8.15/2.97	YES
8.15/2.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
8.15/2.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.15/2.98	
8.15/2.98	
8.15/2.98	Termination of the given RelTRS could be proven:
8.15/2.98	
8.15/2.98	(0) RelTRS
8.15/2.98	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
8.15/2.98	(2) RelTRS
8.15/2.98	(3) RelTRSRRRProof [EQUIVALENT, 105 ms]
8.15/2.98	(4) RelTRS
8.15/2.98	(5) RelTRSRRRProof [EQUIVALENT, 11 ms]
8.15/2.98	(6) RelTRS
8.15/2.98	(7) RelTRSRRRProof [EQUIVALENT, 8 ms]
8.15/2.98	(8) RelTRS
8.15/2.98	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
8.15/2.98	(10) YES
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(0)
8.15/2.98	Obligation:
8.15/2.98	Relative term rewrite system:
8.15/2.98	The relative TRS consists of the following R rules:
8.15/2.98	
8.15/2.98	   a(b(a(x1))) -> b(c(c(x1)))
8.15/2.98	   b(a(c(x1))) -> a(a(c(x1)))
8.15/2.98	   a(b(a(x1))) -> a(c(c(x1)))
8.15/2.98	   b(b(a(x1))) -> a(b(a(x1)))
8.15/2.98	
8.15/2.98	The relative TRS consists of the following S rules:
8.15/2.98	
8.15/2.98	   a(a(c(x1))) -> a(b(c(x1)))
8.15/2.98	   a(c(c(x1))) -> c(b(a(x1)))
8.15/2.98	   b(c(c(x1))) -> c(c(a(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(1) RelTRS Reverse (EQUIVALENT)
8.15/2.98	We have reversed the following relative TRS [REVERSE]:
8.15/2.98	The set of rules R is 
8.15/2.98	   a(b(a(x1))) -> b(c(c(x1)))
8.15/2.98	   b(a(c(x1))) -> a(a(c(x1)))
8.15/2.98	   a(b(a(x1))) -> a(c(c(x1)))
8.15/2.98	   b(b(a(x1))) -> a(b(a(x1)))
8.15/2.98	
8.15/2.98	The set of rules S is 
8.15/2.98	   a(a(c(x1))) -> a(b(c(x1)))
8.15/2.98	   a(c(c(x1))) -> c(b(a(x1)))
8.15/2.98	   b(c(c(x1))) -> c(c(a(x1)))
8.15/2.98	
8.15/2.98	We have obtained the following relative TRS:
8.15/2.98	The set of rules R is 
8.15/2.98	   a(b(a(x1))) -> c(c(b(x1)))
8.15/2.98	   c(a(b(x1))) -> c(a(a(x1)))
8.15/2.98	   a(b(a(x1))) -> c(c(a(x1)))
8.15/2.98	   a(b(b(x1))) -> a(b(a(x1)))
8.15/2.98	
8.15/2.98	The set of rules S is 
8.15/2.98	   c(a(a(x1))) -> c(b(a(x1)))
8.15/2.98	   c(c(a(x1))) -> a(b(c(x1)))
8.15/2.98	   c(c(b(x1))) -> a(c(c(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(2)
8.15/2.98	Obligation:
8.15/2.98	Relative term rewrite system:
8.15/2.98	The relative TRS consists of the following R rules:
8.15/2.98	
8.15/2.98	   a(b(a(x1))) -> c(c(b(x1)))
8.15/2.98	   c(a(b(x1))) -> c(a(a(x1)))
8.15/2.98	   a(b(a(x1))) -> c(c(a(x1)))
8.15/2.98	   a(b(b(x1))) -> a(b(a(x1)))
8.15/2.98	
8.15/2.98	The relative TRS consists of the following S rules:
8.15/2.98	
8.15/2.98	   c(a(a(x1))) -> c(b(a(x1)))
8.15/2.98	   c(c(a(x1))) -> a(b(c(x1)))
8.15/2.98	   c(c(b(x1))) -> a(c(c(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(3) RelTRSRRRProof (EQUIVALENT)
8.15/2.98	We used the following monotonic ordering for rule removal:
8.15/2.98	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(a(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.15/2.98	Rules from R:
8.15/2.98	
8.15/2.98	   a(b(a(x1))) -> c(c(b(x1)))
8.15/2.98	   a(b(a(x1))) -> c(c(a(x1)))
8.15/2.98	Rules from S:
8.15/2.98	
8.15/2.98	   c(c(a(x1))) -> a(b(c(x1)))
8.15/2.98	   c(c(b(x1))) -> a(c(c(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(4)
8.15/2.98	Obligation:
8.15/2.98	Relative term rewrite system:
8.15/2.98	The relative TRS consists of the following R rules:
8.15/2.98	
8.15/2.98	   c(a(b(x1))) -> c(a(a(x1)))
8.15/2.98	   a(b(b(x1))) -> a(b(a(x1)))
8.15/2.98	
8.15/2.98	The relative TRS consists of the following S rules:
8.15/2.98	
8.15/2.98	   c(a(a(x1))) -> c(b(a(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(5) RelTRSRRRProof (EQUIVALENT)
8.15/2.98	We used the following monotonic ordering for rule removal:
8.15/2.98	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.15/2.98	Rules from R:
8.15/2.98	
8.15/2.98	   a(b(b(x1))) -> a(b(a(x1)))
8.15/2.98	Rules from S:
8.15/2.98	none
8.15/2.98	
8.15/2.98	
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(6)
8.15/2.98	Obligation:
8.15/2.98	Relative term rewrite system:
8.15/2.98	The relative TRS consists of the following R rules:
8.15/2.98	
8.15/2.98	   c(a(b(x1))) -> c(a(a(x1)))
8.15/2.98	
8.15/2.98	The relative TRS consists of the following S rules:
8.15/2.98	
8.15/2.98	   c(a(a(x1))) -> c(b(a(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(7) RelTRSRRRProof (EQUIVALENT)
8.15/2.98	We used the following monotonic ordering for rule removal:
8.15/2.98	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	   <<<
8.15/2.98	 POL(b(x_1)) =  	[[2], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.15/2.98	>>>
8.15/2.98	
8.15/2.98	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.15/2.98	Rules from R:
8.15/2.98	
8.15/2.98	   c(a(b(x1))) -> c(a(a(x1)))
8.15/2.98	Rules from S:
8.15/2.98	none
8.15/2.98	
8.15/2.98	
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(8)
8.15/2.98	Obligation:
8.15/2.98	Relative term rewrite system:
8.15/2.98	R is empty.
8.15/2.98	The relative TRS consists of the following S rules:
8.15/2.98	
8.15/2.98	   c(a(a(x1))) -> c(b(a(x1)))
8.15/2.98	
8.15/2.98	
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(9) RIsEmptyProof (EQUIVALENT)
8.15/2.98	The TRS R is empty. Hence, termination is trivially proven.
8.15/2.98	----------------------------------------
8.15/2.98	
8.15/2.98	(10)
8.15/2.98	YES
8.75/3.06	EOF