66.65/17.61	YES
66.65/17.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
66.65/17.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
66.65/17.61	
66.65/17.61	
66.65/17.61	Termination of the given RelTRS could be proven:
66.65/17.61	
66.65/17.61	(0) RelTRS
66.65/17.61	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
66.65/17.61	(2) RelTRS
66.65/17.61	(3) RelTRSRRRProof [EQUIVALENT, 1228 ms]
66.65/17.61	(4) RelTRS
66.65/17.61	(5) RelTRSRRRProof [EQUIVALENT, 877 ms]
66.65/17.61	(6) RelTRS
66.65/17.61	(7) RelTRSRRRProof [EQUIVALENT, 8 ms]
66.65/17.61	(8) RelTRS
66.65/17.61	(9) RelTRSRRRProof [EQUIVALENT, 8 ms]
66.65/17.61	(10) RelTRS
66.65/17.61	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
66.65/17.61	(12) YES
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(0)
66.65/17.61	Obligation:
66.65/17.61	Relative term rewrite system:
66.65/17.61	The relative TRS consists of the following R rules:
66.65/17.61	
66.65/17.61	   a(a(a(x1))) -> b(b(c(x1)))
66.65/17.61	   b(b(b(x1))) -> c(c(b(x1)))
66.65/17.61	
66.65/17.61	The relative TRS consists of the following S rules:
66.65/17.61	
66.65/17.61	   a(c(a(x1))) -> b(c(b(x1)))
66.65/17.61	   b(c(b(x1))) -> a(c(c(x1)))
66.65/17.61	   c(c(a(x1))) -> b(b(a(x1)))
66.65/17.61	   c(a(a(x1))) -> b(b(c(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(1) RelTRS Reverse (EQUIVALENT)
66.65/17.61	We have reversed the following relative TRS [REVERSE]:
66.65/17.61	The set of rules R is 
66.65/17.61	   a(a(a(x1))) -> b(b(c(x1)))
66.65/17.61	   b(b(b(x1))) -> c(c(b(x1)))
66.65/17.61	
66.65/17.61	The set of rules S is 
66.65/17.61	   a(c(a(x1))) -> b(c(b(x1)))
66.65/17.61	   b(c(b(x1))) -> a(c(c(x1)))
66.65/17.61	   c(c(a(x1))) -> b(b(a(x1)))
66.65/17.61	   c(a(a(x1))) -> b(b(c(x1)))
66.65/17.61	
66.65/17.61	We have obtained the following relative TRS:
66.65/17.61	The set of rules R is 
66.65/17.61	   a(a(a(x1))) -> c(b(b(x1)))
66.65/17.61	   b(b(b(x1))) -> b(c(c(x1)))
66.65/17.61	
66.65/17.61	The set of rules S is 
66.65/17.61	   a(c(a(x1))) -> b(c(b(x1)))
66.65/17.61	   b(c(b(x1))) -> c(c(a(x1)))
66.65/17.61	   a(c(c(x1))) -> a(b(b(x1)))
66.65/17.61	   a(a(c(x1))) -> c(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(2)
66.65/17.61	Obligation:
66.65/17.61	Relative term rewrite system:
66.65/17.61	The relative TRS consists of the following R rules:
66.65/17.61	
66.65/17.61	   a(a(a(x1))) -> c(b(b(x1)))
66.65/17.61	   b(b(b(x1))) -> b(c(c(x1)))
66.65/17.61	
66.65/17.61	The relative TRS consists of the following S rules:
66.65/17.61	
66.65/17.61	   a(c(a(x1))) -> b(c(b(x1)))
66.65/17.61	   b(c(b(x1))) -> c(c(a(x1)))
66.65/17.61	   a(c(c(x1))) -> a(b(b(x1)))
66.65/17.61	   a(a(c(x1))) -> c(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(3) RelTRSRRRProof (EQUIVALENT)
66.65/17.61	We used the following monotonic ordering for rule removal:
66.65/17.61	Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(a(x_1)) =  	[[0], [0], [0], [0], [0], [1]] 	 +  	[[1, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(c(x_1)) =  	[[0], [0], [0], [0], [0], [0]] 	 +  	[[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(b(x_1)) =  	[[0], [0], [0], [0], [0], [0]] 	 +  	[[1, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
66.65/17.61	Rules from R:
66.65/17.61	none
66.65/17.61	Rules from S:
66.65/17.61	
66.65/17.61	   a(c(a(x1))) -> b(c(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(4)
66.65/17.61	Obligation:
66.65/17.61	Relative term rewrite system:
66.65/17.61	The relative TRS consists of the following R rules:
66.65/17.61	
66.65/17.61	   a(a(a(x1))) -> c(b(b(x1)))
