19.20/5.64	YES
19.20/5.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
19.20/5.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.20/5.65	
19.20/5.65	
19.20/5.65	Termination of the given RelTRS could be proven:
19.20/5.65	
19.20/5.65	(0) RelTRS
19.20/5.65	(1) RelTRSRRRProof [EQUIVALENT, 653 ms]
19.20/5.65	(2) RelTRS
19.20/5.65	(3) RelTRSRRRProof [EQUIVALENT, 13 ms]
19.20/5.65	(4) RelTRS
19.20/5.65	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
19.20/5.65	(6) YES
19.20/5.65	
19.20/5.65	
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(0)
19.20/5.65	Obligation:
19.20/5.65	Relative term rewrite system:
19.20/5.65	The relative TRS consists of the following R rules:
19.20/5.65	
19.20/5.65	   a(b(a(x1))) -> c(c(c(x1)))
19.20/5.65	   c(c(c(x1))) -> a(c(a(x1)))
19.20/5.65	
19.20/5.65	The relative TRS consists of the following S rules:
19.20/5.65	
19.20/5.65	   a(x1) -> b(c(b(x1)))
19.20/5.65	
19.20/5.65	
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(1) RelTRSRRRProof (EQUIVALENT)
19.20/5.65	We used the following monotonic ordering for rule removal:
19.20/5.65	Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
19.20/5.65	
19.20/5.65	   <<<
19.20/5.65	 POL(a(x_1)) =  	[[0], [0], [1], [0], [0], [0]] 	 +  	[[1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]] 	* 	x_1
19.20/5.65	>>>
19.20/5.65	
19.20/5.65	   <<<
19.20/5.65	 POL(b(x_1)) =  	[[0], [0], [0], [0], [0], [0]] 	 +  	[[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
19.20/5.65	>>>
19.20/5.65	
19.20/5.65	   <<<
19.20/5.65	 POL(c(x_1)) =  	[[0], [0], [0], [0], [0], [1]] 	 +  	[[1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
19.20/5.65	>>>
19.20/5.65	
19.20/5.65	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
19.20/5.65	Rules from R:
19.20/5.65	
19.20/5.65	   a(b(a(x1))) -> c(c(c(x1)))
19.20/5.65	Rules from S:
19.20/5.65	none
19.20/5.65	
19.20/5.65	
19.20/5.65	
19.20/5.65	
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(2)
19.20/5.65	Obligation:
19.20/5.65	Relative term rewrite system:
19.20/5.65	The relative TRS consists of the following R rules:
19.20/5.65	
19.20/5.65	   c(c(c(x1))) -> a(c(a(x1)))
19.20/5.65	
19.20/5.65	The relative TRS consists of the following S rules:
19.20/5.65	
19.20/5.65	   a(x1) -> b(c(b(x1)))
19.20/5.65	
19.20/5.65	
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(3) RelTRSRRRProof (EQUIVALENT)
19.20/5.65	We used the following monotonic ordering for rule removal:
19.20/5.65	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
19.20/5.65	
19.20/5.65	   <<<
19.20/5.65	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
19.20/5.65	>>>
19.20/5.65	
19.20/5.65	   <<<
19.20/5.65	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
19.20/5.65	>>>
19.20/5.65	
19.20/5.65	   <<<
19.20/5.65	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
19.20/5.65	>>>
19.20/5.65	
19.20/5.65	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
19.20/5.65	Rules from R:
19.20/5.65	
19.20/5.65	   c(c(c(x1))) -> a(c(a(x1)))
19.20/5.65	Rules from S:
19.20/5.65	none
19.20/5.65	
19.20/5.65	
19.20/5.65	
19.20/5.65	
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(4)
19.20/5.65	Obligation:
19.20/5.65	Relative term rewrite system:
19.20/5.65	R is empty.
19.20/5.65	The relative TRS consists of the following S rules:
19.20/5.65	
19.20/5.65	   a(x1) -> b(c(b(x1)))
19.20/5.65	
19.20/5.65	
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(5) RIsEmptyProof (EQUIVALENT)
19.20/5.65	The TRS R is empty. Hence, termination is trivially proven.
19.20/5.65	----------------------------------------
19.20/5.65	
19.20/5.65	(6)
19.20/5.65	YES
19.56/5.68	EOF