4.88/2.22	YES
5.62/2.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.62/2.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.62/2.45	
5.62/2.45	
5.62/2.45	Termination of the given RelTRS could be proven:
5.62/2.45	
5.62/2.45	(0) RelTRS
5.62/2.45	(1) RelTRSRRRProof [EQUIVALENT, 57 ms]
5.62/2.45	(2) RelTRS
5.62/2.45	(3) RelTRSRRRProof [EQUIVALENT, 9 ms]
5.62/2.45	(4) RelTRS
5.62/2.45	(5) RelTRSRRRProof [EQUIVALENT, 8 ms]
5.62/2.45	(6) RelTRS
5.62/2.45	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
5.62/2.45	(8) YES
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(0)
5.62/2.45	Obligation:
5.62/2.45	Relative term rewrite system:
5.62/2.45	The relative TRS consists of the following R rules:
5.62/2.45	
5.62/2.45	   f(x, c(y)) -> f(x, s(f(y, y)))
5.62/2.45	   f(s(x), y) -> f(x, s(c(y)))
5.62/2.45	
5.62/2.45	The relative TRS consists of the following S rules:
5.62/2.45	
5.62/2.45	   rand(x) -> x
5.62/2.45	   rand(x) -> rand(s(x))
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(1) RelTRSRRRProof (EQUIVALENT)
5.62/2.45	We used the following monotonic ordering for rule removal:
5.62/2.45	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(f(x_1, x_2)) =  	[[1], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.62/2.45	Rules from R:
5.62/2.45	none
5.62/2.45	Rules from S:
5.62/2.45	
5.62/2.45	   rand(x) -> x
5.62/2.45	
5.62/2.45	
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(2)
5.62/2.45	Obligation:
5.62/2.45	Relative term rewrite system:
5.62/2.45	The relative TRS consists of the following R rules:
5.62/2.45	
5.62/2.45	   f(x, c(y)) -> f(x, s(f(y, y)))
5.62/2.45	   f(s(x), y) -> f(x, s(c(y)))
5.62/2.45	
5.62/2.45	The relative TRS consists of the following S rules:
5.62/2.45	
5.62/2.45	   rand(x) -> rand(s(x))
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(3) RelTRSRRRProof (EQUIVALENT)
5.62/2.45	We used the following monotonic ordering for rule removal:
5.62/2.45	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(f(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.62/2.45	Rules from R:
5.62/2.45	
5.62/2.45	   f(x, c(y)) -> f(x, s(f(y, y)))
5.62/2.45	Rules from S:
5.62/2.45	none
5.62/2.45	
5.62/2.45	
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(4)
5.62/2.45	Obligation:
5.62/2.45	Relative term rewrite system:
5.62/2.45	The relative TRS consists of the following R rules:
5.62/2.45	
5.62/2.45	   f(s(x), y) -> f(x, s(c(y)))
5.62/2.45	
5.62/2.45	The relative TRS consists of the following S rules:
5.62/2.45	
5.62/2.45	   rand(x) -> rand(s(x))
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(5) RelTRSRRRProof (EQUIVALENT)
5.62/2.45	We used the following monotonic ordering for rule removal:
5.62/2.45	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(f(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	   <<<
5.62/2.45	 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
5.62/2.45	>>>
5.62/2.45	
5.62/2.45	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.62/2.45	Rules from R:
5.62/2.45	
5.62/2.45	   f(s(x), y) -> f(x, s(c(y)))
5.62/2.45	Rules from S:
5.62/2.45	none
5.62/2.45	
5.62/2.45	
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(6)
5.62/2.45	Obligation:
5.62/2.45	Relative term rewrite system:
5.62/2.45	R is empty.
5.62/2.45	The relative TRS consists of the following S rules:
5.62/2.45	
5.62/2.45	   rand(x) -> rand(s(x))
5.62/2.45	
5.62/2.45	
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(7) RIsEmptyProof (EQUIVALENT)
5.62/2.45	The TRS R is empty. Hence, termination is trivially proven.
5.62/2.45	----------------------------------------
5.62/2.45	
5.62/2.45	(8)
5.62/2.45	YES
5.96/2.49	EOF