52.12/14.93	YES
52.12/14.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
52.12/14.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
52.12/14.93	
52.12/14.93	
52.12/14.93	Termination of the given RelTRS could be proven:
52.12/14.93	
52.12/14.93	(0) RelTRS
52.12/14.93	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
52.12/14.93	(2) RelTRS
52.12/14.93	(3) RelTRSRRRProof [EQUIVALENT, 572 ms]
52.12/14.93	(4) RelTRS
52.12/14.93	(5) RelTRSRRRProof [EQUIVALENT, 825 ms]
52.12/14.93	(6) RelTRS
52.12/14.93	(7) RelTRSRRRProof [EQUIVALENT, 208 ms]
52.12/14.93	(8) RelTRS
52.12/14.93	(9) RelTRSRRRProof [EQUIVALENT, 10 ms]
52.12/14.93	(10) RelTRS
52.12/14.93	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
52.12/14.93	(12) YES
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(0)
52.12/14.93	Obligation:
52.12/14.93	Relative term rewrite system:
52.12/14.93	The relative TRS consists of the following R rules:
52.12/14.93	
52.12/14.93	   c(c(c(x1))) -> c(a(a(x1)))
52.12/14.93	   b(a(b(x1))) -> b(b(c(x1)))
52.12/14.93	   c(a(a(x1))) -> c(b(a(x1)))
52.12/14.93	   b(a(a(x1))) -> a(b(c(x1)))
52.12/14.93	   b(b(c(x1))) -> a(a(c(x1)))
52.12/14.93	
52.12/14.93	The relative TRS consists of the following S rules:
52.12/14.93	
52.12/14.93	   a(c(c(x1))) -> b(a(c(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(1) RelTRS Reverse (EQUIVALENT)
52.12/14.93	We have reversed the following relative TRS [REVERSE]:
52.12/14.93	The set of rules R is 
52.12/14.93	   c(c(c(x1))) -> c(a(a(x1)))
52.12/14.93	   b(a(b(x1))) -> b(b(c(x1)))
52.12/14.93	   c(a(a(x1))) -> c(b(a(x1)))
52.12/14.93	   b(a(a(x1))) -> a(b(c(x1)))
52.12/14.93	   b(b(c(x1))) -> a(a(c(x1)))
52.12/14.93	
52.12/14.93	The set of rules S is 
52.12/14.93	   a(c(c(x1))) -> b(a(c(x1)))
52.12/14.93	
52.12/14.93	We have obtained the following relative TRS:
52.12/14.93	The set of rules R is 
52.12/14.93	   c(c(c(x1))) -> a(a(c(x1)))
52.12/14.93	   b(a(b(x1))) -> c(b(b(x1)))
52.12/14.93	   a(a(c(x1))) -> a(b(c(x1)))
52.12/14.93	   a(a(b(x1))) -> c(b(a(x1)))
52.12/14.93	   c(b(b(x1))) -> c(a(a(x1)))
52.12/14.93	
52.12/14.93	The set of rules S is 
52.12/14.93	   c(c(a(x1))) -> c(a(b(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(2)
52.12/14.93	Obligation:
52.12/14.93	Relative term rewrite system:
52.12/14.93	The relative TRS consists of the following R rules:
52.12/14.93	
52.12/14.93	   c(c(c(x1))) -> a(a(c(x1)))
52.12/14.93	   b(a(b(x1))) -> c(b(b(x1)))
52.12/14.93	   a(a(c(x1))) -> a(b(c(x1)))
52.12/14.93	   a(a(b(x1))) -> c(b(a(x1)))
52.12/14.93	   c(b(b(x1))) -> c(a(a(x1)))
52.12/14.93	
52.12/14.93	The relative TRS consists of the following S rules:
52.12/14.93	
52.12/14.93	   c(c(a(x1))) -> c(a(b(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(3) RelTRSRRRProof (EQUIVALENT)
52.12/14.93	We used the following monotonic ordering for rule removal:
52.12/14.93	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(c(x_1)) =  	[[1], [1]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(a(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
52.12/14.93	Rules from R:
52.12/14.93	
52.12/14.93	   c(c(c(x1))) -> a(a(c(x1)))
52.12/14.93	Rules from S:
52.12/14.93	none
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(4)
52.12/14.93	Obligation:
52.12/14.93	Relative term rewrite system:
52.12/14.93	The relative TRS consists of the following R rules:
52.12/14.93	
52.12/14.93	   b(a(b(x1))) -> c(b(b(x1)))
52.12/14.93	   a(a(c(x1))) -> a(b(c(x1)))
