6.80/2.59	YES
6.80/2.60	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.80/2.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.80/2.60	
6.80/2.60	
6.80/2.60	Termination of the given RelTRS could be proven:
6.80/2.60	
6.80/2.60	(0) RelTRS
6.80/2.60	(1) RelTRS Reverse [SOUND, 0 ms]
6.80/2.60	(2) RelTRS
6.80/2.60	(3) RelTRSRRRProof [EQUIVALENT, 21 ms]
6.80/2.60	(4) RelTRS
6.80/2.60	(5) RelTRSRRRProof [EQUIVALENT, 5 ms]
6.80/2.60	(6) RelTRS
6.80/2.60	(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
6.80/2.60	(8) RelTRS
6.80/2.60	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
6.80/2.60	(10) YES
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(0)
6.80/2.60	Obligation:
6.80/2.60	Relative term rewrite system:
6.80/2.60	The relative TRS consists of the following R rules:
6.80/2.60	
6.80/2.60	   g(s(x)) -> f(x)
6.80/2.60	   f(0) -> s(0)
6.80/2.60	   f(s(x)) -> s(s(g(x)))
6.80/2.60	   g(0) -> 0
6.80/2.60	
6.80/2.60	The relative TRS consists of the following S rules:
6.80/2.60	
6.80/2.60	   rand(x) -> x
6.80/2.60	   rand(x) -> rand(s(x))
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(1) RelTRS Reverse (SOUND)
6.80/2.60	We have reversed the following relative TRS [REVERSE]:
6.80/2.60	The set of rules R is 
6.80/2.60	   g(s(x)) -> f(x)
6.80/2.60	   f(0) -> s(0)
6.80/2.60	   f(s(x)) -> s(s(g(x)))
6.80/2.60	   g(0) -> 0
6.80/2.60	
6.80/2.60	The set of rules S is 
6.80/2.60	   rand(x) -> x
6.80/2.60	   rand(x) -> rand(s(x))
6.80/2.60	
6.80/2.60	We have obtained the following relative TRS:
6.80/2.60	The set of rules R is 
6.80/2.60	   s(g(x)) -> f(x)
6.80/2.60	   0'(f(x)) -> 0'(s(x))
6.80/2.60	   s(f(x)) -> g(s(s(x)))
6.80/2.60	   0'(g(x)) -> 0'(x)
6.80/2.60	
6.80/2.60	The set of rules S is 
6.80/2.60	   rand(x) -> x
6.80/2.60	   rand(x) -> s(rand(x))
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(2)
6.80/2.60	Obligation:
6.80/2.60	Relative term rewrite system:
6.80/2.60	The relative TRS consists of the following R rules:
6.80/2.60	
6.80/2.60	   s(g(x)) -> f(x)
6.80/2.60	   0'(f(x)) -> 0'(s(x))
6.80/2.60	   s(f(x)) -> g(s(s(x)))
6.80/2.60	   0'(g(x)) -> 0'(x)
6.80/2.60	
6.80/2.60	The relative TRS consists of the following S rules:
6.80/2.60	
6.80/2.60	   rand(x) -> x
6.80/2.60	   rand(x) -> s(rand(x))
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(3) RelTRSRRRProof (EQUIVALENT)
6.80/2.60	We used the following monotonic ordering for rule removal:
6.80/2.60	Polynomial interpretation [POLO]:
6.80/2.60	
6.80/2.60	   POL(0'(x_1)) = x_1
6.80/2.60	   POL(f(x_1)) = x_1
6.80/2.60	   POL(g(x_1)) = x_1
6.80/2.60	   POL(rand(x_1)) = 1 + x_1
6.80/2.60	   POL(s(x_1)) = x_1
6.80/2.60	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.80/2.60	Rules from R:
6.80/2.60	none
6.80/2.60	Rules from S:
6.80/2.60	
6.80/2.60	   rand(x) -> x
6.80/2.60	
6.80/2.60	
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(4)
6.80/2.60	Obligation:
6.80/2.60	Relative term rewrite system:
6.80/2.60	The relative TRS consists of the following R rules:
6.80/2.60	
6.80/2.60	   s(g(x)) -> f(x)
6.80/2.60	   0'(f(x)) -> 0'(s(x))
6.80/2.60	   s(f(x)) -> g(s(s(x)))
6.80/2.60	   0'(g(x)) -> 0'(x)
6.80/2.60	
6.80/2.60	The relative TRS consists of the following S rules:
6.80/2.60	
6.80/2.60	   rand(x) -> s(rand(x))
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(5) RelTRSRRRProof (EQUIVALENT)
6.80/2.60	We used the following monotonic ordering for rule removal:
6.80/2.60	Polynomial interpretation [POLO]:
6.80/2.60	
6.80/2.60	   POL(0'(x_1)) = x_1
6.80/2.60	   POL(f(x_1)) = 1 + x_1
6.80/2.60	   POL(g(x_1)) = 1 + x_1
6.80/2.60	   POL(rand(x_1)) = x_1
6.80/2.60	   POL(s(x_1)) = x_1
6.80/2.60	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.80/2.60	Rules from R:
6.80/2.60	
6.80/2.60	   0'(f(x)) -> 0'(s(x))
6.80/2.60	   0'(g(x)) -> 0'(x)
6.80/2.60	Rules from S:
6.80/2.60	none
6.80/2.60	
6.80/2.60	
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(6)
6.80/2.60	Obligation:
6.80/2.60	Relative term rewrite system:
6.80/2.60	The relative TRS consists of the following R rules:
6.80/2.60	
6.80/2.60	   s(g(x)) -> f(x)
6.80/2.60	   s(f(x)) -> g(s(s(x)))
6.80/2.60	
6.80/2.60	The relative TRS consists of the following S rules:
6.80/2.60	
6.80/2.60	   rand(x) -> s(rand(x))
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(7) RelTRSRRRProof (EQUIVALENT)
6.80/2.60	We used the following monotonic ordering for rule removal:
6.80/2.60	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.80/2.60	
6.80/2.60	   <<<
6.80/2.60	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
6.80/2.60	>>>
6.80/2.60	
6.80/2.60	   <<<
6.80/2.60	 POL(g(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
6.80/2.60	>>>
6.80/2.60	
6.80/2.60	   <<<
6.80/2.60	 POL(f(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
6.80/2.60	>>>
6.80/2.60	
6.80/2.60	   <<<
6.80/2.60	 POL(rand(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
6.80/2.60	>>>
6.80/2.60	
6.80/2.60	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.80/2.60	Rules from R:
6.80/2.60	
6.80/2.60	   s(g(x)) -> f(x)
6.80/2.60	   s(f(x)) -> g(s(s(x)))
6.80/2.60	Rules from S:
6.80/2.60	none
6.80/2.60	
6.80/2.60	
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(8)
6.80/2.60	Obligation:
6.80/2.60	Relative term rewrite system:
6.80/2.60	R is empty.
6.80/2.60	The relative TRS consists of the following S rules:
6.80/2.60	
6.80/2.60	   rand(x) -> s(rand(x))
6.80/2.60	
6.80/2.60	
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(9) RIsEmptyProof (EQUIVALENT)
6.80/2.60	The TRS R is empty. Hence, termination is trivially proven.
6.80/2.60	----------------------------------------
6.80/2.60	
6.80/2.60	(10)
6.80/2.60	YES
6.80/2.62	EOF