5.16/2.19	YES
5.16/2.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.16/2.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.16/2.20	
5.16/2.20	
5.16/2.20	Termination of the given RelTRS could be proven:
5.16/2.20	
5.16/2.20	(0) RelTRS
5.16/2.20	(1) RelTRSRRRProof [EQUIVALENT, 41 ms]
5.16/2.20	(2) RelTRS
5.16/2.20	(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
5.16/2.20	(4) RelTRS
5.16/2.20	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
5.16/2.20	(6) YES
5.16/2.20	
5.16/2.20	
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(0)
5.16/2.20	Obligation:
5.16/2.20	Relative term rewrite system:
5.16/2.20	The relative TRS consists of the following R rules:
5.16/2.20	
5.16/2.20	   a(b(x1)) -> b(a(x1))
5.16/2.20	   d(c(x1)) -> d(a(x1))
5.16/2.20	
5.16/2.20	The relative TRS consists of the following S rules:
5.16/2.20	
5.16/2.20	   a(x1) -> b(c(x1))
5.16/2.20	   b(c(x1)) -> a(c(x1))
5.16/2.20	
5.16/2.20	
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(1) RelTRSRRRProof (EQUIVALENT)
5.16/2.20	We used the following monotonic ordering for rule removal:
5.16/2.20	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(a(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(b(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(d(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.16/2.20	Rules from R:
5.16/2.20	
5.16/2.20	   a(b(x1)) -> b(a(x1))
5.16/2.20	Rules from S:
5.16/2.20	none
5.16/2.20	
5.16/2.20	
5.16/2.20	
5.16/2.20	
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(2)
5.16/2.20	Obligation:
5.16/2.20	Relative term rewrite system:
5.16/2.20	The relative TRS consists of the following R rules:
5.16/2.20	
5.16/2.20	   d(c(x1)) -> d(a(x1))
5.16/2.20	
5.16/2.20	The relative TRS consists of the following S rules:
5.16/2.20	
5.16/2.20	   a(x1) -> b(c(x1))
5.16/2.20	   b(c(x1)) -> a(c(x1))
5.16/2.20	
5.16/2.20	
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(3) RelTRSRRRProof (EQUIVALENT)
5.16/2.20	We used the following monotonic ordering for rule removal:
5.16/2.20	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(d(x_1)) =  	[[2], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(a(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	   <<<
5.16/2.20	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
5.16/2.20	>>>
5.16/2.20	
5.16/2.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.16/2.20	Rules from R:
5.16/2.20	
5.16/2.20	   d(c(x1)) -> d(a(x1))
5.16/2.20	Rules from S:
5.16/2.20	none
5.16/2.20	
5.16/2.20	
5.16/2.20	
5.16/2.20	
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(4)
5.16/2.20	Obligation:
5.16/2.20	Relative term rewrite system:
5.16/2.20	R is empty.
5.16/2.20	The relative TRS consists of the following S rules:
5.16/2.20	
5.16/2.20	   a(x1) -> b(c(x1))
5.16/2.20	   b(c(x1)) -> a(c(x1))
5.16/2.20	
5.16/2.20	
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(5) RIsEmptyProof (EQUIVALENT)
5.16/2.20	The TRS R is empty. Hence, termination is trivially proven.
5.16/2.20	----------------------------------------
5.16/2.20	
5.16/2.20	(6)
5.16/2.20	YES
5.78/2.31	EOF