9.85/3.23	YES
10.38/3.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.38/3.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.38/3.35	
10.38/3.35	
10.38/3.35	Termination of the given RelTRS could be proven:
10.38/3.35	
10.38/3.35	(0) RelTRS
10.38/3.35	(1) RelTRSRRRProof [EQUIVALENT, 190 ms]
10.38/3.35	(2) RelTRS
10.38/3.35	(3) RelTRSRRRProof [EQUIVALENT, 15 ms]
10.38/3.35	(4) RelTRS
10.38/3.35	(5) RIsEmptyProof [EQUIVALENT, 0 ms]
10.38/3.35	(6) YES
10.38/3.35	
10.38/3.35	
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(0)
10.38/3.35	Obligation:
10.38/3.35	Relative term rewrite system:
10.38/3.35	The relative TRS consists of the following R rules:
10.38/3.35	
10.38/3.35	   b(b(x1)) -> c(b(c(x1)))
10.38/3.35	   c(c(c(x1))) -> a(x1)
10.38/3.35	
10.38/3.35	The relative TRS consists of the following S rules:
10.38/3.35	
10.38/3.35	   a(x1) -> a(c(b(x1)))
10.38/3.35	
10.38/3.35	
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(1) RelTRSRRRProof (EQUIVALENT)
10.38/3.35	We used the following monotonic ordering for rule removal:
10.38/3.35	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
10.38/3.35	
10.38/3.35	   <<<
10.38/3.35	 POL(b(x_1)) =  	[[0], [0], [2]] 	 +  	[[1, 0, 1], [0, 0, 2], [0, 1, 2]] 	* 	x_1
10.38/3.35	>>>
10.38/3.35	
10.38/3.35	   <<<
10.38/3.35	 POL(c(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 1, 0], [0, 1, 2], [0, 0, 0]] 	* 	x_1
10.38/3.35	>>>
10.38/3.35	
10.38/3.35	   <<<
10.38/3.35	 POL(a(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 3], [0, 0, 0], [0, 0, 0]] 	* 	x_1
10.38/3.35	>>>
10.38/3.35	
10.38/3.35	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
10.38/3.35	Rules from R:
10.38/3.35	
10.38/3.35	   b(b(x1)) -> c(b(c(x1)))
10.38/3.35	Rules from S:
10.38/3.35	none
10.38/3.35	
10.38/3.35	
10.38/3.35	
10.38/3.35	
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(2)
10.38/3.35	Obligation:
10.38/3.35	Relative term rewrite system:
10.38/3.35	The relative TRS consists of the following R rules:
10.38/3.35	
10.38/3.35	   c(c(c(x1))) -> a(x1)
10.38/3.35	
10.38/3.35	The relative TRS consists of the following S rules:
10.38/3.35	
10.38/3.35	   a(x1) -> a(c(b(x1)))
10.38/3.35	
10.38/3.35	
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(3) RelTRSRRRProof (EQUIVALENT)
10.38/3.35	We used the following monotonic ordering for rule removal:
10.38/3.35	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
10.38/3.35	
10.38/3.35	   <<<
10.38/3.35	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [2, 1]] 	* 	x_1
10.38/3.35	>>>
10.38/3.35	
10.38/3.35	   <<<
10.38/3.35	 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
10.38/3.35	>>>
10.38/3.35	
10.38/3.35	   <<<
10.38/3.35	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
10.38/3.35	>>>
10.38/3.35	
10.38/3.35	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
10.38/3.35	Rules from R:
10.38/3.35	
10.38/3.35	   c(c(c(x1))) -> a(x1)
10.38/3.35	Rules from S:
10.38/3.35	none
10.38/3.35	
10.38/3.35	
10.38/3.35	
10.38/3.35	
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(4)
10.38/3.35	Obligation:
10.38/3.35	Relative term rewrite system:
10.38/3.35	R is empty.
10.38/3.35	The relative TRS consists of the following S rules:
10.38/3.35	
10.38/3.35	   a(x1) -> a(c(b(x1)))
10.38/3.35	
10.38/3.35	
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(5) RIsEmptyProof (EQUIVALENT)
10.38/3.35	The TRS R is empty. Hence, termination is trivially proven.
10.38/3.35	----------------------------------------
10.38/3.35	
10.38/3.35	(6)
10.38/3.35	YES
10.38/3.39	EOF