9.05/3.13	YES
9.05/3.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
9.05/3.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.05/3.14	
9.05/3.14	
9.05/3.14	Termination of the given RelTRS could be proven:
9.05/3.14	
9.05/3.14	(0) RelTRS
9.05/3.14	(1) RelTRSRRRProof [EQUIVALENT, 148 ms]
9.05/3.14	(2) RelTRS
9.05/3.14	(3) RelTRSRRRProof [EQUIVALENT, 7 ms]
9.05/3.14	(4) RelTRS
9.05/3.14	(5) RelTRSRRRProof [EQUIVALENT, 27 ms]
9.05/3.14	(6) RelTRS
9.05/3.14	(7) RelTRSRRRProof [EQUIVALENT, 7 ms]
9.05/3.14	(8) RelTRS
9.05/3.14	(9) RIsEmptyProof [EQUIVALENT, 0 ms]
9.05/3.14	(10) YES
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(0)
9.05/3.14	Obligation:
9.05/3.14	Relative term rewrite system:
9.05/3.14	The relative TRS consists of the following R rules:
9.05/3.14	
9.05/3.14	   a(x1) -> x1
9.05/3.14	   b(x1) -> x1
9.05/3.14	   a(c(x1)) -> b(b(c(a(x1))))
9.05/3.14	   d(b(b(b(x1)))) -> a(d(a(x1)))
9.05/3.14	
9.05/3.14	The relative TRS consists of the following S rules:
9.05/3.14	
9.05/3.14	   a(b(x1)) -> b(a(x1))
9.05/3.14	   b(a(x1)) -> a(b(x1))
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(1) RelTRSRRRProof (EQUIVALENT)
9.05/3.14	We used the following monotonic ordering for rule removal:
9.05/3.14	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(d(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.05/3.14	Rules from R:
9.05/3.14	
9.05/3.14	   d(b(b(b(x1)))) -> a(d(a(x1)))
9.05/3.14	Rules from S:
9.05/3.14	none
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(2)
9.05/3.14	Obligation:
9.05/3.14	Relative term rewrite system:
9.05/3.14	The relative TRS consists of the following R rules:
9.05/3.14	
9.05/3.14	   a(x1) -> x1
9.05/3.14	   b(x1) -> x1
9.05/3.14	   a(c(x1)) -> b(b(c(a(x1))))
9.05/3.14	
9.05/3.14	The relative TRS consists of the following S rules:
9.05/3.14	
9.05/3.14	   a(b(x1)) -> b(a(x1))
9.05/3.14	   b(a(x1)) -> a(b(x1))
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(3) RelTRSRRRProof (EQUIVALENT)
9.05/3.14	We used the following monotonic ordering for rule removal:
9.05/3.14	Polynomial interpretation [POLO]:
9.05/3.14	
9.05/3.14	   POL(a(x_1)) = 1 + x_1
9.05/3.14	   POL(b(x_1)) = x_1
9.05/3.14	   POL(c(x_1)) = x_1
9.05/3.14	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.05/3.14	Rules from R:
9.05/3.14	
9.05/3.14	   a(x1) -> x1
9.05/3.14	Rules from S:
9.05/3.14	none
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(4)
9.05/3.14	Obligation:
9.05/3.14	Relative term rewrite system:
9.05/3.14	The relative TRS consists of the following R rules:
9.05/3.14	
9.05/3.14	   b(x1) -> x1
9.05/3.14	   a(c(x1)) -> b(b(c(a(x1))))
9.05/3.14	
9.05/3.14	The relative TRS consists of the following S rules:
9.05/3.14	
9.05/3.14	   a(b(x1)) -> b(a(x1))
9.05/3.14	   b(a(x1)) -> a(b(x1))
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(5) RelTRSRRRProof (EQUIVALENT)
9.05/3.14	We used the following monotonic ordering for rule removal:
9.05/3.14	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(c(x_1)) =  	[[1], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.05/3.14	Rules from R:
9.05/3.14	
9.05/3.14	   a(c(x1)) -> b(b(c(a(x1))))
9.05/3.14	Rules from S:
9.05/3.14	none
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(6)
9.05/3.14	Obligation:
9.05/3.14	Relative term rewrite system:
9.05/3.14	The relative TRS consists of the following R rules:
9.05/3.14	
9.05/3.14	   b(x1) -> x1
9.05/3.14	
9.05/3.14	The relative TRS consists of the following S rules:
9.05/3.14	
9.05/3.14	   a(b(x1)) -> b(a(x1))
9.05/3.14	   b(a(x1)) -> a(b(x1))
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(7) RelTRSRRRProof (EQUIVALENT)
9.05/3.14	We used the following monotonic ordering for rule removal:
9.05/3.14	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(b(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	   <<<
9.05/3.14	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
9.05/3.14	>>>
9.05/3.14	
9.05/3.14	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
9.05/3.14	Rules from R:
9.05/3.14	
9.05/3.14	   b(x1) -> x1
9.05/3.14	Rules from S:
9.05/3.14	none
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(8)
9.05/3.14	Obligation:
9.05/3.14	Relative term rewrite system:
9.05/3.14	R is empty.
9.05/3.14	The relative TRS consists of the following S rules:
9.05/3.14	
9.05/3.14	   a(b(x1)) -> b(a(x1))
9.05/3.14	   b(a(x1)) -> a(b(x1))
9.05/3.14	
9.05/3.14	
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(9) RIsEmptyProof (EQUIVALENT)
9.05/3.14	The TRS R is empty. Hence, termination is trivially proven.
9.05/3.14	----------------------------------------
9.05/3.14	
9.05/3.14	(10)
9.05/3.14	YES
9.05/3.20	EOF