6.21/2.52	YES
6.21/2.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.21/2.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.21/2.53	
6.21/2.53	
6.21/2.53	Termination of the given RelTRS could be proven:
6.21/2.53	
6.21/2.53	(0) RelTRS
6.21/2.53	(1) RelTRSRRRProof [EQUIVALENT, 24 ms]
6.21/2.53	(2) RelTRS
6.21/2.53	(3) RelTRSRRRProof [EQUIVALENT, 7 ms]
6.21/2.53	(4) RelTRS
6.21/2.53	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
6.21/2.53	(6) YES
6.21/2.53	
6.21/2.53	
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(0)
6.21/2.53	Obligation:
6.21/2.53	Relative term rewrite system:
6.21/2.53	The relative TRS consists of the following R rules:
6.21/2.53	
6.21/2.53	   f(s(x), y, y) -> f(y, x, s(x))
6.21/2.53	
6.21/2.53	The relative TRS consists of the following S rules:
6.21/2.53	
6.21/2.53	   rand(x) -> x
6.21/2.53	   rand(x) -> rand(s(x))
6.21/2.53	
6.21/2.53	
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(1) RelTRSRRRProof (EQUIVALENT)
6.21/2.53	We used the following monotonic ordering for rule removal:
6.21/2.53	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.21/2.53	
6.21/2.53	   <<<
6.21/2.53	 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[3, 0], [0, 0]] 	* 	x_1 	 +  	[[2, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
6.21/2.53	>>>
6.21/2.53	
6.21/2.53	   <<<
6.21/2.53	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
6.21/2.53	>>>
6.21/2.53	
6.21/2.53	   <<<
6.21/2.53	 POL(rand(x_1)) =  	[[3], [3]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
6.21/2.53	>>>
6.21/2.53	
6.21/2.53	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.21/2.53	Rules from R:
6.21/2.53	none
6.21/2.53	Rules from S:
6.21/2.53	
6.21/2.53	   rand(x) -> x
6.21/2.53	
6.21/2.53	
6.21/2.53	
6.21/2.53	
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(2)
6.21/2.53	Obligation:
6.21/2.53	Relative term rewrite system:
6.21/2.53	The relative TRS consists of the following R rules:
6.21/2.53	
6.21/2.53	   f(s(x), y, y) -> f(y, x, s(x))
6.21/2.53	
6.21/2.53	The relative TRS consists of the following S rules:
6.21/2.53	
6.21/2.53	   rand(x) -> rand(s(x))
6.21/2.53	
6.21/2.53	
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(3) RelTRSRRRProof (EQUIVALENT)
6.21/2.53	We used the following monotonic ordering for rule removal:
6.21/2.53	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.21/2.53	
6.21/2.53	   <<<
6.21/2.53	 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 1]] 	* 	x_3
6.21/2.53	>>>
6.21/2.53	
6.21/2.53	   <<<
6.21/2.53	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
6.21/2.53	>>>
6.21/2.53	
6.21/2.53	   <<<
6.21/2.53	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
6.21/2.53	>>>
6.21/2.53	
6.21/2.53	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.21/2.53	Rules from R:
6.21/2.53	
6.21/2.53	   f(s(x), y, y) -> f(y, x, s(x))
6.21/2.53	Rules from S:
6.21/2.53	none
6.21/2.53	
6.21/2.53	
6.21/2.53	
6.21/2.53	
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(4)
6.21/2.53	Obligation:
6.21/2.53	Relative term rewrite system:
6.21/2.53	R is empty.
6.21/2.53	The relative TRS consists of the following S rules:
6.21/2.53	
6.21/2.53	   rand(x) -> rand(s(x))
6.21/2.53	
6.21/2.53	
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(5) RIsEmptyProof (EQUIVALENT)
6.21/2.53	The TRS R is empty. Hence, termination is trivially proven.
6.21/2.53	----------------------------------------
6.21/2.53	
6.21/2.53	(6)
6.21/2.53	YES
6.21/2.57	EOF