5.34/2.08	YES
6.45/2.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.45/2.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.45/2.38	
6.45/2.38	
6.45/2.38	Termination of the given RelTRS could be proven:
6.45/2.38	
6.45/2.38	(0) RelTRS
6.45/2.38	(1) RelTRSRRRProof [EQUIVALENT, 56 ms]
6.45/2.38	(2) RelTRS
6.45/2.38	(3) RIsEmptyProof [EQUIVALENT, 2 ms]
6.45/2.38	(4) YES
6.45/2.38	
6.45/2.38	
6.45/2.38	----------------------------------------
6.45/2.38	
6.45/2.38	(0)
6.45/2.38	Obligation:
6.45/2.38	Relative term rewrite system:
6.45/2.38	The relative TRS consists of the following R rules:
6.45/2.38	
6.45/2.38	   f(el(x), y) -> f(x, el(y))
6.45/2.38	
6.45/2.38	The relative TRS consists of the following S rules:
6.45/2.38	
6.45/2.38	   f(x, y) -> f(l(x), y)
6.45/2.38	   f(x, y) -> f(x, r(y))
6.45/2.38	   l(el(x)) -> el(l(x))
6.45/2.38	   el(r(x)) -> r(el(x))
6.45/2.38	
6.45/2.38	
6.45/2.38	----------------------------------------
6.45/2.38	
6.45/2.38	(1) RelTRSRRRProof (EQUIVALENT)
6.45/2.38	We used the following monotonic ordering for rule removal:
6.45/2.38	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.45/2.38	
6.45/2.38	   <<<
6.45/2.38	 POL(f(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2
6.45/2.38	>>>
6.45/2.38	
6.45/2.38	   <<<
6.45/2.38	 POL(el(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
6.45/2.38	>>>
6.45/2.38	
6.45/2.38	   <<<
6.45/2.38	 POL(l(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
6.45/2.38	>>>
6.45/2.38	
6.45/2.38	   <<<
6.45/2.38	 POL(r(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
6.45/2.38	>>>
6.45/2.38	
6.45/2.38	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.45/2.38	Rules from R:
6.45/2.38	
6.45/2.38	   f(el(x), y) -> f(x, el(y))
6.45/2.38	Rules from S:
6.45/2.38	none
6.45/2.38	
6.45/2.38	
6.45/2.38	
6.45/2.38	
6.45/2.38	----------------------------------------
6.45/2.38	
6.45/2.38	(2)
6.45/2.38	Obligation:
6.45/2.38	Relative term rewrite system:
6.45/2.38	R is empty.
6.45/2.38	The relative TRS consists of the following S rules:
6.45/2.38	
6.45/2.38	   f(x, y) -> f(l(x), y)
6.45/2.38	   f(x, y) -> f(x, r(y))
6.45/2.38	   l(el(x)) -> el(l(x))
6.45/2.38	   el(r(x)) -> r(el(x))
6.45/2.38	
6.45/2.38	
6.45/2.38	----------------------------------------
6.45/2.38	
6.45/2.38	(3) RIsEmptyProof (EQUIVALENT)
6.45/2.38	The TRS R is empty. Hence, termination is trivially proven.
6.45/2.38	----------------------------------------
6.45/2.38	
6.45/2.38	(4)
6.45/2.38	YES
6.62/2.45	EOF