48.31/13.18	YES
48.31/13.18	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
48.31/13.18	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
48.31/13.18	
48.31/13.18	
48.31/13.18	Termination of the given RelTRS could be proven:
48.31/13.18	
48.31/13.18	(0) RelTRS
48.31/13.18	(1) RelTRSRRRProof [EQUIVALENT, 3695 ms]
48.31/13.18	(2) RelTRS
48.31/13.18	(3) RelTRSRRRProof [EQUIVALENT, 81 ms]
48.31/13.18	(4) RelTRS
48.31/13.18	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
48.31/13.18	(6) YES
48.31/13.18	
48.31/13.18	
48.31/13.18	----------------------------------------
48.31/13.18	
48.31/13.18	(0)
48.31/13.18	Obligation:
48.31/13.18	Relative term rewrite system:
48.31/13.18	The relative TRS consists of the following R rules:
48.31/13.18	
48.31/13.18	   a(a(c(c(b(b(x1)))))) -> b(b(a(a(b(b(a(a(x1))))))))
48.31/13.18	   a(a(a(a(x1)))) -> a(a(b(b(a(a(x1))))))
48.31/13.18	
48.31/13.18	The relative TRS consists of the following S rules:
48.31/13.18	
48.31/13.18	   b(b(x1)) -> b(b(c(c(x1))))
48.31/13.18	
48.31/13.18	
48.31/13.18	----------------------------------------
48.31/13.18	
48.31/13.18	(1) RelTRSRRRProof (EQUIVALENT)
48.31/13.18	We used the following monotonic ordering for rule removal:
48.31/13.18	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
48.31/13.18	
48.31/13.18	   <<<
48.31/13.18	 POL(a(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 2], [0, 0, 3], [0, 0, 1]] 	* 	x_1
48.31/13.18	>>>
48.31/13.18	
48.31/13.18	   <<<
48.31/13.18	 POL(c(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 1, 1]] 	* 	x_1
48.31/13.19	>>>
48.31/13.19	
48.31/13.19	   <<<
48.31/13.19	 POL(b(x_1)) =  	[[0], [2], [0]] 	 +  	[[1, 0, 0], [0, 2, 2], [0, 0, 0]] 	* 	x_1
48.31/13.19	>>>
48.31/13.19	
48.31/13.19	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
48.31/13.19	Rules from R:
48.31/13.19	
48.31/13.19	   a(a(c(c(b(b(x1)))))) -> b(b(a(a(b(b(a(a(x1))))))))
48.31/13.19	Rules from S:
48.31/13.19	none
48.31/13.19	
48.31/13.19	
48.31/13.19	
48.31/13.19	
48.31/13.19	----------------------------------------
48.31/13.19	
48.31/13.19	(2)
48.31/13.19	Obligation:
48.31/13.19	Relative term rewrite system:
48.31/13.19	The relative TRS consists of the following R rules:
48.31/13.19	
48.31/13.19	   a(a(a(a(x1)))) -> a(a(b(b(a(a(x1))))))
48.31/13.19	
48.31/13.19	The relative TRS consists of the following S rules:
48.31/13.19	
48.31/13.19	   b(b(x1)) -> b(b(c(c(x1))))
48.31/13.19	
48.31/13.19	
48.31/13.19	----------------------------------------
48.31/13.19	
48.31/13.19	(3) RelTRSRRRProof (EQUIVALENT)
48.31/13.19	We used the following monotonic ordering for rule removal:
48.31/13.19	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
48.31/13.19	
48.31/13.19	   <<<
48.31/13.19	 POL(a(x_1)) =  	[[2], [1]] 	 +  	[[2, 2], [0, 2]] 	* 	x_1
48.31/13.19	>>>
48.31/13.19	
48.31/13.19	   <<<
48.31/13.19	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
48.31/13.19	>>>
48.31/13.19	
48.31/13.19	   <<<
48.31/13.19	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
48.31/13.19	>>>
48.31/13.19	
48.31/13.19	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
48.31/13.19	Rules from R:
48.31/13.19	
48.31/13.19	   a(a(a(a(x1)))) -> a(a(b(b(a(a(x1))))))
48.31/13.19	Rules from S:
48.31/13.19	none
48.31/13.19	
48.31/13.19	
48.31/13.19	
48.31/13.19	
48.31/13.19	----------------------------------------
48.31/13.19	
48.31/13.19	(4)
48.31/13.19	Obligation:
48.31/13.19	Relative term rewrite system:
48.31/13.19	R is empty.
48.31/13.19	The relative TRS consists of the following S rules:
48.31/13.19	
48.31/13.19	   b(b(x1)) -> b(b(c(c(x1))))
48.31/13.19	
48.31/13.19	
48.31/13.19	----------------------------------------
48.31/13.19	
48.31/13.19	(5) RIsEmptyProof (EQUIVALENT)
48.31/13.19	The TRS R is empty. Hence, termination is trivially proven.
48.31/13.19	----------------------------------------
48.31/13.19	
48.31/13.19	(6)
48.31/13.19	YES
48.71/13.37	EOF