5.56/2.21	YES
5.56/2.26	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.56/2.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.56/2.26	
5.56/2.26	
5.56/2.26	Termination of the given RelTRS could be proven:
5.56/2.26	
5.56/2.26	(0) RelTRS
5.56/2.26	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
5.56/2.26	(2) RelTRS
5.56/2.26	(3) RelTRSRRRProof [EQUIVALENT, 163 ms]
5.56/2.26	(4) RelTRS
5.56/2.26	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
5.56/2.26	(6) YES
5.56/2.26	
5.56/2.26	
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(0)
5.56/2.26	Obligation:
5.56/2.26	Relative term rewrite system:
5.56/2.26	The relative TRS consists of the following R rules:
5.56/2.26	
5.56/2.26	   b(b(c(x1))) -> a(a(b(x1)))
5.56/2.26	   b(b(a(x1))) -> c(b(b(x1)))
5.56/2.26	
5.56/2.26	The relative TRS consists of the following S rules:
5.56/2.26	
5.56/2.26	   a(c(c(x1))) -> c(a(b(x1)))
5.56/2.26	   a(b(a(x1))) -> b(c(c(x1)))
5.56/2.26	   b(a(a(x1))) -> c(b(b(x1)))
5.56/2.26	
5.56/2.26	
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(1) RelTRS Reverse (EQUIVALENT)
5.56/2.26	We have reversed the following relative TRS [REVERSE]:
5.56/2.26	The set of rules R is 
5.56/2.26	   b(b(c(x1))) -> a(a(b(x1)))
5.56/2.26	   b(b(a(x1))) -> c(b(b(x1)))
5.56/2.26	
5.56/2.26	The set of rules S is 
5.56/2.26	   a(c(c(x1))) -> c(a(b(x1)))
5.56/2.26	   a(b(a(x1))) -> b(c(c(x1)))
5.56/2.26	   b(a(a(x1))) -> c(b(b(x1)))
5.56/2.26	
5.56/2.26	We have obtained the following relative TRS:
5.56/2.26	The set of rules R is 
5.56/2.26	   c(b(b(x1))) -> b(a(a(x1)))
5.56/2.26	   a(b(b(x1))) -> b(b(c(x1)))
5.56/2.26	
5.56/2.26	The set of rules S is 
5.56/2.26	   c(c(a(x1))) -> b(a(c(x1)))
5.56/2.26	   a(b(a(x1))) -> c(c(b(x1)))
5.56/2.26	   a(a(b(x1))) -> b(b(c(x1)))
5.56/2.26	
5.56/2.26	
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(2)
5.56/2.26	Obligation:
5.56/2.26	Relative term rewrite system:
5.56/2.26	The relative TRS consists of the following R rules:
5.56/2.26	
5.56/2.26	   c(b(b(x1))) -> b(a(a(x1)))
5.56/2.26	   a(b(b(x1))) -> b(b(c(x1)))
5.56/2.26	
5.56/2.26	The relative TRS consists of the following S rules:
5.56/2.26	
5.56/2.26	   c(c(a(x1))) -> b(a(c(x1)))
5.56/2.26	   a(b(a(x1))) -> c(c(b(x1)))
5.56/2.26	   a(a(b(x1))) -> b(b(c(x1)))
5.56/2.26	
5.56/2.26	
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(3) RelTRSRRRProof (EQUIVALENT)
5.56/2.26	We used the following monotonic ordering for rule removal:
5.56/2.26	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.56/2.26	
5.56/2.26	   <<<
5.56/2.26	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
5.56/2.26	>>>
5.56/2.26	
5.56/2.26	   <<<
5.56/2.26	 POL(b(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
5.56/2.26	>>>
5.56/2.26	
5.56/2.26	   <<<
5.56/2.26	 POL(a(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
5.56/2.26	>>>
5.56/2.26	
5.56/2.26	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.56/2.26	Rules from R:
5.56/2.26	
5.56/2.26	   c(b(b(x1))) -> b(a(a(x1)))
5.56/2.26	   a(b(b(x1))) -> b(b(c(x1)))
5.56/2.26	Rules from S:
5.56/2.26	
5.56/2.26	   a(b(a(x1))) -> c(c(b(x1)))
5.56/2.26	   a(a(b(x1))) -> b(b(c(x1)))
5.56/2.26	
5.56/2.26	
5.56/2.26	
5.56/2.26	
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(4)
5.56/2.26	Obligation:
5.56/2.26	Relative term rewrite system:
5.56/2.26	R is empty.
5.56/2.26	The relative TRS consists of the following S rules:
5.56/2.26	
5.56/2.26	   c(c(a(x1))) -> b(a(c(x1)))
5.56/2.26	
5.56/2.26	
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(5) RIsEmptyProof (EQUIVALENT)
5.56/2.26	The TRS R is empty. Hence, termination is trivially proven.
5.56/2.26	----------------------------------------
5.56/2.26	
5.56/2.26	(6)
5.56/2.26	YES
6.62/2.55	EOF