7.96/2.78	YES
8.26/2.80	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.26/2.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.26/2.80	
8.26/2.80	
8.26/2.80	Termination of the given RelTRS could be proven:
8.26/2.80	
8.26/2.80	(0) RelTRS
8.26/2.80	(1) RelTRSRRRProof [EQUIVALENT, 178 ms]
8.26/2.80	(2) RelTRS
8.26/2.80	(3) RelTRSRRRProof [EQUIVALENT, 19 ms]
8.26/2.80	(4) RelTRS
8.26/2.80	(5) RIsEmptyProof [EQUIVALENT, 3 ms]
8.26/2.80	(6) YES
8.26/2.80	
8.26/2.80	
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(0)
8.26/2.80	Obligation:
8.26/2.80	Relative term rewrite system:
8.26/2.80	The relative TRS consists of the following R rules:
8.26/2.80	
8.26/2.80	   a(a(b(x1))) -> a(c(b(x1)))
8.26/2.80	   c(c(c(x1))) -> a(b(a(x1)))
8.26/2.80	   b(c(b(x1))) -> b(a(c(x1)))
8.26/2.80	
8.26/2.80	The relative TRS consists of the following S rules:
8.26/2.80	
8.26/2.80	   b(b(c(x1))) -> c(a(b(x1)))
8.26/2.80	   b(b(b(x1))) -> a(b(b(x1)))
8.26/2.80	   b(b(c(x1))) -> b(c(c(x1)))
8.26/2.80	   b(a(a(x1))) -> a(c(c(x1)))
8.26/2.80	
8.26/2.80	
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(1) RelTRSRRRProof (EQUIVALENT)
8.26/2.80	We used the following monotonic ordering for rule removal:
8.26/2.80	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.26/2.80	
8.26/2.80	   <<<
8.26/2.80	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.26/2.80	>>>
8.26/2.80	
8.26/2.80	   <<<
8.26/2.80	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
8.26/2.80	>>>
8.26/2.80	
8.26/2.80	   <<<
8.26/2.80	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
8.26/2.80	>>>
8.26/2.80	
8.26/2.80	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.26/2.80	Rules from R:
8.26/2.80	
8.26/2.80	   b(c(b(x1))) -> b(a(c(x1)))
8.26/2.80	Rules from S:
8.26/2.80	
8.26/2.80	   b(b(c(x1))) -> c(a(b(x1)))
8.26/2.80	   b(b(b(x1))) -> a(b(b(x1)))
8.26/2.80	   b(b(c(x1))) -> b(c(c(x1)))
8.26/2.80	
8.26/2.80	
8.26/2.80	
8.26/2.80	
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(2)
8.26/2.80	Obligation:
8.26/2.80	Relative term rewrite system:
8.26/2.80	The relative TRS consists of the following R rules:
8.26/2.80	
8.26/2.80	   a(a(b(x1))) -> a(c(b(x1)))
8.26/2.80	   c(c(c(x1))) -> a(b(a(x1)))
8.26/2.80	
8.26/2.80	The relative TRS consists of the following S rules:
8.26/2.80	
8.26/2.80	   b(a(a(x1))) -> a(c(c(x1)))
8.26/2.80	
8.26/2.80	
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(3) RelTRSRRRProof (EQUIVALENT)
8.26/2.80	We used the following monotonic ordering for rule removal:
8.26/2.80	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.26/2.80	
8.26/2.80	   <<<
8.26/2.80	 POL(a(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
8.26/2.80	>>>
8.26/2.80	
8.26/2.80	   <<<
8.26/2.80	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
8.26/2.80	>>>
8.26/2.80	
8.26/2.80	   <<<
8.26/2.80	 POL(c(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
8.26/2.80	>>>
8.26/2.80	
8.26/2.80	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.26/2.80	Rules from R:
8.26/2.80	
8.26/2.80	   a(a(b(x1))) -> a(c(b(x1)))
8.26/2.80	   c(c(c(x1))) -> a(b(a(x1)))
8.26/2.80	Rules from S:
8.26/2.80	
8.26/2.80	   b(a(a(x1))) -> a(c(c(x1)))
8.26/2.80	
8.26/2.80	
8.26/2.80	
8.26/2.80	
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(4)
8.26/2.80	Obligation:
8.26/2.80	Relative term rewrite system:
8.26/2.80	R is empty.
8.26/2.80	S is empty.
8.26/2.80	
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(5) RIsEmptyProof (EQUIVALENT)
8.26/2.80	The TRS R is empty. Hence, termination is trivially proven.
8.26/2.80	----------------------------------------
8.26/2.80	
8.26/2.80	(6)
8.26/2.80	YES
8.31/2.86	EOF