66.65/17.61	   b(b(b(x1))) -> b(c(c(x1)))
66.65/17.61	
66.65/17.61	The relative TRS consists of the following S rules:
66.65/17.61	
66.65/17.61	   b(c(b(x1))) -> c(c(a(x1)))
66.65/17.61	   a(c(c(x1))) -> a(b(b(x1)))
66.65/17.61	   a(a(c(x1))) -> c(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(5) RelTRSRRRProof (EQUIVALENT)
66.65/17.61	We used the following monotonic ordering for rule removal:
66.65/17.61	Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(a(x_1)) =  	[[0], [0], [0], [0], [0], [1]] 	 +  	[[1, 0, 1, 0, 0, 0], [0, 1, 1, 1, 0, 1], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, 1, 1, 0, 0, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(c(x_1)) =  	[[0], [0], [0], [0], [0], [0]] 	 +  	[[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(b(x_1)) =  	[[0], [0], [0], [0], [1], [0]] 	 +  	[[1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [1, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
66.65/17.61	Rules from R:
66.65/17.61	none
66.65/17.61	Rules from S:
66.65/17.61	
66.65/17.61	   b(c(b(x1))) -> c(c(a(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(6)
66.65/17.61	Obligation:
66.65/17.61	Relative term rewrite system:
66.65/17.61	The relative TRS consists of the following R rules:
66.65/17.61	
66.65/17.61	   a(a(a(x1))) -> c(b(b(x1)))
66.65/17.61	   b(b(b(x1))) -> b(c(c(x1)))
66.65/17.61	
66.65/17.61	The relative TRS consists of the following S rules:
66.65/17.61	
66.65/17.61	   a(c(c(x1))) -> a(b(b(x1)))
66.65/17.61	   a(a(c(x1))) -> c(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(7) RelTRSRRRProof (EQUIVALENT)
66.65/17.61	We used the following monotonic ordering for rule removal:
66.65/17.61	Polynomial interpretation [POLO]:
66.65/17.61	
66.65/17.61	   POL(a(x_1)) = 1 + x_1
66.65/17.61	   POL(b(x_1)) = x_1
66.65/17.61	   POL(c(x_1)) = x_1
66.65/17.61	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
66.65/17.61	Rules from R:
66.65/17.61	
66.65/17.61	   a(a(a(x1))) -> c(b(b(x1)))
66.65/17.61	Rules from S:
66.65/17.61	
66.65/17.61	   a(a(c(x1))) -> c(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(8)
66.65/17.61	Obligation:
66.65/17.61	Relative term rewrite system:
66.65/17.61	The relative TRS consists of the following R rules:
66.65/17.61	
66.65/17.61	   b(b(b(x1))) -> b(c(c(x1)))
66.65/17.61	
66.65/17.61	The relative TRS consists of the following S rules:
66.65/17.61	
66.65/17.61	   a(c(c(x1))) -> a(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(9) RelTRSRRRProof (EQUIVALENT)
66.65/17.61	We used the following monotonic ordering for rule removal:
66.65/17.61	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 2]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(c(x_1)) =  	[[1], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	   <<<
66.65/17.61	 POL(a(x_1)) =  	[[2], [0]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
66.65/17.61	>>>
66.65/17.61	
66.65/17.61	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
66.65/17.61	Rules from R:
66.65/17.61	
66.65/17.61	   b(b(b(x1))) -> b(c(c(x1)))
66.65/17.61	Rules from S:
66.65/17.61	
66.65/17.61	   a(c(c(x1))) -> a(b(b(x1)))
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(10)
66.65/17.61	Obligation:
66.65/17.61	Relative term rewrite system:
66.65/17.61	R is empty.
66.65/17.61	S is empty.
66.65/17.61	
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(11) RIsEmptyProof (EQUIVALENT)
66.65/17.61	The TRS R is empty. Hence, termination is trivially proven.
66.65/17.61	----------------------------------------
66.65/17.61	
66.65/17.61	(12)
66.65/17.61	YES
66.81/17.65	EOF