3.98/1.77	YES
4.24/1.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.24/1.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.24/1.79	
4.24/1.79	
4.24/1.79	Termination of the given RelTRS could be proven:
4.24/1.79	
4.24/1.79	(0) RelTRS
4.24/1.79	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
4.24/1.79	(2) RelTRS
4.24/1.79	(3) RelTRSRRRProof [EQUIVALENT, 5 ms]
4.24/1.79	(4) RelTRS
4.24/1.79	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
4.24/1.79	(6) YES
4.24/1.79	
4.24/1.79	
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(0)
4.24/1.79	Obligation:
4.24/1.79	Relative term rewrite system:
4.24/1.79	The relative TRS consists of the following R rules:
4.24/1.79	
4.24/1.79	   b(b(b(x1))) -> b(a(b(x1)))
4.24/1.79	   c(c(c(x1))) -> c(b(a(x1)))
4.24/1.79	
4.24/1.79	The relative TRS consists of the following S rules:
4.24/1.79	
4.24/1.79	   c(a(b(x1))) -> a(c(c(x1)))
4.24/1.79	   a(b(c(x1))) -> b(c(c(x1)))
4.24/1.79	   c(a(c(x1))) -> a(c(a(x1)))
4.24/1.79	   b(a(c(x1))) -> a(a(a(x1)))
4.24/1.79	
4.24/1.79	
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(1) RelTRS Reverse (EQUIVALENT)
4.24/1.79	We have reversed the following relative TRS [REVERSE]:
4.24/1.79	The set of rules R is 
4.24/1.79	   b(b(b(x1))) -> b(a(b(x1)))
4.24/1.79	   c(c(c(x1))) -> c(b(a(x1)))
4.24/1.79	
4.24/1.79	The set of rules S is 
4.24/1.79	   c(a(b(x1))) -> a(c(c(x1)))
4.24/1.79	   a(b(c(x1))) -> b(c(c(x1)))
4.24/1.79	   c(a(c(x1))) -> a(c(a(x1)))
4.24/1.79	   b(a(c(x1))) -> a(a(a(x1)))
4.24/1.79	
4.24/1.79	We have obtained the following relative TRS:
4.24/1.79	The set of rules R is 
4.24/1.79	   b(b(b(x1))) -> b(a(b(x1)))
4.24/1.79	   c(c(c(x1))) -> a(b(c(x1)))
4.24/1.79	
4.24/1.79	The set of rules S is 
4.24/1.79	   b(a(c(x1))) -> c(c(a(x1)))
4.24/1.79	   c(b(a(x1))) -> c(c(b(x1)))
4.24/1.79	   c(a(c(x1))) -> a(c(a(x1)))
4.24/1.79	   c(a(b(x1))) -> a(a(a(x1)))
4.24/1.79	
4.24/1.79	
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(2)
4.24/1.79	Obligation:
4.24/1.79	Relative term rewrite system:
4.24/1.79	The relative TRS consists of the following R rules:
4.24/1.79	
4.24/1.79	   b(b(b(x1))) -> b(a(b(x1)))
4.24/1.79	   c(c(c(x1))) -> a(b(c(x1)))
4.24/1.79	
4.24/1.79	The relative TRS consists of the following S rules:
4.24/1.79	
4.24/1.79	   b(a(c(x1))) -> c(c(a(x1)))
4.24/1.79	   c(b(a(x1))) -> c(c(b(x1)))
4.24/1.79	   c(a(c(x1))) -> a(c(a(x1)))
4.24/1.79	   c(a(b(x1))) -> a(a(a(x1)))
4.24/1.79	
4.24/1.79	
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(3) RelTRSRRRProof (EQUIVALENT)
4.24/1.79	We used the following monotonic ordering for rule removal:
4.24/1.79	Knuth-Bendix order [KBO] with precedence:b_1 > c_1 > a_1
4.24/1.79	
4.24/1.79	and weight map:
4.24/1.79	
4.24/1.79	   b_1=1
4.24/1.79	   a_1=1
4.24/1.79	   c_1=1
4.24/1.79	
4.24/1.79	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.24/1.79	Rules from R:
4.24/1.79	
4.24/1.79	   b(b(b(x1))) -> b(a(b(x1)))
4.24/1.79	   c(c(c(x1))) -> a(b(c(x1)))
4.24/1.79	Rules from S:
4.24/1.79	
4.24/1.79	   b(a(c(x1))) -> c(c(a(x1)))
4.24/1.79	   c(b(a(x1))) -> c(c(b(x1)))
4.24/1.79	   c(a(c(x1))) -> a(c(a(x1)))
4.24/1.79	   c(a(b(x1))) -> a(a(a(x1)))
4.24/1.79	
4.24/1.79	
4.24/1.79	
4.24/1.79	
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(4)
4.24/1.79	Obligation:
4.24/1.79	Relative term rewrite system:
4.24/1.79	R is empty.
4.24/1.79	S is empty.
4.24/1.79	
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(5) RIsEmptyProof (EQUIVALENT)
4.24/1.79	The TRS R is empty. Hence, termination is trivially proven.
4.24/1.79	----------------------------------------
4.24/1.79	
4.24/1.79	(6)
4.24/1.79	YES
4.24/1.87	EOF