20.75/6.19	YES
20.75/6.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.75/6.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.75/6.20	
20.75/6.20	
20.75/6.20	Termination of the given RelTRS could be proven:
20.75/6.20	
20.75/6.20	(0) RelTRS
20.75/6.20	(1) RelTRSRRRProof [EQUIVALENT, 471 ms]
20.75/6.20	(2) RelTRS
20.75/6.20	(3) RelTRSRRRProof [EQUIVALENT, 295 ms]
20.75/6.20	(4) RelTRS
20.75/6.20	(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
20.75/6.20	(6) RelTRS
20.75/6.20	(7) RelTRSRRRProof [EQUIVALENT, 2 ms]
20.75/6.20	(8) RelTRS
20.75/6.20	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
20.75/6.20	(10) YES
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(0)
20.75/6.20	Obligation:
20.75/6.20	Relative term rewrite system:
20.75/6.20	The relative TRS consists of the following R rules:
20.75/6.20	
20.75/6.20	   a(a(c(x1))) -> c(c(b(x1)))
20.75/6.20	   b(b(b(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	The relative TRS consists of the following S rules:
20.75/6.20	
20.75/6.20	   b(b(a(x1))) -> a(a(c(x1)))
20.75/6.20	   c(c(b(x1))) -> b(b(c(x1)))
20.75/6.20	   b(b(a(x1))) -> a(b(b(x1)))
20.75/6.20	   a(b(a(x1))) -> c(b(a(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(1) RelTRSRRRProof (EQUIVALENT)
20.75/6.20	We used the following monotonic ordering for rule removal:
20.75/6.20	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(a(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
20.75/6.20	Rules from R:
20.75/6.20	none
20.75/6.20	Rules from S:
20.75/6.20	
20.75/6.20	   b(b(a(x1))) -> a(a(c(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(2)
20.75/6.20	Obligation:
20.75/6.20	Relative term rewrite system:
20.75/6.20	The relative TRS consists of the following R rules:
20.75/6.20	
20.75/6.20	   a(a(c(x1))) -> c(c(b(x1)))
20.75/6.20	   b(b(b(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	The relative TRS consists of the following S rules:
20.75/6.20	
20.75/6.20	   c(c(b(x1))) -> b(b(c(x1)))
20.75/6.20	   b(b(a(x1))) -> a(b(b(x1)))
20.75/6.20	   a(b(a(x1))) -> c(b(a(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(3) RelTRSRRRProof (EQUIVALENT)
20.75/6.20	We used the following monotonic ordering for rule removal:
20.75/6.20	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(a(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
20.75/6.20	Rules from R:
20.75/6.20	none
20.75/6.20	Rules from S:
20.75/6.20	
20.75/6.20	   a(b(a(x1))) -> c(b(a(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(4)
20.75/6.20	Obligation:
20.75/6.20	Relative term rewrite system:
20.75/6.20	The relative TRS consists of the following R rules:
20.75/6.20	
20.75/6.20	   a(a(c(x1))) -> c(c(b(x1)))
20.75/6.20	   b(b(b(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	The relative TRS consists of the following S rules:
20.75/6.20	
20.75/6.20	   c(c(b(x1))) -> b(b(c(x1)))
20.75/6.20	   b(b(a(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(5) RelTRSRRRProof (EQUIVALENT)
20.75/6.20	We used the following monotonic ordering for rule removal:
20.75/6.20	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 2]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	   <<<
20.75/6.20	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
20.75/6.20	>>>
20.75/6.20	
20.75/6.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
20.75/6.20	Rules from R:
20.75/6.20	
20.75/6.20	   a(a(c(x1))) -> c(c(b(x1)))
20.75/6.20	Rules from S:
20.75/6.20	none
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(6)
20.75/6.20	Obligation:
20.75/6.20	Relative term rewrite system:
20.75/6.20	The relative TRS consists of the following R rules:
20.75/6.20	
20.75/6.20	   b(b(b(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	The relative TRS consists of the following S rules:
20.75/6.20	
20.75/6.20	   c(c(b(x1))) -> b(b(c(x1)))
20.75/6.20	   b(b(a(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(7) RelTRSRRRProof (EQUIVALENT)
20.75/6.20	We used the following monotonic ordering for rule removal:
20.75/6.20	Polynomial interpretation [POLO]:
20.75/6.20	
20.75/6.20	   POL(a(x_1)) = x_1
20.75/6.20	   POL(b(x_1)) = 1 + x_1
20.75/6.20	   POL(c(x_1)) = 1 + x_1
20.75/6.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
20.75/6.20	Rules from R:
20.75/6.20	
20.75/6.20	   b(b(b(x1))) -> a(b(b(x1)))
20.75/6.20	Rules from S:
20.75/6.20	none
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(8)
20.75/6.20	Obligation:
20.75/6.20	Relative term rewrite system:
20.75/6.20	R is empty.
20.75/6.20	The relative TRS consists of the following S rules:
20.75/6.20	
20.75/6.20	   c(c(b(x1))) -> b(b(c(x1)))
20.75/6.20	   b(b(a(x1))) -> a(b(b(x1)))
20.75/6.20	
20.75/6.20	
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(9) RIsEmptyProof (EQUIVALENT)
20.75/6.20	The TRS R is empty. Hence, termination is trivially proven.
20.75/6.20	----------------------------------------
20.75/6.20	
20.75/6.20	(10)
20.75/6.20	YES
20.75/6.24	EOF