4.14/1.85	YES
4.14/1.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.14/1.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.14/1.86	
4.14/1.86	
4.14/1.86	Termination of the given RelTRS could be proven:
4.14/1.86	
4.14/1.86	(0) RelTRS
4.14/1.86	(1) RelTRSRRRProof [EQUIVALENT, 57 ms]
4.14/1.86	(2) RelTRS
4.14/1.86	(3) RelTRSRRRProof [EQUIVALENT, 10 ms]
4.14/1.86	(4) RelTRS
4.14/1.86	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
4.14/1.86	(6) YES
4.14/1.86	
4.14/1.86	
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(0)
4.14/1.86	Obligation:
4.14/1.86	Relative term rewrite system:
4.14/1.86	The relative TRS consists of the following R rules:
4.14/1.86	
4.14/1.86	   f(b(x), y) -> f(x, b(y))
4.14/1.86	   f(x, a(y)) -> f(a(x), y)
4.14/1.86	
4.14/1.86	The relative TRS consists of the following S rules:
4.14/1.86	
4.14/1.86	   f(x, y) -> f(a(x), y)
4.14/1.86	   f(x, y) -> f(x, b(y))
4.14/1.86	
4.14/1.86	
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(1) RelTRSRRRProof (EQUIVALENT)
4.14/1.86	We used the following monotonic ordering for rule removal:
4.14/1.86	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.14/1.86	
4.14/1.86	   <<<
4.14/1.86	 POL(f(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2
4.14/1.86	>>>
4.14/1.86	
4.14/1.86	   <<<
4.14/1.86	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
4.14/1.86	>>>
4.14/1.86	
4.14/1.86	   <<<
4.14/1.86	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
4.14/1.86	>>>
4.14/1.86	
4.14/1.86	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.14/1.86	Rules from R:
4.14/1.86	
4.14/1.86	   f(b(x), y) -> f(x, b(y))
4.14/1.86	Rules from S:
4.14/1.86	none
4.14/1.86	
4.14/1.86	
4.14/1.86	
4.14/1.86	
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(2)
4.14/1.86	Obligation:
4.14/1.86	Relative term rewrite system:
4.14/1.86	The relative TRS consists of the following R rules:
4.14/1.86	
4.14/1.86	   f(x, a(y)) -> f(a(x), y)
4.14/1.86	
4.14/1.86	The relative TRS consists of the following S rules:
4.14/1.86	
4.14/1.86	   f(x, y) -> f(a(x), y)
4.14/1.86	   f(x, y) -> f(x, b(y))
4.14/1.86	
4.14/1.86	
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(3) RelTRSRRRProof (EQUIVALENT)
4.14/1.86	We used the following monotonic ordering for rule removal:
4.14/1.86	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.14/1.86	
4.14/1.86	   <<<
4.14/1.86	 POL(f(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.14/1.86	>>>
4.14/1.86	
4.14/1.86	   <<<
4.14/1.86	 POL(a(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
4.14/1.86	>>>
4.14/1.86	
4.14/1.86	   <<<
4.14/1.86	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
4.14/1.86	>>>
4.14/1.86	
4.14/1.86	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.14/1.86	Rules from R:
4.14/1.86	
4.14/1.86	   f(x, a(y)) -> f(a(x), y)
4.14/1.86	Rules from S:
4.14/1.86	none
4.14/1.86	
4.14/1.86	
4.14/1.86	
4.14/1.86	
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(4)
4.14/1.86	Obligation:
4.14/1.86	Relative term rewrite system:
4.14/1.86	R is empty.
4.14/1.86	The relative TRS consists of the following S rules:
4.14/1.86	
4.14/1.86	   f(x, y) -> f(a(x), y)
4.14/1.86	   f(x, y) -> f(x, b(y))
4.14/1.86	
4.14/1.86	
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(5) RIsEmptyProof (EQUIVALENT)
4.14/1.86	The TRS R is empty. Hence, termination is trivially proven.
4.14/1.86	----------------------------------------
4.14/1.86	
4.14/1.86	(6)
4.14/1.86	YES
4.14/1.89	EOF