6.10/2.38	YES
6.87/2.59	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.87/2.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.87/2.59	
6.87/2.59	
6.87/2.59	Termination of the given RelTRS could be proven:
6.87/2.59	
6.87/2.59	(0) RelTRS
6.87/2.59	(1) RelTRSRRRProof [EQUIVALENT, 48 ms]
6.87/2.59	(2) RelTRS
6.87/2.59	(3) RIsEmptyProof [EQUIVALENT, 0 ms]
6.87/2.59	(4) YES
6.87/2.59	
6.87/2.59	
6.87/2.59	----------------------------------------
6.87/2.59	
6.87/2.59	(0)
6.87/2.59	Obligation:
6.87/2.59	Relative term rewrite system:
6.87/2.59	The relative TRS consists of the following R rules:
6.87/2.59	
6.87/2.59	   a -> b
6.87/2.59	
6.87/2.59	The relative TRS consists of the following S rules:
6.87/2.59	
6.87/2.59	   f(s(x)) -> c(x, f(x))
6.87/2.59	   c(x, c(y, z)) -> c(y, c(x, z))
6.87/2.59	
6.87/2.59	
6.87/2.59	----------------------------------------
6.87/2.59	
6.87/2.59	(1) RelTRSRRRProof (EQUIVALENT)
6.87/2.59	We used the following monotonic ordering for rule removal:
6.87/2.59	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(a) =  	[[1], [1]]
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(b) =  	[[0], [1]]
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(c(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(y) =  	[[0], [0]]
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	   <<<
6.87/2.59	 POL(z) =  	[[0], [1]]
6.87/2.59	>>>
6.87/2.59	
6.87/2.59	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.87/2.59	Rules from R:
6.87/2.59	
6.87/2.59	   a -> b
6.87/2.59	Rules from S:
6.87/2.59	none
6.87/2.59	
6.87/2.59	
6.87/2.59	
6.87/2.59	
6.87/2.59	----------------------------------------
6.87/2.59	
6.87/2.59	(2)
6.87/2.59	Obligation:
6.87/2.59	Relative term rewrite system:
6.87/2.59	R is empty.
6.87/2.59	The relative TRS consists of the following S rules:
6.87/2.59	
6.87/2.59	   f(s(x)) -> c(x, f(x))
6.87/2.59	   c(x, c(y, z)) -> c(y, c(x, z))
6.87/2.59	
6.87/2.59	
6.87/2.59	----------------------------------------
6.87/2.59	
6.87/2.59	(3) RIsEmptyProof (EQUIVALENT)
6.87/2.59	The TRS R is empty. Hence, termination is trivially proven.
6.87/2.59	----------------------------------------
6.87/2.59	
6.87/2.59	(4)
6.87/2.59	YES
6.87/2.62	EOF