52.12/14.93	   a(a(b(x1))) -> c(b(a(x1)))
52.12/14.93	   c(b(b(x1))) -> c(a(a(x1)))
52.12/14.93	
52.12/14.93	The relative TRS consists of the following S rules:
52.12/14.93	
52.12/14.93	   c(c(a(x1))) -> c(a(b(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(5) RelTRSRRRProof (EQUIVALENT)
52.12/14.93	We used the following monotonic ordering for rule removal:
52.12/14.93	Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(b(x_1)) =  	[[0], [0], [0], [0], [0], [1]] 	 +  	[[1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(a(x_1)) =  	[[0], [0], [0], [0], [0], [0]] 	 +  	[[1, 1, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 1], [0, 0, 1, 0, 0, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(c(x_1)) =  	[[0], [0], [1], [0], [1], [0]] 	 +  	[[1, 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, 0, 0, 0, 0, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
52.12/14.93	Rules from R:
52.12/14.93	
52.12/14.93	   a(a(c(x1))) -> a(b(c(x1)))
52.12/14.93	Rules from S:
52.12/14.93	none
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(6)
52.12/14.93	Obligation:
52.12/14.93	Relative term rewrite system:
52.12/14.93	The relative TRS consists of the following R rules:
52.12/14.93	
52.12/14.93	   b(a(b(x1))) -> c(b(b(x1)))
52.12/14.93	   a(a(b(x1))) -> c(b(a(x1)))
52.12/14.93	   c(b(b(x1))) -> c(a(a(x1)))
52.12/14.93	
52.12/14.93	The relative TRS consists of the following S rules:
52.12/14.93	
52.12/14.93	   c(c(a(x1))) -> c(a(b(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(7) RelTRSRRRProof (EQUIVALENT)
52.12/14.93	We used the following monotonic ordering for rule removal:
52.12/14.93	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 2]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
52.12/14.93	Rules from R:
52.12/14.93	
52.12/14.93	   a(a(b(x1))) -> c(b(a(x1)))
52.12/14.93	   c(b(b(x1))) -> c(a(a(x1)))
52.12/14.93	Rules from S:
52.12/14.93	none
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(8)
52.12/14.93	Obligation:
52.12/14.93	Relative term rewrite system:
52.12/14.93	The relative TRS consists of the following R rules:
52.12/14.93	
52.12/14.93	   b(a(b(x1))) -> c(b(b(x1)))
52.12/14.93	
52.12/14.93	The relative TRS consists of the following S rules:
52.12/14.93	
52.12/14.93	   c(c(a(x1))) -> c(a(b(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(9) RelTRSRRRProof (EQUIVALENT)
52.12/14.93	We used the following monotonic ordering for rule removal:
52.12/14.93	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(a(x_1)) =  	[[2], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	   <<<
52.12/14.93	 POL(c(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
52.12/14.93	>>>
52.12/14.93	
52.12/14.93	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
52.12/14.93	Rules from R:
52.12/14.93	
52.12/14.93	   b(a(b(x1))) -> c(b(b(x1)))
52.12/14.93	Rules from S:
52.12/14.93	none
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(10)
52.12/14.93	Obligation:
52.12/14.93	Relative term rewrite system:
52.12/14.93	R is empty.
52.12/14.93	The relative TRS consists of the following S rules:
52.12/14.93	
52.12/14.93	   c(c(a(x1))) -> c(a(b(x1)))
52.12/14.93	
52.12/14.93	
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(11) RIsEmptyProof (EQUIVALENT)
52.12/14.93	The TRS R is empty. Hence, termination is trivially proven.
52.12/14.93	----------------------------------------
52.12/14.93	
52.12/14.93	(12)
52.12/14.93	YES
52.12/14.96	EOF