28.20/7.89	YES
28.20/7.90	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.20/7.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.20/7.90	
28.20/7.90	
28.20/7.90	Termination of the given RelTRS could be proven:
28.20/7.90	
28.20/7.90	(0) RelTRS
28.20/7.90	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
28.20/7.90	(2) RelTRS
28.20/7.90	(3) RelTRSRRRProof [EQUIVALENT, 346 ms]
28.20/7.90	(4) RelTRS
28.20/7.90	(5) RelTRSRRRProof [EQUIVALENT, 351 ms]
28.20/7.90	(6) RelTRS
28.20/7.90	(7) RelTRSRRRProof [EQUIVALENT, 1 ms]
28.20/7.90	(8) RelTRS
28.20/7.90	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
28.20/7.90	(10) YES
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(0)
28.20/7.90	Obligation:
28.20/7.90	Relative term rewrite system:
28.20/7.90	The relative TRS consists of the following R rules:
28.20/7.90	
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	
28.20/7.90	The relative TRS consists of the following S rules:
28.20/7.90	
28.20/7.90	   b(a(a(x1))) -> a(b(c(x1)))
28.20/7.90	   b(b(b(x1))) -> a(a(b(x1)))
28.20/7.90	   b(c(a(x1))) -> a(c(a(x1)))
28.20/7.90	   a(c(b(x1))) -> c(a(a(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(1) RelTRS Reverse (EQUIVALENT)
28.20/7.90	We have reversed the following relative TRS [REVERSE]:
28.20/7.90	The set of rules R is 
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	
28.20/7.90	The set of rules S is 
28.20/7.90	   b(a(a(x1))) -> a(b(c(x1)))
28.20/7.90	   b(b(b(x1))) -> a(a(b(x1)))
28.20/7.90	   b(c(a(x1))) -> a(c(a(x1)))
28.20/7.90	   a(c(b(x1))) -> c(a(a(x1)))
28.20/7.90	
28.20/7.90	We have obtained the following relative TRS:
28.20/7.90	The set of rules R is 
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	
28.20/7.90	The set of rules S is 
28.20/7.90	   a(a(b(x1))) -> c(b(a(x1)))
28.20/7.90	   b(b(b(x1))) -> b(a(a(x1)))
28.20/7.90	   a(c(b(x1))) -> a(c(a(x1)))
28.20/7.90	   b(c(a(x1))) -> a(a(c(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(2)
28.20/7.90	Obligation:
28.20/7.90	Relative term rewrite system:
28.20/7.90	The relative TRS consists of the following R rules:
28.20/7.90	
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	
28.20/7.90	The relative TRS consists of the following S rules:
28.20/7.90	
28.20/7.90	   a(a(b(x1))) -> c(b(a(x1)))
28.20/7.90	   b(b(b(x1))) -> b(a(a(x1)))
28.20/7.90	   a(c(b(x1))) -> a(c(a(x1)))
28.20/7.90	   b(c(a(x1))) -> a(a(c(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(3) RelTRSRRRProof (EQUIVALENT)
28.20/7.90	We used the following monotonic ordering for rule removal:
28.20/7.90	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
28.20/7.90	
28.20/7.90	   <<<
28.20/7.90	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
28.20/7.90	>>>
28.20/7.90	
28.20/7.90	   <<<
28.20/7.90	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
28.20/7.90	>>>
28.20/7.90	
28.20/7.90	   <<<
28.20/7.90	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
28.20/7.90	>>>
28.20/7.90	
28.20/7.90	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
28.20/7.90	Rules from R:
28.20/7.90	none
28.20/7.90	Rules from S:
28.20/7.90	
28.20/7.90	   b(b(b(x1))) -> b(a(a(x1)))
28.20/7.90	   a(c(b(x1))) -> a(c(a(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(4)
28.20/7.90	Obligation:
28.20/7.90	Relative term rewrite system:
28.20/7.90	The relative TRS consists of the following R rules:
28.20/7.90	
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	
28.20/7.90	The relative TRS consists of the following S rules:
28.20/7.90	
28.20/7.90	   a(a(b(x1))) -> c(b(a(x1)))
28.20/7.90	   b(c(a(x1))) -> a(a(c(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(5) RelTRSRRRProof (EQUIVALENT)
28.20/7.90	We used the following monotonic ordering for rule removal:
28.20/7.90	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
28.20/7.90	
28.20/7.90	   <<<
28.20/7.90	 POL(c(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 1, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
28.20/7.90	>>>
28.20/7.90	
28.20/7.90	   <<<
28.20/7.90	 POL(b(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 3], [0, 0, 0]] 	* 	x_1
28.20/7.90	>>>
28.20/7.90	
28.20/7.90	   <<<
28.20/7.90	 POL(a(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 1, 0], [0, 0, 0]] 	* 	x_1
28.20/7.90	>>>
28.20/7.90	
28.20/7.90	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
28.20/7.90	Rules from R:
28.20/7.90	none
28.20/7.90	Rules from S:
28.20/7.90	
28.20/7.90	   b(c(a(x1))) -> a(a(c(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(6)
28.20/7.90	Obligation:
28.20/7.90	Relative term rewrite system:
28.20/7.90	The relative TRS consists of the following R rules:
28.20/7.90	
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	
28.20/7.90	The relative TRS consists of the following S rules:
28.20/7.90	
28.20/7.90	   a(a(b(x1))) -> c(b(a(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(7) RelTRSRRRProof (EQUIVALENT)
28.20/7.90	We used the following monotonic ordering for rule removal:
28.20/7.90	Polynomial interpretation [POLO]:
28.20/7.90	
28.20/7.90	   POL(a(x_1)) = 1 + x_1
28.20/7.90	   POL(b(x_1)) = x_1
28.20/7.90	   POL(c(x_1)) = 1 + x_1
28.20/7.90	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
28.20/7.90	Rules from R:
28.20/7.90	
28.20/7.90	   c(c(c(x1))) -> b(b(b(x1)))
28.20/7.90	Rules from S:
28.20/7.90	none
28.20/7.90	
28.20/7.90	
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(8)
28.20/7.90	Obligation:
28.20/7.90	Relative term rewrite system:
28.20/7.90	R is empty.
28.20/7.90	The relative TRS consists of the following S rules:
28.20/7.90	
28.20/7.90	   a(a(b(x1))) -> c(b(a(x1)))
28.20/7.90	
28.20/7.90	
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(9) RIsEmptyProof (EQUIVALENT)
28.20/7.90	The TRS R is empty. Hence, termination is trivially proven.
28.20/7.90	----------------------------------------
28.20/7.90	
28.20/7.90	(10)
28.20/7.90	YES
28.47/7.94	EOF