5.12/2.03	YES
5.12/2.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.12/2.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.12/2.03	
5.12/2.03	
5.12/2.03	Termination of the given RelTRS could be proven:
5.12/2.03	
5.12/2.03	(0) RelTRS
5.12/2.03	(1) RelTRSRRRProof [EQUIVALENT, 31 ms]
5.12/2.03	(2) RelTRS
5.12/2.03	(3) RelTRSRRRProof [EQUIVALENT, 5 ms]
5.12/2.03	(4) RelTRS
5.12/2.03	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
5.12/2.03	(6) YES
5.12/2.03	
5.12/2.03	
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(0)
5.12/2.03	Obligation:
5.12/2.03	Relative term rewrite system:
5.12/2.03	The relative TRS consists of the following R rules:
5.12/2.03	
5.12/2.03	   g(x, y) -> x
5.12/2.03	   g(x, y) -> y
5.12/2.03	   f(s(x), y, y) -> f(y, x, s(x))
5.12/2.03	
5.12/2.03	The relative TRS consists of the following S rules:
5.12/2.03	
5.12/2.03	   rand(x) -> x
5.12/2.03	   rand(x) -> rand(s(x))
5.12/2.03	
5.12/2.03	
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(1) RelTRSRRRProof (EQUIVALENT)
5.12/2.03	We used the following monotonic ordering for rule removal:
5.12/2.03	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 POL(g(x_1, x_2)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1 	 +  	[[1, 0], [0, 1]] 	* 	x_2
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 POL(rand(x_1)) =  	[[3], [3]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.12/2.03	Rules from R:
5.12/2.03	
5.12/2.03	   g(x, y) -> x
5.12/2.03	   g(x, y) -> y
5.12/2.03	Rules from S:
5.12/2.03	
5.12/2.03	   rand(x) -> x
5.12/2.03	
5.12/2.03	
5.12/2.03	
5.12/2.03	
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(2)
5.12/2.03	Obligation:
5.12/2.03	Relative term rewrite system:
5.12/2.03	The relative TRS consists of the following R rules:
5.12/2.03	
5.12/2.03	   f(s(x), y, y) -> f(y, x, s(x))
5.12/2.03	
5.12/2.03	The relative TRS consists of the following S rules:
5.12/2.03	
5.12/2.03	   rand(x) -> rand(s(x))
5.12/2.03	
5.12/2.03	
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(3) RelTRSRRRProof (EQUIVALENT)
5.12/2.03	We used the following monotonic ordering for rule removal:
5.12/2.03	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 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
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	   <<<
5.12/2.03	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
5.12/2.03	>>>
5.12/2.03	
5.12/2.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.12/2.03	Rules from R:
5.12/2.03	
5.12/2.03	   f(s(x), y, y) -> f(y, x, s(x))
5.12/2.03	Rules from S:
5.12/2.03	none
5.12/2.03	
5.12/2.03	
5.12/2.03	
5.12/2.03	
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(4)
5.12/2.03	Obligation:
5.12/2.03	Relative term rewrite system:
5.12/2.03	R is empty.
5.12/2.03	The relative TRS consists of the following S rules:
5.12/2.03	
5.12/2.03	   rand(x) -> rand(s(x))
5.12/2.03	
5.12/2.03	
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(5) RIsEmptyProof (EQUIVALENT)
5.12/2.03	The TRS R is empty. Hence, termination is trivially proven.
5.12/2.03	----------------------------------------
5.12/2.03	
5.12/2.03	(6)
5.12/2.03	YES
5.39/2.09	EOF