16.88/5.16	YES
17.22/5.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.22/5.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.22/5.22	
17.22/5.22	
17.22/5.22	Termination of the given RelTRS could be proven:
17.22/5.22	
17.22/5.22	(0) RelTRS
17.22/5.22	(1) RelTRSRRRProof [EQUIVALENT, 872 ms]
17.22/5.22	(2) RelTRS
17.22/5.22	(3) RelTRSRRRProof [EQUIVALENT, 8 ms]
17.22/5.22	(4) RelTRS
17.22/5.22	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
17.22/5.22	(6) YES
17.22/5.22	
17.22/5.22	
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(0)
17.22/5.22	Obligation:
17.22/5.22	Relative term rewrite system:
17.22/5.22	The relative TRS consists of the following R rules:
17.22/5.22	
17.22/5.22	   c(b(a(x1))) -> a(b(b(x1)))
17.22/5.22	   b(a(c(x1))) -> b(b(b(x1)))
17.22/5.22	   a(c(c(x1))) -> a(b(b(x1)))
17.22/5.22	
17.22/5.22	The relative TRS consists of the following S rules:
17.22/5.22	
17.22/5.22	   c(c(a(x1))) -> c(a(c(x1)))
17.22/5.22	   a(a(a(x1))) -> a(a(b(x1)))
17.22/5.22	   a(b(a(x1))) -> b(b(a(x1)))
17.22/5.22	   b(b(b(x1))) -> a(b(c(x1)))
17.22/5.22	
17.22/5.22	
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(1) RelTRSRRRProof (EQUIVALENT)
17.22/5.22	We used the following monotonic ordering for rule removal:
17.22/5.22	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
17.22/5.22	
17.22/5.22	   <<<
17.22/5.22	 POL(c(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
17.22/5.22	>>>
17.22/5.22	
17.22/5.22	   <<<
17.22/5.22	 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
17.22/5.22	>>>
17.22/5.22	
17.22/5.22	   <<<
17.22/5.22	 POL(a(x_1)) =  	[[2], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
17.22/5.22	>>>
17.22/5.22	
17.22/5.22	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
17.22/5.22	Rules from R:
17.22/5.22	
17.22/5.22	   c(b(a(x1))) -> a(b(b(x1)))
17.22/5.22	   b(a(c(x1))) -> b(b(b(x1)))
17.22/5.22	Rules from S:
17.22/5.22	
17.22/5.22	   a(a(a(x1))) -> a(a(b(x1)))
17.22/5.22	   a(b(a(x1))) -> b(b(a(x1)))
17.22/5.22	   b(b(b(x1))) -> a(b(c(x1)))
17.22/5.22	
17.22/5.22	
17.22/5.22	
17.22/5.22	
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(2)
17.22/5.22	Obligation:
17.22/5.22	Relative term rewrite system:
17.22/5.22	The relative TRS consists of the following R rules:
17.22/5.22	
17.22/5.22	   a(c(c(x1))) -> a(b(b(x1)))
17.22/5.22	
17.22/5.22	The relative TRS consists of the following S rules:
17.22/5.22	
17.22/5.22	   c(c(a(x1))) -> c(a(c(x1)))
17.22/5.22	
17.22/5.22	
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(3) RelTRSRRRProof (EQUIVALENT)
17.22/5.22	We used the following monotonic ordering for rule removal:
17.22/5.22	Polynomial interpretation [POLO]:
17.22/5.22	
17.22/5.22	   POL(a(x_1)) = x_1
17.22/5.22	   POL(b(x_1)) = x_1
17.22/5.22	   POL(c(x_1)) = 1 + x_1
17.22/5.22	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
17.22/5.22	Rules from R:
17.22/5.22	
17.22/5.22	   a(c(c(x1))) -> a(b(b(x1)))
17.22/5.22	Rules from S:
17.22/5.22	none
17.22/5.22	
17.22/5.22	
17.22/5.22	
17.22/5.22	
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(4)
17.22/5.22	Obligation:
17.22/5.22	Relative term rewrite system:
17.22/5.22	R is empty.
17.22/5.22	The relative TRS consists of the following S rules:
17.22/5.22	
17.22/5.22	   c(c(a(x1))) -> c(a(c(x1)))
17.22/5.22	
17.22/5.22	
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(5) RIsEmptyProof (EQUIVALENT)
17.22/5.22	The TRS R is empty. Hence, termination is trivially proven.
17.22/5.22	----------------------------------------
17.22/5.22	
17.22/5.22	(6)
17.22/5.22	YES
17.46/5.32	EOF