8.13/2.86	YES
8.13/2.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.13/2.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.13/2.87	
8.13/2.87	
8.13/2.87	Termination of the given RelTRS could be proven:
8.13/2.87	
8.13/2.87	(0) RelTRS
8.13/2.87	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
8.13/2.87	(2) RelTRS
8.13/2.87	(3) RelTRSRRRProof [EQUIVALENT, 460 ms]
8.13/2.87	(4) RelTRS
8.13/2.87	(5) RelTRSRRRProof [EQUIVALENT, 6 ms]
8.13/2.87	(6) RelTRS
8.13/2.87	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
8.13/2.87	(8) YES
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(0)
8.13/2.87	Obligation:
8.13/2.87	Relative term rewrite system:
8.13/2.87	The relative TRS consists of the following R rules:
8.13/2.87	
8.13/2.87	   a(a(c(x1))) -> b(a(a(x1)))
8.13/2.87	   b(c(a(x1))) -> a(a(a(x1)))
8.13/2.87	
8.13/2.87	The relative TRS consists of the following S rules:
8.13/2.87	
8.13/2.87	   c(a(a(x1))) -> b(c(a(x1)))
8.13/2.87	   c(c(a(x1))) -> c(a(c(x1)))
8.13/2.87	   a(b(c(x1))) -> a(c(b(x1)))
8.13/2.87	   a(b(b(x1))) -> b(a(c(x1)))
8.13/2.87	   a(a(c(x1))) -> a(c(a(x1)))
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(1) RelTRS Reverse (EQUIVALENT)
8.13/2.87	We have reversed the following relative TRS [REVERSE]:
8.13/2.87	The set of rules R is 
8.13/2.87	   a(a(c(x1))) -> b(a(a(x1)))
8.13/2.87	   b(c(a(x1))) -> a(a(a(x1)))
8.13/2.87	
8.13/2.87	The set of rules S is 
8.13/2.87	   c(a(a(x1))) -> b(c(a(x1)))
8.13/2.87	   c(c(a(x1))) -> c(a(c(x1)))
8.13/2.87	   a(b(c(x1))) -> a(c(b(x1)))
8.13/2.87	   a(b(b(x1))) -> b(a(c(x1)))
8.13/2.87	   a(a(c(x1))) -> a(c(a(x1)))
8.13/2.87	
8.13/2.87	We have obtained the following relative TRS:
8.13/2.87	The set of rules R is 
8.13/2.87	   c(a(a(x1))) -> a(a(b(x1)))
8.13/2.87	   a(c(b(x1))) -> a(a(a(x1)))
8.13/2.87	
8.13/2.87	The set of rules S is 
8.13/2.87	   a(a(c(x1))) -> a(c(b(x1)))
8.13/2.87	   a(c(c(x1))) -> c(a(c(x1)))
8.13/2.87	   c(b(a(x1))) -> b(c(a(x1)))
8.13/2.87	   b(b(a(x1))) -> c(a(b(x1)))
8.13/2.87	   c(a(a(x1))) -> a(c(a(x1)))
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(2)
8.13/2.87	Obligation:
8.13/2.87	Relative term rewrite system:
8.13/2.87	The relative TRS consists of the following R rules:
8.13/2.87	
8.13/2.87	   c(a(a(x1))) -> a(a(b(x1)))
8.13/2.87	   a(c(b(x1))) -> a(a(a(x1)))
8.13/2.87	
8.13/2.87	The relative TRS consists of the following S rules:
8.13/2.87	
8.13/2.87	   a(a(c(x1))) -> a(c(b(x1)))
8.13/2.87	   a(c(c(x1))) -> c(a(c(x1)))
8.13/2.87	   c(b(a(x1))) -> b(c(a(x1)))
8.13/2.87	   b(b(a(x1))) -> c(a(b(x1)))
8.13/2.87	   c(a(a(x1))) -> a(c(a(x1)))
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(3) RelTRSRRRProof (EQUIVALENT)
8.13/2.87	We used the following monotonic ordering for rule removal:
8.13/2.87	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.13/2.87	
8.13/2.87	   <<<
8.13/2.87	 POL(c(x_1)) =  	[[1], [0]] 	 +  	[[2, 2], [0, 2]] 	* 	x_1
8.13/2.87	>>>
8.13/2.87	
8.13/2.87	   <<<
8.13/2.87	 POL(a(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.13/2.87	>>>
8.13/2.87	
8.13/2.87	   <<<
8.13/2.87	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.13/2.87	>>>
8.13/2.87	
8.13/2.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.13/2.87	Rules from R:
8.13/2.87	
8.13/2.87	   c(a(a(x1))) -> a(a(b(x1)))
8.13/2.87	Rules from S:
8.13/2.87	
8.13/2.87	   c(b(a(x1))) -> b(c(a(x1)))
8.13/2.87	   b(b(a(x1))) -> c(a(b(x1)))
8.13/2.87	
8.13/2.87	
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(4)
8.13/2.87	Obligation:
8.13/2.87	Relative term rewrite system:
8.13/2.87	The relative TRS consists of the following R rules:
8.13/2.87	
8.13/2.87	   a(c(b(x1))) -> a(a(a(x1)))
8.13/2.87	
8.13/2.87	The relative TRS consists of the following S rules:
8.13/2.87	
8.13/2.87	   a(a(c(x1))) -> a(c(b(x1)))
8.13/2.87	   a(c(c(x1))) -> c(a(c(x1)))
8.13/2.87	   c(a(a(x1))) -> a(c(a(x1)))
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(5) RelTRSRRRProof (EQUIVALENT)
8.13/2.87	We used the following monotonic ordering for rule removal:
8.13/2.87	Polynomial interpretation [POLO]:
8.13/2.87	
8.13/2.87	   POL(a(x_1)) = x_1
8.13/2.87	   POL(b(x_1)) = x_1
8.13/2.87	   POL(c(x_1)) = 1 + x_1
8.13/2.87	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.13/2.87	Rules from R:
8.13/2.87	
8.13/2.87	   a(c(b(x1))) -> a(a(a(x1)))
8.13/2.87	Rules from S:
8.13/2.87	none
8.13/2.87	
8.13/2.87	
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(6)
8.13/2.87	Obligation:
8.13/2.87	Relative term rewrite system:
8.13/2.87	R is empty.
8.13/2.87	The relative TRS consists of the following S rules:
8.13/2.87	
8.13/2.87	   a(a(c(x1))) -> a(c(b(x1)))
8.13/2.87	   a(c(c(x1))) -> c(a(c(x1)))
8.13/2.87	   c(a(a(x1))) -> a(c(a(x1)))
8.13/2.87	
8.13/2.87	
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(7) RIsEmptyProof (EQUIVALENT)
8.13/2.87	The TRS R is empty. Hence, termination is trivially proven.
8.13/2.87	----------------------------------------
8.13/2.87	
8.13/2.87	(8)
8.13/2.87	YES
8.32/3.02	EOF