981.74/249.19	YES
981.74/249.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
981.74/249.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
981.74/249.20	
981.74/249.20	
981.74/249.20	Termination of the given RelTRS could be proven:
981.74/249.20	
981.74/249.20	(0) RelTRS
981.74/249.20	(1) RelTRSRRRProof [EQUIVALENT, 72.3 s]
981.74/249.20	(2) RelTRS
981.74/249.20	(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
981.74/249.20	(4) RelTRS
981.74/249.20	(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
981.74/249.20	(6) RelTRS
981.74/249.20	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
981.74/249.20	(8) YES
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(0)
981.74/249.20	Obligation:
981.74/249.20	Relative term rewrite system:
981.74/249.20	The relative TRS consists of the following R rules:
981.74/249.20	
981.74/249.20	   b(c(b(x1))) -> a(b(b(x1)))
981.74/249.20	   a(c(c(x1))) -> c(a(a(x1)))
981.74/249.20	   b(c(a(x1))) -> a(b(a(x1)))
981.74/249.20	
981.74/249.20	The relative TRS consists of the following S rules:
981.74/249.20	
981.74/249.20	   b(c(b(x1))) -> b(c(c(x1)))
981.74/249.20	   a(b(c(x1))) -> b(b(b(x1)))
981.74/249.20	   c(a(c(x1))) -> c(b(a(x1)))
981.74/249.20	   a(a(a(x1))) -> a(c(c(x1)))
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(1) RelTRSRRRProof (EQUIVALENT)
981.74/249.20	We used the following monotonic ordering for rule removal:
981.74/249.20	Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :
981.74/249.20	
981.74/249.20	   <<<
981.74/249.20	 POL(b(x_1)) =  	[[1], [1], [1]] 	 +  	[[1, 1, 0], [1, 0, 1], [1, 0, 1]] 	* 	x_1
981.74/249.20	>>>
981.74/249.20	
981.74/249.20	   <<<
981.74/249.20	 POL(c(x_1)) =  	[[1], [2], [0]] 	 +  	[[2, 0, 0], [0, 1, 1], [0, 1, 1]] 	* 	x_1
981.74/249.20	>>>
981.74/249.20	
981.74/249.20	   <<<
981.74/249.20	 POL(a(x_1)) =  	[[1], [0], [0]] 	 +  	[[2, 0, 0], [2, 0, 0], [2, 0, 0]] 	* 	x_1
981.74/249.20	>>>
981.74/249.20	
981.74/249.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
981.74/249.20	Rules from R:
981.74/249.20	
981.74/249.20	   b(c(b(x1))) -> a(b(b(x1)))
981.74/249.20	   b(c(a(x1))) -> a(b(a(x1)))
981.74/249.20	Rules from S:
981.74/249.20	
981.74/249.20	   a(b(c(x1))) -> b(b(b(x1)))
981.74/249.20	   c(a(c(x1))) -> c(b(a(x1)))
981.74/249.20	
981.74/249.20	
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(2)
981.74/249.20	Obligation:
981.74/249.20	Relative term rewrite system:
981.74/249.20	The relative TRS consists of the following R rules:
981.74/249.20	
981.74/249.20	   a(c(c(x1))) -> c(a(a(x1)))
981.74/249.20	
981.74/249.20	The relative TRS consists of the following S rules:
981.74/249.20	
981.74/249.20	   b(c(b(x1))) -> b(c(c(x1)))
981.74/249.20	   a(a(a(x1))) -> a(c(c(x1)))
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(3) RelTRSRRRProof (EQUIVALENT)
981.74/249.20	We used the following monotonic ordering for rule removal:
981.74/249.20	Polynomial interpretation [POLO]:
981.74/249.20	
981.74/249.20	   POL(a(x_1)) = x_1
981.74/249.20	   POL(b(x_1)) = 1 + x_1
981.74/249.20	   POL(c(x_1)) = x_1
981.74/249.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
981.74/249.20	Rules from R:
981.74/249.20	none
981.74/249.20	Rules from S:
981.74/249.20	
981.74/249.20	   b(c(b(x1))) -> b(c(c(x1)))
981.74/249.20	
981.74/249.20	
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(4)
981.74/249.20	Obligation:
981.74/249.20	Relative term rewrite system:
981.74/249.20	The relative TRS consists of the following R rules:
981.74/249.20	
981.74/249.20	   a(c(c(x1))) -> c(a(a(x1)))
981.74/249.20	
981.74/249.20	The relative TRS consists of the following S rules:
981.74/249.20	
981.74/249.20	   a(a(a(x1))) -> a(c(c(x1)))
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(5) RelTRSRRRProof (EQUIVALENT)
981.74/249.20	We used the following monotonic ordering for rule removal:
981.74/249.20	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
981.74/249.20	
981.74/249.20	   <<<
981.74/249.20	 POL(a(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 2]] 	* 	x_1
981.74/249.20	>>>
981.74/249.20	
981.74/249.20	   <<<
981.74/249.20	 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
981.74/249.20	>>>
981.74/249.20	
981.74/249.20	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
981.74/249.20	Rules from R:
981.74/249.20	
981.74/249.20	   a(c(c(x1))) -> c(a(a(x1)))
981.74/249.20	Rules from S:
981.74/249.20	
981.74/249.20	   a(a(a(x1))) -> a(c(c(x1)))
981.74/249.20	
981.74/249.20	
981.74/249.20	
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(6)
981.74/249.20	Obligation:
981.74/249.20	Relative term rewrite system:
981.74/249.20	R is empty.
981.74/249.20	S is empty.
981.74/249.20	
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(7) RIsEmptyProof (EQUIVALENT)
981.74/249.20	The TRS R is empty. Hence, termination is trivially proven.
981.74/249.20	----------------------------------------
981.74/249.20	
981.74/249.20	(8)
981.74/249.20	YES
982.00/249.30	EOF