4.42/1.95	YES
4.42/1.97	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.42/1.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.42/1.97	
4.42/1.97	
4.42/1.97	Termination of the given RelTRS could be proven:
4.42/1.97	
4.42/1.97	(0) RelTRS
4.42/1.97	(1) RelTRSRRRProof [EQUIVALENT, 53 ms]
4.42/1.97	(2) RelTRS
4.42/1.97	(3) RelTRSRRRProof [EQUIVALENT, 5 ms]
4.42/1.97	(4) RelTRS
4.42/1.97	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
4.42/1.97	(6) YES
4.42/1.97	
4.42/1.97	
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(0)
4.42/1.97	Obligation:
4.42/1.97	Relative term rewrite system:
4.42/1.97	The relative TRS consists of the following R rules:
4.42/1.97	
4.42/1.97	   f(g(x), y, z) -> f(x, y, g(z))
4.42/1.97	
4.42/1.97	The relative TRS consists of the following S rules:
4.42/1.97	
4.42/1.97	   f(x, y, g(z)) -> f(x, g(y), z)
4.42/1.97	   f(x, a, z) -> f(x, g(a), z)
4.42/1.97	   f(x, y, z) -> f(x, y, g(z))
4.42/1.97	
4.42/1.97	
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(1) RelTRSRRRProof (EQUIVALENT)
4.42/1.97	We used the following monotonic ordering for rule removal:
4.42/1.97	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.42/1.97	
4.42/1.97	   <<<
4.42/1.97	 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
4.42/1.97	>>>
4.42/1.97	
4.42/1.97	   <<<
4.42/1.97	 POL(g(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
4.42/1.97	>>>
4.42/1.97	
4.42/1.97	   <<<
4.42/1.97	 POL(a) =  	[[0], [1]]
4.42/1.97	>>>
4.42/1.97	
4.42/1.97	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.42/1.97	Rules from R:
4.42/1.97	none
4.42/1.97	Rules from S:
4.42/1.97	
4.42/1.97	   f(x, a, z) -> f(x, g(a), z)
4.42/1.97	
4.42/1.97	
4.42/1.97	
4.42/1.97	
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(2)
4.42/1.97	Obligation:
4.42/1.97	Relative term rewrite system:
4.42/1.97	The relative TRS consists of the following R rules:
4.42/1.97	
4.42/1.97	   f(g(x), y, z) -> f(x, y, g(z))
4.42/1.97	
4.42/1.97	The relative TRS consists of the following S rules:
4.42/1.97	
4.42/1.97	   f(x, y, g(z)) -> f(x, g(y), z)
4.42/1.97	   f(x, y, z) -> f(x, y, g(z))
4.42/1.97	
4.42/1.97	
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(3) RelTRSRRRProof (EQUIVALENT)
4.42/1.97	We used the following monotonic ordering for rule removal:
4.42/1.97	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.42/1.97	
4.42/1.97	   <<<
4.42/1.97	 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
4.42/1.97	>>>
4.42/1.97	
4.42/1.97	   <<<
4.42/1.97	 POL(g(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
4.42/1.97	>>>
4.42/1.97	
4.42/1.97	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.42/1.97	Rules from R:
4.42/1.97	
4.42/1.97	   f(g(x), y, z) -> f(x, y, g(z))
4.42/1.97	Rules from S:
4.42/1.97	none
4.42/1.97	
4.42/1.97	
4.42/1.97	
4.42/1.97	
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(4)
4.42/1.97	Obligation:
4.42/1.97	Relative term rewrite system:
4.42/1.97	R is empty.
4.42/1.97	The relative TRS consists of the following S rules:
4.42/1.97	
4.42/1.97	   f(x, y, g(z)) -> f(x, g(y), z)
4.42/1.97	   f(x, y, z) -> f(x, y, g(z))
4.42/1.97	
4.42/1.97	
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(5) RIsEmptyProof (EQUIVALENT)
4.42/1.97	The TRS R is empty. Hence, termination is trivially proven.
4.42/1.97	----------------------------------------
4.42/1.97	
4.42/1.97	(6)
4.42/1.97	YES
4.61/1.99	EOF