49/1.95	
4.49/1.95	Termination of the given RelTRS could be proven:
4.49/1.95	
4.49/1.95	(0) RelTRS
4.49/1.95	(1) RelTRSRRRProof [EQUIVALENT, 137 ms]
4.49/1.95	(2) RelTRS
4.49/1.95	(3) RelTRSRRRProof [EQUIVALENT, 24 ms]
4.49/1.95	(4) RelTRS
4.49/1.95	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
4.49/1.95	(6) YES
4.49/1.95	
4.49/1.95	
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(0)
4.49/1.95	Obligation:
4.49/1.95	Relative term rewrite system:
4.49/1.95	The relative TRS consists of the following R rules:
4.49/1.95	
4.49/1.95	   top(left(car(x, y), car(old, z))) -> top(right(y, car(old, z)))
4.49/1.95	   top(left(car(x, car(old, y)), z)) -> top(right(car(old, y), z))
4.49/1.95	   top(right(x, car(y, car(old, z)))) -> top(left(x, car(old, z)))
4.49/1.95	   top(right(car(old, x), car(y, z))) -> top(left(car(old, x), z))
4.49/1.95	   top(left(bot, car(old, x))) -> top(right(bot, car(old, x)))
4.49/1.95	   top(right(car(old, x), bot)) -> top(left(car(old, x), bot))
4.49/1.95	
4.49/1.95	The relative TRS consists of the following S rules:
4.49/1.95	
4.49/1.95	   top(left(car(x, y), z)) -> top(left(y, z))
4.49/1.95	   top(right(x, car(y, z))) -> top(right(x, z))
4.49/1.95	   bot -> car(new, bot)
4.49/1.95	
4.49/1.95	
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(1) RelTRSRRRProof (EQUIVALENT)
4.49/1.95	We used the following monotonic ordering for rule removal:
4.49/1.95	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(top(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(left(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 1]] 	* 	x_2
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(car(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(old) =  	[[0], [1]]
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(right(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(bot) =  	[[0], [0]]
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(new) =  	[[0], [0]]
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.49/1.95	Rules from R:
4.49/1.95	
4.49/1.95	   top(left(car(x, y), car(old, z))) -> top(right(y, car(old, z)))
4.49/1.95	   top(left(car(x, car(old, y)), z)) -> top(right(car(old, y), z))
4.49/1.95	   top(left(bot, car(old, x))) -> top(right(bot, car(old, x)))
4.49/1.95	Rules from S:
4.49/1.95	none
4.49/1.95	
4.49/1.95	
4.49/1.95	
4.49/1.95	
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(2)
4.49/1.95	Obligation:
4.49/1.95	Relative term rewrite system:
4.49/1.95	The relative TRS consists of the following R rules:
4.49/1.95	
4.49/1.95	   top(right(x, car(y, car(old, z)))) -> top(left(x, car(old, z)))
4.49/1.95	   top(right(car(old, x), car(y, z))) -> top(left(car(old, x), z))
4.49/1.95	   top(right(car(old, x), bot)) -> top(left(car(old, x), bot))
4.49/1.95	
4.49/1.95	The relative TRS consists of the following S rules:
4.49/1.95	
4.49/1.95	   top(left(car(x, y), z)) -> top(left(y, z))
4.49/1.95	   top(right(x, car(y, z))) -> top(right(x, z))
4.49/1.95	   bot -> car(new, bot)
4.49/1.95	
4.49/1.95	
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(3) RelTRSRRRProof (EQUIVALENT)
4.49/1.95	We used the following monotonic ordering for rule removal:
4.49/1.95	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(top(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(right(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(car(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(old) =  	[[1], [1]]
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(left(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(bot) =  	[[0], [0]]
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	   <<<
4.49/1.95	 POL(new) =  	[[0], [0]]
4.49/1.95	>>>
4.49/1.95	
4.49/1.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.49/1.95	Rules from R:
4.49/1.95	
4.49/1.95	   top(right(x, car(y, car(old, z)))) -> top(left(x, car(old, z)))
4.49/1.95	   top(right(car(old, x), car(y, z))) -> top(left(car(old, x), z))
4.49/1.95	   top(right(car(old, x), bot)) -> top(left(car(old, x), bot))
4.49/1.95	Rules from S:
4.49/1.95	none
4.49/1.95	
4.49/1.95	
4.49/1.95	
4.49/1.95	
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(4)
4.49/1.95	Obligation:
4.49/1.95	Relative term rewrite system:
4.49/1.95	R is empty.
4.49/1.95	The relative TRS consists of the following S rules:
4.49/1.95	
4.49/1.95	   top(left(car(x, y), z)) -> top(left(y, z))
4.49/1.95	   top(right(x, car(y, z))) -> top(right(x, z))
4.49/1.95	   bot -> car(new, bot)
4.49/1.95	
4.49/1.95	
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(5) RIsEmptyProof (EQUIVALENT)
4.49/1.95	The TRS R is empty. Hence, termination is trivially proven.
4.49/1.95	----------------------------------------
4.49/1.95	
4.49/1.95	(6)
4.49/1.95	YES
4.88/2.02	EOF