8.42/2.95	YES
8.42/3.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.42/3.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.42/3.03	
8.42/3.03	
8.42/3.03	Termination of the given RelTRS could be proven:
8.42/3.03	
8.42/3.03	(0) RelTRS
8.42/3.03	(1) RelTRSRRRProof [EQUIVALENT, 111 ms]
8.42/3.03	(2) RelTRS
8.42/3.03	(3) RelTRSRRRProof [EQUIVALENT, 15 ms]
8.42/3.03	(4) RelTRS
8.42/3.03	(5) RIsEmptyProof [EQUIVALENT, 2 ms]
8.42/3.03	(6) YES
8.42/3.03	
8.42/3.03	
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(0)
8.42/3.03	Obligation:
8.42/3.03	Relative term rewrite system:
8.42/3.03	The relative TRS consists of the following R rules:
8.42/3.03	
8.42/3.03	   a(c(b(x1))) -> b(a(b(a(x1))))
8.42/3.03	   a(a(x1)) -> a(b(a(x1)))
8.42/3.03	
8.42/3.03	The relative TRS consists of the following S rules:
8.42/3.03	
8.42/3.03	   b(x1) -> b(c(x1))
8.42/3.03	
8.42/3.03	
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(1) RelTRSRRRProof (EQUIVALENT)
8.42/3.03	We used the following monotonic ordering for rule removal:
8.42/3.03	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
8.42/3.03	
8.42/3.03	   <<<
8.42/3.03	 POL(a(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 1], [0, 0, 1], [0, 0, 0]] 	* 	x_1
8.42/3.03	>>>
8.42/3.03	
8.42/3.03	   <<<
8.42/3.03	 POL(c(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
8.42/3.03	>>>
8.42/3.03	
8.42/3.03	   <<<
8.42/3.03	 POL(b(x_1)) =  	[[0], [2], [0]] 	 +  	[[1, 0, 0], [0, 2, 1], [0, 0, 0]] 	* 	x_1
8.42/3.03	>>>
8.42/3.03	
8.42/3.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.42/3.03	Rules from R:
8.42/3.03	
8.42/3.03	   a(c(b(x1))) -> b(a(b(a(x1))))
8.42/3.03	Rules from S:
8.42/3.03	none
8.42/3.03	
8.42/3.03	
8.42/3.03	
8.42/3.03	
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(2)
8.42/3.03	Obligation:
8.42/3.03	Relative term rewrite system:
8.42/3.03	The relative TRS consists of the following R rules:
8.42/3.03	
8.42/3.03	   a(a(x1)) -> a(b(a(x1)))
8.42/3.03	
8.42/3.03	The relative TRS consists of the following S rules:
8.42/3.03	
8.42/3.03	   b(x1) -> b(c(x1))
8.42/3.03	
8.42/3.03	
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(3) RelTRSRRRProof (EQUIVALENT)
8.42/3.03	We used the following monotonic ordering for rule removal:
8.42/3.03	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.42/3.03	
8.42/3.03	   <<<
8.42/3.03	 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
8.42/3.03	>>>
8.42/3.03	
8.42/3.03	   <<<
8.42/3.03	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.42/3.03	>>>
8.42/3.03	
8.42/3.03	   <<<
8.42/3.03	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.42/3.03	>>>
8.42/3.03	
8.42/3.03	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.42/3.03	Rules from R:
8.42/3.03	
8.42/3.03	   a(a(x1)) -> a(b(a(x1)))
8.42/3.03	Rules from S:
8.42/3.03	none
8.42/3.03	
8.42/3.03	
8.42/3.03	
8.42/3.03	
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(4)
8.42/3.03	Obligation:
8.42/3.03	Relative term rewrite system:
8.42/3.03	R is empty.
8.42/3.03	The relative TRS consists of the following S rules:
8.42/3.03	
8.42/3.03	   b(x1) -> b(c(x1))
8.42/3.03	
8.42/3.03	
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(5) RIsEmptyProof (EQUIVALENT)
8.42/3.03	The TRS R is empty. Hence, termination is trivially proven.
8.42/3.03	----------------------------------------
8.42/3.03	
8.42/3.03	(6)
8.42/3.03	YES
8.89/3.08	EOF