7.78/2.77	YES
7.78/2.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.78/2.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.78/2.78	
7.78/2.78	
7.78/2.78	Termination of the given RelTRS could be proven:
7.78/2.78	
7.78/2.78	(0) RelTRS
7.78/2.78	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
7.78/2.78	(2) RelTRS
7.78/2.78	(3) RelTRSRRRProof [EQUIVALENT, 415 ms]
7.78/2.78	(4) RelTRS
7.78/2.78	(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
7.78/2.78	(6) RelTRS
7.78/2.78	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
7.78/2.78	(8) YES
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(0)
7.78/2.78	Obligation:
7.78/2.78	Relative term rewrite system:
7.78/2.78	The relative TRS consists of the following R rules:
7.78/2.78	
7.78/2.78	   a(c(a(x1))) -> c(b(b(x1)))
7.78/2.78	   a(a(b(x1))) -> b(c(c(x1)))
7.78/2.78	   a(c(a(x1))) -> a(b(b(x1)))
7.78/2.78	
7.78/2.78	The relative TRS consists of the following S rules:
7.78/2.78	
7.78/2.78	   c(a(a(x1))) -> a(c(c(x1)))
7.78/2.78	   c(a(a(x1))) -> b(b(b(x1)))
7.78/2.78	   c(b(c(x1))) -> a(c(a(x1)))
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(1) RelTRS Reverse (EQUIVALENT)
7.78/2.78	We have reversed the following relative TRS [REVERSE]:
7.78/2.78	The set of rules R is 
7.78/2.78	   a(c(a(x1))) -> c(b(b(x1)))
7.78/2.78	   a(a(b(x1))) -> b(c(c(x1)))
7.78/2.78	   a(c(a(x1))) -> a(b(b(x1)))
7.78/2.78	
7.78/2.78	The set of rules S is 
7.78/2.78	   c(a(a(x1))) -> a(c(c(x1)))
7.78/2.78	   c(a(a(x1))) -> b(b(b(x1)))
7.78/2.78	   c(b(c(x1))) -> a(c(a(x1)))
7.78/2.78	
7.78/2.78	We have obtained the following relative TRS:
7.78/2.78	The set of rules R is 
7.78/2.78	   a(c(a(x1))) -> b(b(c(x1)))
7.78/2.78	   b(a(a(x1))) -> c(c(b(x1)))
7.78/2.78	   a(c(a(x1))) -> b(b(a(x1)))
7.78/2.78	
7.78/2.78	The set of rules S is 
7.78/2.78	   a(a(c(x1))) -> c(c(a(x1)))
7.78/2.78	   a(a(c(x1))) -> b(b(b(x1)))
7.78/2.78	   c(b(c(x1))) -> a(c(a(x1)))
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(2)
7.78/2.78	Obligation:
7.78/2.78	Relative term rewrite system:
7.78/2.78	The relative TRS consists of the following R rules:
7.78/2.78	
7.78/2.78	   a(c(a(x1))) -> b(b(c(x1)))
7.78/2.78	   b(a(a(x1))) -> c(c(b(x1)))
7.78/2.78	   a(c(a(x1))) -> b(b(a(x1)))
7.78/2.78	
7.78/2.78	The relative TRS consists of the following S rules:
7.78/2.78	
7.78/2.78	   a(a(c(x1))) -> c(c(a(x1)))
7.78/2.78	   a(a(c(x1))) -> b(b(b(x1)))
7.78/2.78	   c(b(c(x1))) -> a(c(a(x1)))
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(3) RelTRSRRRProof (EQUIVALENT)
7.78/2.78	We used the following monotonic ordering for rule removal:
7.78/2.78	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
7.78/2.78	
7.78/2.78	   <<<
7.78/2.78	 POL(a(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
7.78/2.78	>>>
7.78/2.78	
7.78/2.78	   <<<
7.78/2.78	 POL(c(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
7.78/2.78	>>>
7.78/2.78	
7.78/2.78	   <<<
7.78/2.78	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
7.78/2.78	>>>
7.78/2.78	
7.78/2.78	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.78/2.78	Rules from R:
7.78/2.78	
7.78/2.78	   a(c(a(x1))) -> b(b(a(x1)))
7.78/2.78	Rules from S:
7.78/2.78	
7.78/2.78	   a(a(c(x1))) -> b(b(b(x1)))
7.78/2.78	   c(b(c(x1))) -> a(c(a(x1)))
7.78/2.78	
7.78/2.78	
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(4)
7.78/2.78	Obligation:
7.78/2.78	Relative term rewrite system:
7.78/2.78	The relative TRS consists of the following R rules:
7.78/2.78	
7.78/2.78	   a(c(a(x1))) -> b(b(c(x1)))
7.78/2.78	   b(a(a(x1))) -> c(c(b(x1)))
7.78/2.78	
7.78/2.78	The relative TRS consists of the following S rules:
7.78/2.78	
7.78/2.78	   a(a(c(x1))) -> c(c(a(x1)))
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(5) RelTRSRRRProof (EQUIVALENT)
7.78/2.78	We used the following monotonic ordering for rule removal:
7.78/2.78	Knuth-Bendix order [KBO] with precedence:a_1 > b_1 > c_1
7.78/2.78	
7.78/2.78	and weight map:
7.78/2.78	
7.78/2.78	   a_1=2
7.78/2.78	   c_1=1
7.78/2.78	   b_1=2
7.78/2.78	
7.78/2.78	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.78/2.78	Rules from R:
7.78/2.78	
7.78/2.78	   a(c(a(x1))) -> b(b(c(x1)))
7.78/2.78	   b(a(a(x1))) -> c(c(b(x1)))
7.78/2.78	Rules from S:
7.78/2.78	
7.78/2.78	   a(a(c(x1))) -> c(c(a(x1)))
7.78/2.78	
7.78/2.78	
7.78/2.78	
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(6)
7.78/2.78	Obligation:
7.78/2.78	Relative term rewrite system:
7.78/2.78	R is empty.
7.78/2.78	S is empty.
7.78/2.78	
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(7) RIsEmptyProof (EQUIVALENT)
7.78/2.78	The TRS R is empty. Hence, termination is trivially proven.
7.78/2.78	----------------------------------------
7.78/2.78	
7.78/2.78	(8)
7.78/2.78	YES
7.97/2.88	EOF