6.28/2.55	YES
6.28/2.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.28/2.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.28/2.55	
6.28/2.55	
6.28/2.55	Termination of the given RelTRS could be proven:
6.28/2.55	
6.28/2.55	(0) RelTRS
6.28/2.55	(1) RelTRSRRRProof [EQUIVALENT, 39 ms]
6.28/2.55	(2) RelTRS
6.28/2.55	(3) RelTRSRRRProof [EQUIVALENT, 6 ms]
6.28/2.55	(4) RelTRS
6.28/2.55	(5) RelTRSRRRProof [EQUIVALENT, 12 ms]
6.28/2.55	(6) RelTRS
6.28/2.55	(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
6.28/2.55	(8) RelTRS
6.28/2.55	(9) RIsEmptyProof [EQUIVALENT, 4 ms]
6.28/2.55	(10) YES
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(0)
6.28/2.55	Obligation:
6.28/2.55	Relative term rewrite system:
6.28/2.55	The relative TRS consists of the following R rules:
6.28/2.55	
6.28/2.55	   average(s(x), y) -> average(x, s(y))
6.28/2.55	   average(x, s(s(s(y)))) -> s(average(s(x), y))
6.28/2.55	   average(0, 0) -> 0
6.28/2.55	   average(0, s(0)) -> 0
6.28/2.55	   average(0, s(s(0))) -> s(0)
6.28/2.55	
6.28/2.55	The relative TRS consists of the following S rules:
6.28/2.55	
6.28/2.55	   rand(x) -> x
6.28/2.55	   rand(x) -> rand(s(x))
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(1) RelTRSRRRProof (EQUIVALENT)
6.28/2.55	We used the following monotonic ordering for rule removal:
6.28/2.55	Polynomial interpretation [POLO]:
6.28/2.55	
6.28/2.55	   POL(0) = 0
6.28/2.55	   POL(average(x_1, x_2)) = x_1 + x_2
6.28/2.55	   POL(rand(x_1)) = 1 + x_1
6.28/2.55	   POL(s(x_1)) = x_1
6.28/2.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.28/2.55	Rules from R:
6.28/2.55	none
6.28/2.55	Rules from S:
6.28/2.55	
6.28/2.55	   rand(x) -> x
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(2)
6.28/2.55	Obligation:
6.28/2.55	Relative term rewrite system:
6.28/2.55	The relative TRS consists of the following R rules:
6.28/2.55	
6.28/2.55	   average(s(x), y) -> average(x, s(y))
6.28/2.55	   average(x, s(s(s(y)))) -> s(average(s(x), y))
6.28/2.55	   average(0, 0) -> 0
6.28/2.55	   average(0, s(0)) -> 0
6.28/2.55	   average(0, s(s(0))) -> s(0)
6.28/2.55	
6.28/2.55	The relative TRS consists of the following S rules:
6.28/2.55	
6.28/2.55	   rand(x) -> rand(s(x))
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(3) RelTRSRRRProof (EQUIVALENT)
6.28/2.55	We used the following monotonic ordering for rule removal:
6.28/2.55	Polynomial interpretation [POLO]:
6.28/2.55	
6.28/2.55	   POL(0) = 0
6.28/2.55	   POL(average(x_1, x_2)) = 1 + x_1 + x_2
6.28/2.55	   POL(rand(x_1)) = x_1
6.28/2.55	   POL(s(x_1)) = x_1
6.28/2.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.28/2.55	Rules from R:
6.28/2.55	
6.28/2.55	   average(0, 0) -> 0
6.28/2.55	   average(0, s(0)) -> 0
6.28/2.55	   average(0, s(s(0))) -> s(0)
6.28/2.55	Rules from S:
6.28/2.55	none
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(4)
6.28/2.55	Obligation:
6.28/2.55	Relative term rewrite system:
6.28/2.55	The relative TRS consists of the following R rules:
6.28/2.55	
6.28/2.55	   average(s(x), y) -> average(x, s(y))
6.28/2.55	   average(x, s(s(s(y)))) -> s(average(s(x), y))
6.28/2.55	
6.28/2.55	The relative TRS consists of the following S rules:
6.28/2.55	
6.28/2.55	   rand(x) -> rand(s(x))
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(5) RelTRSRRRProof (EQUIVALENT)
6.28/2.55	We used the following monotonic ordering for rule removal:
6.28/2.55	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.28/2.55	
6.28/2.55	   <<<
6.28/2.55	 POL(average(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [0, 1]] 	* 	x_2
6.28/2.55	>>>
6.28/2.55	
6.28/2.55	   <<<
6.28/2.55	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
6.28/2.55	>>>
6.28/2.55	
6.28/2.55	   <<<
6.28/2.55	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
6.28/2.55	>>>
6.28/2.55	
6.28/2.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.28/2.55	Rules from R:
6.28/2.55	
6.28/2.55	   average(x, s(s(s(y)))) -> s(average(s(x), y))
6.28/2.55	Rules from S:
6.28/2.55	none
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(6)
6.28/2.55	Obligation:
6.28/2.55	Relative term rewrite system:
6.28/2.55	The relative TRS consists of the following R rules:
6.28/2.55	
6.28/2.55	   average(s(x), y) -> average(x, s(y))
6.28/2.55	
6.28/2.55	The relative TRS consists of the following S rules:
6.28/2.55	
6.28/2.55	   rand(x) -> rand(s(x))
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(7) RelTRSRRRProof (EQUIVALENT)
6.28/2.55	We used the following monotonic ordering for rule removal:
6.28/2.55	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.28/2.55	
6.28/2.55	   <<<
6.28/2.55	 POL(average(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
6.28/2.55	>>>
6.28/2.55	
6.28/2.55	   <<<
6.28/2.55	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
6.28/2.55	>>>
6.28/2.55	
6.28/2.55	   <<<
6.28/2.55	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
6.28/2.55	>>>
6.28/2.55	
6.28/2.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.28/2.55	Rules from R:
6.28/2.55	
6.28/2.55	   average(s(x), y) -> average(x, s(y))
6.28/2.55	Rules from S:
6.28/2.55	none
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(8)
6.28/2.55	Obligation:
6.28/2.55	Relative term rewrite system:
6.28/2.55	R is empty.
6.28/2.55	The relative TRS consists of the following S rules:
6.28/2.55	
6.28/2.55	   rand(x) -> rand(s(x))
6.28/2.55	
6.28/2.55	
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(9) RIsEmptyProof (EQUIVALENT)
6.28/2.55	The TRS R is empty. Hence, termination is trivially proven.
6.28/2.55	----------------------------------------
6.28/2.55	
6.28/2.55	(10)
6.28/2.55	YES
6.28/2.60	EOF