8.23/2.97	YES
8.23/2.99	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
8.23/2.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.23/2.99	
8.23/2.99	
8.23/2.99	Termination of the given RelTRS could be proven:
8.23/2.99	
8.23/2.99	(0) RelTRS
8.23/2.99	(1) RelTRSRRRProof [EQUIVALENT, 78 ms]
8.23/2.99	(2) RelTRS
8.23/2.99	(3) RelTRSRRRProof [EQUIVALENT, 14 ms]
8.23/2.99	(4) RelTRS
8.23/2.99	(5) RelTRSRRRProof [EQUIVALENT, 13 ms]
8.23/2.99	(6) RelTRS
8.23/2.99	(7) RelTRSRRRProof [EQUIVALENT, 9 ms]
8.23/2.99	(8) RelTRS
8.23/2.99	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
8.23/2.99	(10) YES
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(0)
8.23/2.99	Obligation:
8.23/2.99	Relative term rewrite system:
8.23/2.99	The relative TRS consists of the following R rules:
8.23/2.99	
8.23/2.99	   f(0) -> s(0)
8.23/2.99	   f(s(0)) -> s(0)
8.23/2.99	   f(s(s(x))) -> f(f(s(x)))
8.23/2.99	
8.23/2.99	The relative TRS consists of the following S rules:
8.23/2.99	
8.23/2.99	   rand(x) -> x
8.23/2.99	   rand(x) -> rand(s(x))
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(1) RelTRSRRRProof (EQUIVALENT)
8.23/2.99	We used the following monotonic ordering for rule removal:
8.23/2.99	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(0) =  	[[0], [0]]
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(rand(x_1)) =  	[[2], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.23/2.99	Rules from R:
8.23/2.99	none
8.23/2.99	Rules from S:
8.23/2.99	
8.23/2.99	   rand(x) -> x
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(2)
8.23/2.99	Obligation:
8.23/2.99	Relative term rewrite system:
8.23/2.99	The relative TRS consists of the following R rules:
8.23/2.99	
8.23/2.99	   f(0) -> s(0)
8.23/2.99	   f(s(0)) -> s(0)
8.23/2.99	   f(s(s(x))) -> f(f(s(x)))
8.23/2.99	
8.23/2.99	The relative TRS consists of the following S rules:
8.23/2.99	
8.23/2.99	   rand(x) -> rand(s(x))
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(3) RelTRSRRRProof (EQUIVALENT)
8.23/2.99	We used the following monotonic ordering for rule removal:
8.23/2.99	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(0) =  	[[0], [2]]
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(rand(x_1)) =  	[[2], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.23/2.99	Rules from R:
8.23/2.99	
8.23/2.99	   f(0) -> s(0)
8.23/2.99	Rules from S:
8.23/2.99	none
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(4)
8.23/2.99	Obligation:
8.23/2.99	Relative term rewrite system:
8.23/2.99	The relative TRS consists of the following R rules:
8.23/2.99	
8.23/2.99	   f(s(0)) -> s(0)
8.23/2.99	   f(s(s(x))) -> f(f(s(x)))
8.23/2.99	
8.23/2.99	The relative TRS consists of the following S rules:
8.23/2.99	
8.23/2.99	   rand(x) -> rand(s(x))
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(5) RelTRSRRRProof (EQUIVALENT)
8.23/2.99	We used the following monotonic ordering for rule removal:
8.23/2.99	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(f(x_1)) =  	[[0], [2]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(s(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 2]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(0) =  	[[0], [0]]
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(rand(x_1)) =  	[[1], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.23/2.99	Rules from R:
8.23/2.99	
8.23/2.99	   f(s(0)) -> s(0)
8.23/2.99	Rules from S:
8.23/2.99	none
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(6)
8.23/2.99	Obligation:
8.23/2.99	Relative term rewrite system:
8.23/2.99	The relative TRS consists of the following R rules:
8.23/2.99	
8.23/2.99	   f(s(s(x))) -> f(f(s(x)))
8.23/2.99	
8.23/2.99	The relative TRS consists of the following S rules:
8.23/2.99	
8.23/2.99	   rand(x) -> rand(s(x))
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(7) RelTRSRRRProof (EQUIVALENT)
8.23/2.99	We used the following monotonic ordering for rule removal:
8.23/2.99	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	   <<<
8.23/2.99	 POL(rand(x_1)) =  	[[1], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.23/2.99	>>>
8.23/2.99	
8.23/2.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.23/2.99	Rules from R:
8.23/2.99	
8.23/2.99	   f(s(s(x))) -> f(f(s(x)))
8.23/2.99	Rules from S:
8.23/2.99	none
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(8)
8.23/2.99	Obligation:
8.23/2.99	Relative term rewrite system:
8.23/2.99	R is empty.
8.23/2.99	The relative TRS consists of the following S rules:
8.23/2.99	
8.23/2.99	   rand(x) -> rand(s(x))
8.23/2.99	
8.23/2.99	
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(9) RIsEmptyProof (EQUIVALENT)
8.23/2.99	The TRS R is empty. Hence, termination is trivially proven.
8.23/2.99	----------------------------------------
8.23/2.99	
8.23/2.99	(10)
8.23/2.99	YES
8.38/3.03	EOF