8.39/2.93	YES
8.47/2.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.47/2.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.47/2.95	
8.47/2.95	
8.47/2.95	Termination of the given RelTRS could be proven:
8.47/2.95	
8.47/2.95	(0) RelTRS
8.47/2.95	(1) RelTRSRRRProof [EQUIVALENT, 25 ms]
8.47/2.95	(2) RelTRS
8.47/2.95	(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
8.47/2.95	(4) RelTRS
8.47/2.95	(5) RelTRSRRRProof [EQUIVALENT, 17 ms]
8.47/2.95	(6) RelTRS
8.47/2.95	(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
8.47/2.95	(8) RelTRS
8.47/2.95	(9) RelTRSRRRProof [EQUIVALENT, 0 ms]
8.47/2.95	(10) RelTRS
8.47/2.95	(11) RIsEmptyProof [EQUIVALENT, 3 ms]
8.47/2.95	(12) YES
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(0)
8.47/2.95	Obligation:
8.47/2.95	Relative term rewrite system:
8.47/2.95	The relative TRS consists of the following R rules:
8.47/2.95	
8.47/2.95	   half(0) -> 0
8.47/2.95	   half(s(s(x))) -> s(half(x))
8.47/2.95	   log(s(0)) -> 0
8.47/2.95	   log(s(s(x))) -> s(log(s(half(x))))
8.47/2.95	
8.47/2.95	The relative TRS consists of the following S rules:
8.47/2.95	
8.47/2.95	   rand(x) -> x
8.47/2.95	   rand(x) -> rand(s(x))
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(1) RelTRSRRRProof (EQUIVALENT)
8.47/2.95	We used the following monotonic ordering for rule removal:
8.47/2.95	Polynomial interpretation [POLO]:
8.47/2.95	
8.47/2.95	   POL(0) = 0
8.47/2.95	   POL(half(x_1)) = x_1
8.47/2.95	   POL(log(x_1)) = x_1
8.47/2.95	   POL(rand(x_1)) = 1 + x_1
8.47/2.95	   POL(s(x_1)) = x_1
8.47/2.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.47/2.95	Rules from R:
8.47/2.95	none
8.47/2.95	Rules from S:
8.47/2.95	
8.47/2.95	   rand(x) -> x
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(2)
8.47/2.95	Obligation:
8.47/2.95	Relative term rewrite system:
8.47/2.95	The relative TRS consists of the following R rules:
8.47/2.95	
8.47/2.95	   half(0) -> 0
8.47/2.95	   half(s(s(x))) -> s(half(x))
8.47/2.95	   log(s(0)) -> 0
8.47/2.95	   log(s(s(x))) -> s(log(s(half(x))))
8.47/2.95	
8.47/2.95	The relative TRS consists of the following S rules:
8.47/2.95	
8.47/2.95	   rand(x) -> rand(s(x))
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(3) RelTRSRRRProof (EQUIVALENT)
8.47/2.95	We used the following monotonic ordering for rule removal:
8.47/2.95	Polynomial interpretation [POLO]:
8.47/2.95	
8.47/2.95	   POL(0) = 0
8.47/2.95	   POL(half(x_1)) = x_1
8.47/2.95	   POL(log(x_1)) = 1 + x_1
8.47/2.95	   POL(rand(x_1)) = x_1
8.47/2.95	   POL(s(x_1)) = x_1
8.47/2.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.47/2.95	Rules from R:
8.47/2.95	
8.47/2.95	   log(s(0)) -> 0
8.47/2.95	Rules from S:
8.47/2.95	none
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(4)
8.47/2.95	Obligation:
8.47/2.95	Relative term rewrite system:
8.47/2.95	The relative TRS consists of the following R rules:
8.47/2.95	
8.47/2.95	   half(0) -> 0
8.47/2.95	   half(s(s(x))) -> s(half(x))
8.47/2.95	   log(s(s(x))) -> s(log(s(half(x))))
8.47/2.95	
8.47/2.95	The relative TRS consists of the following S rules:
8.47/2.95	
8.47/2.95	   rand(x) -> rand(s(x))
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(5) RelTRSRRRProof (EQUIVALENT)
8.47/2.95	We used the following monotonic ordering for rule removal:
8.47/2.95	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(half(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(0) =  	[[0], [1]]
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(log(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(rand(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.47/2.95	Rules from R:
8.47/2.95	
8.47/2.95	   half(0) -> 0
8.47/2.95	Rules from S:
8.47/2.95	none
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(6)
8.47/2.95	Obligation:
8.47/2.95	Relative term rewrite system:
8.47/2.95	The relative TRS consists of the following R rules:
8.47/2.95	
8.47/2.95	   half(s(s(x))) -> s(half(x))
8.47/2.95	   log(s(s(x))) -> s(log(s(half(x))))
8.47/2.95	
8.47/2.95	The relative TRS consists of the following S rules:
8.47/2.95	
8.47/2.95	   rand(x) -> rand(s(x))
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(7) RelTRSRRRProof (EQUIVALENT)
8.47/2.95	We used the following monotonic ordering for rule removal:
8.47/2.95	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(half(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(s(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(log(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 1]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(rand(x_1)) =  	[[1], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.47/2.95	Rules from R:
8.47/2.95	
8.47/2.95	   log(s(s(x))) -> s(log(s(half(x))))
8.47/2.95	Rules from S:
8.47/2.95	none
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(8)
8.47/2.95	Obligation:
8.47/2.95	Relative term rewrite system:
8.47/2.95	The relative TRS consists of the following R rules:
8.47/2.95	
8.47/2.95	   half(s(s(x))) -> s(half(x))
8.47/2.95	
8.47/2.95	The relative TRS consists of the following S rules:
8.47/2.95	
8.47/2.95	   rand(x) -> rand(s(x))
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(9) RelTRSRRRProof (EQUIVALENT)
8.47/2.95	We used the following monotonic ordering for rule removal:
8.47/2.95	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(half(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(s(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	   <<<
8.47/2.95	 POL(rand(x_1)) =  	[[1], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.47/2.95	>>>
8.47/2.95	
8.47/2.95	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.47/2.95	Rules from R:
8.47/2.95	
8.47/2.95	   half(s(s(x))) -> s(half(x))
8.47/2.95	Rules from S:
8.47/2.95	none
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(10)
8.47/2.95	Obligation:
8.47/2.95	Relative term rewrite system:
8.47/2.95	R is empty.
8.47/2.95	The relative TRS consists of the following S rules:
8.47/2.95	
8.47/2.95	   rand(x) -> rand(s(x))
8.47/2.95	
8.47/2.95	
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(11) RIsEmptyProof (EQUIVALENT)
8.47/2.95	The TRS R is empty. Hence, termination is trivially proven.
8.47/2.95	----------------------------------------
8.47/2.95	
8.47/2.95	(12)
8.47/2.95	YES
8.66/3.10	EOF