5.34/2.15	YES
6.36/2.42	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.36/2.42	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.36/2.42	
6.36/2.42	
6.36/2.42	Termination of the given RelTRS could be proven:
6.36/2.42	
6.36/2.42	(0) RelTRS
6.36/2.42	(1) RelTRSRRRProof [EQUIVALENT, 35 ms]
6.36/2.42	(2) RelTRS
6.36/2.42	(3) RelTRSRRRProof [EQUIVALENT, 10 ms]
6.36/2.42	(4) RelTRS
6.36/2.42	(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
6.36/2.42	(6) RelTRS
6.36/2.42	(7) RIsEmptyProof [EQUIVALENT, 2 ms]
6.36/2.42	(8) YES
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(0)
6.36/2.42	Obligation:
6.36/2.42	Relative term rewrite system:
6.36/2.42	The relative TRS consists of the following R rules:
6.36/2.42	
6.36/2.42	   f(a, g(y), z) -> f(a, y, g(y))
6.36/2.42	   f(b, g(y), z) -> f(a, y, z)
6.36/2.42	   a -> b
6.36/2.42	
6.36/2.42	The relative TRS consists of the following S rules:
6.36/2.42	
6.36/2.42	   f(x, y, z) -> f(x, y, g(z))
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(1) RelTRSRRRProof (EQUIVALENT)
6.36/2.42	We used the following monotonic ordering for rule removal:
6.36/2.42	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(f(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 1]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(a) =  	[[0], [1]]
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(g(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(b) =  	[[0], [0]]
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.36/2.42	Rules from R:
6.36/2.42	
6.36/2.42	   f(a, g(y), z) -> f(a, y, g(y))
6.36/2.42	Rules from S:
6.36/2.42	none
6.36/2.42	
6.36/2.42	
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(2)
6.36/2.42	Obligation:
6.36/2.42	Relative term rewrite system:
6.36/2.42	The relative TRS consists of the following R rules:
6.36/2.42	
6.36/2.42	   f(b, g(y), z) -> f(a, y, z)
6.36/2.42	   a -> b
6.36/2.42	
6.36/2.42	The relative TRS consists of the following S rules:
6.36/2.42	
6.36/2.42	   f(x, y, z) -> f(x, y, g(z))
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(3) RelTRSRRRProof (EQUIVALENT)
6.36/2.42	We used the following monotonic ordering for rule removal:
6.36/2.42	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(f(x_1, x_2, x_3)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2 	 +  	[[1, 0], [1, 0]] 	* 	x_3
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(b) =  	[[0], [1]]
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(g(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	   <<<
6.36/2.42	 POL(a) =  	[[1], [1]]
6.36/2.42	>>>
6.36/2.42	
6.36/2.42	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.36/2.42	Rules from R:
6.36/2.42	
6.36/2.42	   a -> b
6.36/2.42	Rules from S:
6.36/2.42	none
6.36/2.42	
6.36/2.42	
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(4)
6.36/2.42	Obligation:
6.36/2.42	Relative term rewrite system:
6.36/2.42	The relative TRS consists of the following R rules:
6.36/2.42	
6.36/2.42	   f(b, g(y), z) -> f(a, y, z)
6.36/2.42	
6.36/2.42	The relative TRS consists of the following S rules:
6.36/2.42	
6.36/2.42	   f(x, y, z) -> f(x, y, g(z))
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(5) RelTRSRRRProof (EQUIVALENT)
6.36/2.42	We used the following monotonic ordering for rule removal:
6.36/2.42	Polynomial interpretation [POLO]:
6.36/2.42	
6.36/2.42	   POL(a) = 0
6.36/2.42	   POL(b) = 1
6.36/2.42	   POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
6.36/2.42	   POL(g(x_1)) = x_1
6.36/2.42	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.36/2.42	Rules from R:
6.36/2.42	
6.36/2.42	   f(b, g(y), z) -> f(a, y, z)
6.36/2.42	Rules from S:
6.36/2.42	none
6.36/2.42	
6.36/2.42	
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(6)
6.36/2.42	Obligation:
6.36/2.42	Relative term rewrite system:
6.36/2.42	R is empty.
6.36/2.42	The relative TRS consists of the following S rules:
6.36/2.42	
6.36/2.42	   f(x, y, z) -> f(x, y, g(z))
6.36/2.42	
6.36/2.42	
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(7) RIsEmptyProof (EQUIVALENT)
6.36/2.42	The TRS R is empty. Hence, termination is trivially proven.
6.36/2.42	----------------------------------------
6.36/2.42	
6.36/2.42	(8)
6.36/2.42	YES
6.41/2.45	EOF